Introduction
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Linear Equations- One Variable
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Linear One Unknown, 0 Or Infinite Solutions
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Linear One Unknown Denom
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Linear Two Unknowns
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Linear Two Unknowns, 0 Or Infinite Solutions
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Applications of Linear Equations
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Quadratic Equations
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Disarranged Quadratic
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Quadratics in Disguise
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Nonlinear Equations
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Equations with Roots
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Equations with Absolute Value
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{"Free":0,"Sample":1,"Paid":2}

[{"Name":"Introduction","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"7m 57s","ChapterTopicVideoID":4931,"CourseChapterTopicPlaylistID":3861,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"I\u0027m going to be introducing equations in college algebra."},{"Start":"00:04.050 ","End":"00:06.915","Text":"I\u0027d like to start with an informal introduction."},{"Start":"00:06.915 ","End":"00:08.880","Text":"I\u0027m not going to define what an equation is."},{"Start":"00:08.880 ","End":"00:10.635","Text":"I\u0027ll just show you a few examples."},{"Start":"00:10.635 ","End":"00:17.505","Text":"Let\u0027s start straight away with an example of an equation, x+2=5."},{"Start":"00:17.505 ","End":"00:20.100","Text":"Notice that it has an equal sign in it."},{"Start":"00:20.100 ","End":"00:23.205","Text":"That\u0027s very important for an equation to have an equality."},{"Start":"00:23.205 ","End":"00:25.635","Text":"Notice that there\u0027s a letter in it."},{"Start":"00:25.635 ","End":"00:27.870","Text":"It\u0027s not just numbers."},{"Start":"00:27.870 ","End":"00:30.995","Text":"This letter is a variable or an unknown,"},{"Start":"00:30.995 ","End":"00:34.805","Text":"and x is a very common letter in algebra."},{"Start":"00:34.805 ","End":"00:37.385","Text":"X is a favorite variable name."},{"Start":"00:37.385 ","End":"00:40.910","Text":"The idea of an equation usually is to solve it."},{"Start":"00:40.910 ","End":"00:42.500","Text":"What does it mean to solve it?"},{"Start":"00:42.500 ","End":"00:47.270","Text":"It means to find what value of x will make this true."},{"Start":"00:47.270 ","End":"00:49.550","Text":"In other words, what can I put instead of x?"},{"Start":"00:49.550 ","End":"00:53.135","Text":"I\u0027d like to substitute x equals something to make it true."},{"Start":"00:53.135 ","End":"00:59.360","Text":"Now, I could do it by guesswork and let\u0027s say I try x=1."},{"Start":"00:59.360 ","End":"01:04.115","Text":"Then I put x=1 and I get 1+2=5."},{"Start":"01:04.115 ","End":"01:08.155","Text":"That\u0027s false. Then I could try x=2."},{"Start":"01:08.155 ","End":"01:10.900","Text":"I would get 2+2=5."},{"Start":"01:10.900 ","End":"01:15.535","Text":"Still wrong. Then I could try x=3,"},{"Start":"01:15.535 ","End":"01:17.745","Text":"and I\u0027d get 3+2=5."},{"Start":"01:17.745 ","End":"01:21.931","Text":"Yes, it happens to be true, bingo."},{"Start":"01:21.931 ","End":"01:24.019","Text":"So x=3 is the solution."},{"Start":"01:24.019 ","End":"01:26.240","Text":"Of course, we\u0027re not going to be doing it by guesswork,"},{"Start":"01:26.240 ","End":"01:27.865","Text":"but I just wanted to show you,"},{"Start":"01:27.865 ","End":"01:32.240","Text":"this actually leads to a more formal definition of what is a solution."},{"Start":"01:32.240 ","End":"01:35.675","Text":"It\u0027s a value that when you substitute it for the variable,"},{"Start":"01:35.675 ","End":"01:38.840","Text":"gives you a true statement,"},{"Start":"01:38.840 ","End":"01:42.050","Text":"a true expression, rather than one that\u0027s false."},{"Start":"01:42.050 ","End":"01:44.210","Text":"If I substitute anything other than 3,"},{"Start":"01:44.210 ","End":"01:45.440","Text":"I\u0027ll get something that\u0027s false."},{"Start":"01:45.440 ","End":"01:48.140","Text":"I could try x=4, 4+2=5."},{"Start":"01:48.140 ","End":"01:54.395","Text":"No. We want to find out what values of x we can substitute to make this thing true."},{"Start":"01:54.395 ","End":"01:55.910","Text":"That\u0027s the first example."},{"Start":"01:55.910 ","End":"01:58.325","Text":"I\u0027ll just give you another example."},{"Start":"01:58.325 ","End":"02:04.170","Text":"Suppose I take 2x+3=11."},{"Start":"02:05.300 ","End":"02:09.540","Text":"Now 2x, as you remember, means 2*x."},{"Start":"02:09.540 ","End":"02:12.600","Text":"Once again, I could try substituting x=1."},{"Start":"02:12.600 ","End":"02:17.705","Text":"I\u0027d get 2*1+3=5, it wouldn\u0027t work."},{"Start":"02:17.705 ","End":"02:21.350","Text":"Finally, I\u0027d hit upon x=4."},{"Start":"02:21.350 ","End":"02:23.510","Text":"If I tried x=4,"},{"Start":"02:23.510 ","End":"02:28.770","Text":"I\u0027d get 2*4+3 really is equal to 11."},{"Start":"02:28.770 ","End":"02:32.055","Text":"Because 2*4 is 8 plus 3 is 11 so that works."},{"Start":"02:32.055 ","End":"02:34.925","Text":"We would say that x=4 is a solution."},{"Start":"02:34.925 ","End":"02:37.790","Text":"But for example, x=1 is not a solution,"},{"Start":"02:37.790 ","End":"02:41.270","Text":"because 2*1+3 is not equal to 11."},{"Start":"02:41.270 ","End":"02:44.345","Text":"This is a general idea of equations."},{"Start":"02:44.345 ","End":"02:51.320","Text":"Now, I want to show you an example of an equation that has more than one solution."},{"Start":"02:51.320 ","End":"02:58.580","Text":"Suppose I said, I want x such that x^2=9."},{"Start":"02:58.580 ","End":"03:02.105","Text":"This time I have an x squared in the equation."},{"Start":"03:02.105 ","End":"03:06.305","Text":"Notice that if I put x=3,"},{"Start":"03:06.305 ","End":"03:07.940","Text":"I will get a solution."},{"Start":"03:07.940 ","End":"03:10.265","Text":"Because if I substitute it,"},{"Start":"03:10.265 ","End":"03:12.350","Text":"I get 3^2=9,"},{"Start":"03:12.350 ","End":"03:13.970","Text":"which is a true statement."},{"Start":"03:13.970 ","End":"03:16.270","Text":"If I put, for example,"},{"Start":"03:16.270 ","End":"03:19.260","Text":"x=2, 2^2=9,"},{"Start":"03:19.260 ","End":"03:21.075","Text":"no, that\u0027s not a true equation."},{"Start":"03:21.075 ","End":"03:22.635","Text":"2 is not a solution,"},{"Start":"03:22.635 ","End":"03:24.485","Text":"3 is a solution."},{"Start":"03:24.485 ","End":"03:28.895","Text":"But what happens if I substitute x=-3?"},{"Start":"03:28.895 ","End":"03:33.390","Text":"I\u0027ll get -3^2=9. I ask myself,"},{"Start":"03:33.390 ","End":"03:35.235","Text":"is this a true statement?"},{"Start":"03:35.235 ","End":"03:36.910","Text":"The answer is, yes,"},{"Start":"03:36.910 ","End":"03:40.085","Text":"because you learned that minus times minus is plus."},{"Start":"03:40.085 ","End":"03:43.700","Text":"Sometimes an equation can have more than one solution."},{"Start":"03:43.700 ","End":"03:47.150","Text":"Carrying on this idea of a square,"},{"Start":"03:47.150 ","End":"03:53.390","Text":"I could ask x^2=-4 as another equation."},{"Start":"03:53.390 ","End":"03:55.670","Text":"In this case, I can guess all I want."},{"Start":"03:55.670 ","End":"03:57.725","Text":"I can try 0, I can try 1."},{"Start":"03:57.725 ","End":"04:01.380","Text":"Nothing will work. Well,"},{"Start":"04:01.380 ","End":"04:03.300","Text":"let\u0027s just try. Suppose I took x=0."},{"Start":"04:03.300 ","End":"04:06.390","Text":"0^2, is that equal to -4?"},{"Start":"04:06.390 ","End":"04:09.690","Text":"No. Try x=2. 2^2,"},{"Start":"04:09.690 ","End":"04:11.475","Text":"is that equal to -4?"},{"Start":"04:11.475 ","End":"04:15.555","Text":"No. You quickly see that 0^2 or 2^2,"},{"Start":"04:15.555 ","End":"04:16.770","Text":"whatever I square,"},{"Start":"04:16.770 ","End":"04:18.720","Text":"I\u0027ll never get a negative number."},{"Start":"04:18.720 ","End":"04:22.570","Text":"This time the equation has no solution."},{"Start":"04:22.570 ","End":"04:24.620","Text":"That\u0027s also possible."},{"Start":"04:24.620 ","End":"04:26.990","Text":"Sometimes an equation has one solution."},{"Start":"04:26.990 ","End":"04:28.700","Text":"Sometimes it has more than one solution."},{"Start":"04:28.700 ","End":"04:30.740","Text":"Sometimes it has no solution."},{"Start":"04:30.740 ","End":"04:33.680","Text":"All these things are possible with an equation."},{"Start":"04:33.680 ","End":"04:38.450","Text":"Just to give you one more example of crazy things that can happen."},{"Start":"04:38.450 ","End":"04:40.685","Text":"Suppose I took the example,"},{"Start":"04:40.685 ","End":"04:47.810","Text":"I\u0027ll write it over here, x+x=2x. I just had an idea."},{"Start":"04:47.810 ","End":"04:52.085","Text":"Why don\u0027t I just highlight all the equations so we can see examples of equations?"},{"Start":"04:52.085 ","End":"04:54.275","Text":"This was our first example."},{"Start":"04:54.275 ","End":"04:56.365","Text":"This was our second example."},{"Start":"04:56.365 ","End":"04:58.035","Text":"Third, fourth,"},{"Start":"04:58.035 ","End":"04:59.655","Text":"and now this example."},{"Start":"04:59.655 ","End":"05:03.050","Text":"We\u0027ve seen quite a lot of equations and they all have an equal sign in them."},{"Start":"05:03.050 ","End":"05:04.850","Text":"That\u0027s the important thing."},{"Start":"05:04.850 ","End":"05:06.620","Text":"I\u0027m just using x,"},{"Start":"05:06.620 ","End":"05:08.480","Text":"though any other letter could\u0027ve done,"},{"Start":"05:08.480 ","End":"05:10.835","Text":"but x as I said, is very popular."},{"Start":"05:10.835 ","End":"05:12.545","Text":"Back to this one."},{"Start":"05:12.545 ","End":"05:18.245","Text":"What would I get if I substituted x=5, for example?"},{"Start":"05:18.245 ","End":"05:19.820","Text":"I would get 5+5=2(5)."},{"Start":"05:19.820 ","End":"05:24.980","Text":"2(5) is 10,"},{"Start":"05:24.980 ","End":"05:28.264","Text":"and 5 plus 5 is 10. x=5 is a solution."},{"Start":"05:28.264 ","End":"05:31.010","Text":"But you quickly realize that almost anything,"},{"Start":"05:31.010 ","End":"05:33.320","Text":"not almost anything, anything I try as x,"},{"Start":"05:33.320 ","End":"05:35.690","Text":"I could try x=11,"},{"Start":"05:35.690 ","End":"05:40.025","Text":"and I\u0027d get 11+11 equals 2(11)."},{"Start":"05:40.025 ","End":"05:41.990","Text":"Is this a true statement?"},{"Start":"05:41.990 ","End":"05:44.555","Text":"The answer is yes so this is a solution."},{"Start":"05:44.555 ","End":"05:46.940","Text":"In fact, every x is a solution,"},{"Start":"05:46.940 ","End":"05:55.195","Text":"and I have an infinite number of solutions so every number is a solution."},{"Start":"05:55.195 ","End":"05:57.510","Text":"Let me rephrase that."},{"Start":"05:57.510 ","End":"05:59.400","Text":"Let\u0027s say every x is a solution."},{"Start":"05:59.400 ","End":"06:01.530","Text":"Whatever value I put for x is a solution."},{"Start":"06:01.530 ","End":"06:03.090","Text":"Here\u0027s a few examples."},{"Start":"06:03.090 ","End":"06:06.380","Text":"We can also have more than one variable."},{"Start":"06:06.380 ","End":"06:09.425","Text":"Let me give an example of that."},{"Start":"06:09.425 ","End":"06:16.760","Text":"I could take something like x+y=10."},{"Start":"06:16.760 ","End":"06:18.890","Text":"That\u0027s an equation that has two variables,"},{"Start":"06:18.890 ","End":"06:23.630","Text":"x and y. I\u0027ll give you an example of what I mean by a solution."},{"Start":"06:23.630 ","End":"06:28.265","Text":"A solution has to be a value of x and a value of y."},{"Start":"06:28.265 ","End":"06:29.900","Text":"Let me give an example."},{"Start":"06:29.900 ","End":"06:35.360","Text":"I could say take x=3 and y=7,"},{"Start":"06:35.360 ","End":"06:36.890","Text":"and ask if it\u0027s a solution."},{"Start":"06:36.890 ","End":"06:39.440","Text":"The reason I\u0027m putting braces here is to say that,"},{"Start":"06:39.440 ","End":"06:44.465","Text":"this pair of values together may or may not be a solution."},{"Start":"06:44.465 ","End":"06:45.740","Text":"In this case, it is,"},{"Start":"06:45.740 ","End":"06:50.090","Text":"because if I put x=3 and y=7,"},{"Start":"06:50.090 ","End":"06:53.405","Text":"3+7=10, is a true statement."},{"Start":"06:53.405 ","End":"06:55.905","Text":"But if I ask you, if,"},{"Start":"06:55.905 ","End":"06:59.615","Text":"let\u0027s say x=1,"},{"Start":"06:59.615 ","End":"07:02.229","Text":"y=6 is a solution,"},{"Start":"07:02.229 ","End":"07:07.145","Text":"then you would have to say no because 1+6=10,"},{"Start":"07:07.145 ","End":"07:08.944","Text":"which is what I get when I substitute,"},{"Start":"07:08.944 ","End":"07:10.820","Text":"is not a true statement."},{"Start":"07:10.820 ","End":"07:13.369","Text":"So this is not a solution this is a solution."},{"Start":"07:13.369 ","End":"07:16.730","Text":"This equation has also many solutions."},{"Start":"07:16.730 ","End":"07:19.415","Text":"For example, if I took,"},{"Start":"07:19.415 ","End":"07:27.140","Text":"let\u0027s say, x=4,"},{"Start":"07:27.140 ","End":"07:28.710","Text":"y=6, would this be a solution?"},{"Start":"07:28.710 ","End":"07:30.735","Text":"I substitute and ask,"},{"Start":"07:30.735 ","End":"07:35.265","Text":"is 4+6 equal 10?"},{"Start":"07:35.265 ","End":"07:37.875","Text":"The answer is, yes."},{"Start":"07:37.875 ","End":"07:39.965","Text":"So this is a solution,"},{"Start":"07:39.965 ","End":"07:41.270","Text":"and this is a solution,"},{"Start":"07:41.270 ","End":"07:43.150","Text":"but this is not a solution."},{"Start":"07:43.150 ","End":"07:47.705","Text":"This is just to show you all kinds of equations."},{"Start":"07:47.705 ","End":"07:50.800","Text":"I just forgot to highlight this one."},{"Start":"07:50.800 ","End":"07:53.310","Text":"Let\u0027s see how many equations have we looked at?"},{"Start":"07:53.310 ","End":"07:54.905","Text":"We have six examples."},{"Start":"07:54.905 ","End":"07:57.900","Text":"Now I\u0027ll get a little bit more formal."}],"ID":4929},{"Watched":false,"Name":"Exercise 1","Duration":"6m 39s","ChapterTopicVideoID":4930,"CourseChapterTopicPlaylistID":3861,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this exercise there are six sections and in each one of them we have to"},{"Start":"00:04.320 ","End":"00:09.870","Text":"check if a given number is a solution to the equation in different letters,"},{"Start":"00:09.870 ","End":"00:12.390","Text":"a couple of them in x, couple of them in z,"},{"Start":"00:12.390 ","End":"00:15.630","Text":"there\u0027s a y and there\u0027s a w. What we have to do is"},{"Start":"00:15.630 ","End":"00:19.122","Text":"substitute and see if we get an equality,"},{"Start":"00:19.122 ","End":"00:21.405","Text":"if the left-hand side equals the right-hand side."},{"Start":"00:21.405 ","End":"00:27.045","Text":"Let\u0027s start with the first one and I\u0027ve copied them again with a bit more space."},{"Start":"00:27.045 ","End":"00:31.400","Text":"All we do is substitute in each of the sides,"},{"Start":"00:31.400 ","End":"00:34.460","Text":"left-hand side and right-hand side and see if we get the same thing."},{"Start":"00:34.460 ","End":"00:36.230","Text":"Sometimes I start with"},{"Start":"00:36.230 ","End":"00:38.960","Text":"the left-hand side and then see if we can reach the right-hand side,"},{"Start":"00:38.960 ","End":"00:42.455","Text":"and sometimes you can substitute in both simultaneously."},{"Start":"00:42.455 ","End":"00:44.555","Text":"It depends what\u0027s more convenient."},{"Start":"00:44.555 ","End":"00:47.510","Text":"Here I\u0027m going to substitute simultaneously."},{"Start":"00:47.510 ","End":"00:53.615","Text":"What we have to ask this question mark is important because if I let x equals 6,"},{"Start":"00:53.615 ","End":"00:56.885","Text":"I get twice 6 plus 1."},{"Start":"00:56.885 ","End":"01:03.900","Text":"The parentheses of are emphasis equals 3 times 6 minus 2."},{"Start":"01:03.900 ","End":"01:06.950","Text":"This is the 6 from here, minus 5."},{"Start":"01:06.950 ","End":"01:08.845","Text":"Now, I don\u0027t just write equals,"},{"Start":"01:08.845 ","End":"01:12.470","Text":"I write equals with a question mark because that\u0027s what I want to check."},{"Start":"01:12.470 ","End":"01:14.480","Text":"It may be equal and it may be not."},{"Start":"01:14.480 ","End":"01:16.040","Text":"If it is and it is a solution."},{"Start":"01:16.040 ","End":"01:19.550","Text":"If not, then not. Let us see what is the left-hand side equal?"},{"Start":"01:19.550 ","End":"01:24.605","Text":"The left-hand side is twice 6 is 12 plus 1 is 13."},{"Start":"01:24.605 ","End":"01:27.710","Text":"The right-hand side, order of operations,"},{"Start":"01:27.710 ","End":"01:30.005","Text":"the brackets first that\u0027s 4,"},{"Start":"01:30.005 ","End":"01:39.090","Text":"then 3 times 4 is 12 and 12 minus 5 is 7."},{"Start":"01:39.090 ","End":"01:42.544","Text":"These are not equal."},{"Start":"01:42.544 ","End":"01:46.350","Text":"No. This is not equal."},{"Start":"01:46.580 ","End":"01:50.090","Text":"The answer is no, so it\u0027s not a solution."},{"Start":"01:50.090 ","End":"01:52.340","Text":"The next one is very similar to this."},{"Start":"01:52.340 ","End":"01:55.775","Text":"I see that the only difference is instead of a minus 5 there\u0027s a plus 1."},{"Start":"01:55.775 ","End":"01:58.790","Text":"But just copying the bits that are similar,"},{"Start":"01:58.790 ","End":"02:03.540","Text":"equals question mark 3 times 6 minus 2."},{"Start":"02:03.540 ","End":"02:05.505","Text":"Then instead of the minus 5 here,"},{"Start":"02:05.505 ","End":"02:07.265","Text":"I\u0027ve got a plus 1 here."},{"Start":"02:07.265 ","End":"02:09.680","Text":"Now we\u0027ve already computed this as 13."},{"Start":"02:09.680 ","End":"02:11.885","Text":"Let\u0027s just see this time the right-hand side,"},{"Start":"02:11.885 ","End":"02:17.310","Text":"3 times 4 is 12 as before 12 plus 1 is 13."},{"Start":"02:17.310 ","End":"02:21.105","Text":"The answer is 13 equal 13."},{"Start":"02:21.105 ","End":"02:23.285","Text":"The answer is yes."},{"Start":"02:23.285 ","End":"02:26.800","Text":"Put a checkmark and I write a yes."},{"Start":"02:26.800 ","End":"02:29.390","Text":"The answer is yes it is a solution."},{"Start":"02:29.390 ","End":"02:31.760","Text":"Next one."},{"Start":"02:31.760 ","End":"02:36.485","Text":"This time we\u0027ll move into letter z or z if you\u0027re in the UK."},{"Start":"02:36.485 ","End":"02:41.090","Text":"Is z equal minus 2 a solution to this equation?"},{"Start":"02:41.090 ","End":"02:45.620","Text":"Once again, we\u0027ll substitute in each side and the left-hand side."},{"Start":"02:45.620 ","End":"02:49.460","Text":"Let me just copy the equal here and put a question mark."},{"Start":"02:49.460 ","End":"02:57.710","Text":"We\u0027re asking is minus 2 squared plus minus 2 plus 1,"},{"Start":"02:57.710 ","End":"03:02.780","Text":"does this equal twice minus 2 plus 5?"},{"Start":"03:02.780 ","End":"03:07.520","Text":"Let\u0027s see. Here we have 4 because the whole thing is squared."},{"Start":"03:07.520 ","End":"03:12.905","Text":"4 minus 2 plus 1 is 3 and is this"},{"Start":"03:12.905 ","End":"03:18.625","Text":"equal to minus 4 plus 5 is 1?"},{"Start":"03:18.625 ","End":"03:21.485","Text":"I would say that the answer is no."},{"Start":"03:21.485 ","End":"03:23.875","Text":"It\u0027s not a solution."},{"Start":"03:23.875 ","End":"03:26.345","Text":"In this one, again,"},{"Start":"03:26.345 ","End":"03:28.100","Text":"similar to the previous one,"},{"Start":"03:28.100 ","End":"03:31.780","Text":"the only difference is that we have a plus 7 at the end."},{"Start":"03:31.780 ","End":"03:34.710","Text":"Just copy this bit again,"},{"Start":"03:34.710 ","End":"03:39.110","Text":"minus 2 squared plus minus 2 plus one."},{"Start":"03:39.110 ","End":"03:44.570","Text":"Is this equal to twice minus 2 this time plus 7?"},{"Start":"03:44.570 ","End":"03:53.295","Text":"In other words, is 3 equal to twice minus 2 plus 7?"},{"Start":"03:53.295 ","End":"03:59.265","Text":"This time, minus 4 plus 7 is,"},{"Start":"03:59.265 ","End":"04:02.160","Text":"just erase this, 3."},{"Start":"04:02.160 ","End":"04:04.425","Text":"Yes, it is."},{"Start":"04:04.425 ","End":"04:08.030","Text":"This time this is a solution to this."},{"Start":"04:08.030 ","End":"04:10.055","Text":"A couple more to go,"},{"Start":"04:10.055 ","End":"04:13.105","Text":"this time with a division sign."},{"Start":"04:13.105 ","End":"04:17.660","Text":"Let me just say that when you have a division example,"},{"Start":"04:17.660 ","End":"04:20.295","Text":"you have to make sure denominator is not 0,"},{"Start":"04:20.295 ","End":"04:22.190","Text":"otherwise the thing doesn\u0027t make sense."},{"Start":"04:22.190 ","End":"04:23.525","Text":"First one."},{"Start":"04:23.525 ","End":"04:26.194","Text":"Is y equals 1 a solution to this?"},{"Start":"04:26.194 ","End":"04:31.070","Text":"Let\u0027s substitute in the left-hand side and in the right-hand side."},{"Start":"04:31.070 ","End":"04:32.375","Text":"Right-hand side is easier,"},{"Start":"04:32.375 ","End":"04:35.300","Text":"0, left-hand side we\u0027re asking,"},{"Start":"04:35.300 ","End":"04:42.050","Text":"does 1 minus 1/1^2 plus 1 equals 0?"},{"Start":"04:42.050 ","End":"04:44.405","Text":"The numerator is 0,"},{"Start":"04:44.405 ","End":"04:47.315","Text":"and 0 over anything is 0."},{"Start":"04:47.315 ","End":"04:50.164","Text":"The answer is yes,"},{"Start":"04:50.164 ","End":"04:51.860","Text":"it is equal to."},{"Start":"04:51.860 ","End":"04:56.975","Text":"But if I put this thing in here,"},{"Start":"04:56.975 ","End":"04:59.120","Text":"there\u0027s a contrast with this question."},{"Start":"04:59.120 ","End":"05:01.610","Text":"I also get 0 in the numerator as we\u0027ll see."},{"Start":"05:01.610 ","End":"05:03.484","Text":"Okay, let\u0027s just get started."},{"Start":"05:03.484 ","End":"05:09.660","Text":"I\u0027m asking is minus 4 squared plus 5"},{"Start":"05:09.660 ","End":"05:17.160","Text":"times minus 4 plus 4/minus 4 plus 4."},{"Start":"05:17.160 ","End":"05:20.685","Text":"Is this equal to 0?"},{"Start":"05:20.685 ","End":"05:23.080","Text":"If you just did the numerator,"},{"Start":"05:23.080 ","End":"05:26.590","Text":"you would say minus 4 squared is 16."},{"Start":"05:26.590 ","End":"05:28.300","Text":"Let me just do this at the side here."},{"Start":"05:28.300 ","End":"05:32.825","Text":"16, 5 times minus 4 is minus 20,"},{"Start":"05:32.825 ","End":"05:34.580","Text":"and then plus 4,"},{"Start":"05:34.580 ","End":"05:38.260","Text":"you would say that this is equal to 0."},{"Start":"05:38.260 ","End":"05:40.135","Text":"If you just stopped there and said,"},{"Start":"05:40.135 ","End":"05:43.210","Text":"like I did before, 0 over anything is 0,"},{"Start":"05:43.210 ","End":"05:48.775","Text":"that would not be quite right because here the denominator is also 0."},{"Start":"05:48.775 ","End":"05:52.820","Text":"Really, I should have checked in the previous question,"},{"Start":"05:52.820 ","End":"05:54.845","Text":"I shouldn\u0027t have just said 0 over anything."},{"Start":"05:54.845 ","End":"05:57.715","Text":"I should have said 0 over 2."},{"Start":"05:57.715 ","End":"05:59.450","Text":"Not that the 2 is important,"},{"Start":"05:59.450 ","End":"06:01.670","Text":"but it\u0027s important that it\u0027s not 0."},{"Start":"06:01.670 ","End":"06:04.475","Text":"0 over non-zero is 0,"},{"Start":"06:04.475 ","End":"06:08.465","Text":"but 0 over 0 is undefined."},{"Start":"06:08.465 ","End":"06:10.925","Text":"It\u0027s just undefined."},{"Start":"06:10.925 ","End":"06:12.500","Text":"It doesn\u0027t make sense,"},{"Start":"06:12.500 ","End":"06:15.124","Text":"not a number, not defined."},{"Start":"06:15.124 ","End":"06:18.740","Text":"We\u0027re not allowed to substitute w equals minus 4."},{"Start":"06:18.740 ","End":"06:20.630","Text":"If we can\u0027t substitute it,"},{"Start":"06:20.630 ","End":"06:22.310","Text":"It\u0027s certainly not equal to 0."},{"Start":"06:22.310 ","End":"06:24.200","Text":"It\u0027s not defined and therefore it\u0027s not"},{"Start":"06:24.200 ","End":"06:26.470","Text":"equal to 0 because this whole thing doesn\u0027t make sense,"},{"Start":"06:26.470 ","End":"06:29.345","Text":"and so we would say the answer is no,"},{"Start":"06:29.345 ","End":"06:34.370","Text":"not a solution because it\u0027s undefined on the left-hand side,"},{"Start":"06:34.370 ","End":"06:36.710","Text":"which is defined on the right-hand side."},{"Start":"06:36.710 ","End":"06:39.240","Text":"That\u0027s the last one."}],"ID":4930},{"Watched":false,"Name":"Definition, Solving","Duration":"9m 22s","ChapterTopicVideoID":4929,"CourseChapterTopicPlaylistID":3861,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.620","Text":"In the previous clip,"},{"Start":"00:01.620 ","End":"00:04.845","Text":"I gave an informal introduction to equations."},{"Start":"00:04.845 ","End":"00:08.070","Text":"In this clip, I\u0027m going to give a definition of what an equation"},{"Start":"00:08.070 ","End":"00:11.910","Text":"is and also some basic techniques for solving equations."},{"Start":"00:11.910 ","End":"00:14.040","Text":"Let\u0027s start with the definition of an equation."},{"Start":"00:14.040 ","End":"00:21.420","Text":"An equation is when one algebraic expression is equal to another algebraic expression."},{"Start":"00:21.420 ","End":"00:23.955","Text":"That\u0027s basically what an equation is."},{"Start":"00:23.955 ","End":"00:27.150","Text":"It\u0027s just one thing that the algebraic expressions,"},{"Start":"00:27.150 ","End":"00:30.135","Text":"one of the other should contain at least one variable."},{"Start":"00:30.135 ","End":"00:33.970","Text":"Because also an arithmetic expression is also an algebraic expression."},{"Start":"00:33.970 ","End":"00:36.320","Text":"We have to have at least one variable, usually x,"},{"Start":"00:36.320 ","End":"00:38.945","Text":"but it could be more, and let me give an example."},{"Start":"00:38.945 ","End":"00:40.130","Text":"In the previous clip,"},{"Start":"00:40.130 ","End":"00:42.260","Text":"we did this equation,"},{"Start":"00:42.260 ","End":"00:43.895","Text":"which I copy pasted."},{"Start":"00:43.895 ","End":"00:45.965","Text":"If you look at it,"},{"Start":"00:45.965 ","End":"00:49.250","Text":"it does fit the pattern that"},{"Start":"00:49.250 ","End":"00:55.235","Text":"one algebraic expression is equal to another algebraic expression."},{"Start":"00:55.235 ","End":"00:59.855","Text":"It also satisfies the extra condition I didn\u0027t write, I should write it."},{"Start":"00:59.855 ","End":"01:03.785","Text":"That is, it must contain at least 1 variable,"},{"Start":"01:03.785 ","End":"01:06.875","Text":"and here is our variable x."},{"Start":"01:06.875 ","End":"01:09.935","Text":"Now last time we guessed the solution."},{"Start":"01:09.935 ","End":"01:16.970","Text":"We somehow guessed that the solution was x=3."},{"Start":"01:17.020 ","End":"01:20.210","Text":"Even if we don\u0027t know how to solve equations,"},{"Start":"01:20.210 ","End":"01:23.090","Text":"we can certainly check if something is a solution."},{"Start":"01:23.090 ","End":"01:25.865","Text":"For example, if I put x=3 here,"},{"Start":"01:25.865 ","End":"01:30.140","Text":"I get that 3 plus 2=5 and I ask myself,"},{"Start":"01:30.140 ","End":"01:31.670","Text":"is this a true statement?"},{"Start":"01:31.670 ","End":"01:33.980","Text":"If it is, and it is,"},{"Start":"01:33.980 ","End":"01:36.905","Text":"then we say that this is a solution."},{"Start":"01:36.905 ","End":"01:39.185","Text":"I don\u0027t want to guess solutions,"},{"Start":"01:39.185 ","End":"01:42.605","Text":"and I want to know how to get from here to here."},{"Start":"01:42.605 ","End":"01:44.285","Text":"How do I get there?"},{"Start":"01:44.285 ","End":"01:46.440","Text":"I\u0027m deliberately starting with easy examples,"},{"Start":"01:46.440 ","End":"01:48.020","Text":"and of course in your head,"},{"Start":"01:48.020 ","End":"01:51.680","Text":"you could tell me what number plus 2=5,"},{"Start":"01:51.680 ","End":"01:54.470","Text":"and here you\u0027d immediately say, \"Yeah, 3.\""},{"Start":"01:54.470 ","End":"01:58.415","Text":"Somehow in your head you\u0027d be subtracting 5 less 2."},{"Start":"01:58.415 ","End":"02:00.580","Text":"Let\u0027s try and make this more formal."},{"Start":"02:00.580 ","End":"02:04.850","Text":"I\u0027m going to erase whatever it is that I don\u0027t need."},{"Start":"02:04.850 ","End":"02:09.890","Text":"Let\u0027s say we don\u0027t know the solution and we have x plus 2=5,"},{"Start":"02:09.890 ","End":"02:11.945","Text":"and we want to solve it."},{"Start":"02:11.945 ","End":"02:13.610","Text":"The basic principle is,"},{"Start":"02:13.610 ","End":"02:15.395","Text":"if I have an equation,"},{"Start":"02:15.395 ","End":"02:17.165","Text":"something equals something,"},{"Start":"02:17.165 ","End":"02:19.655","Text":"if I do the same thing to both sides,"},{"Start":"02:19.655 ","End":"02:21.050","Text":"they will still be equal."},{"Start":"02:21.050 ","End":"02:23.885","Text":"Let\u0027s say I want to subtract 2 from both sides."},{"Start":"02:23.885 ","End":"02:26.690","Text":"What I will get will still be an equation."},{"Start":"02:26.690 ","End":"02:27.935","Text":"They\u0027ll still be equal."},{"Start":"02:27.935 ","End":"02:30.335","Text":"I didn\u0027t just randomly say subtract 2."},{"Start":"02:30.335 ","End":"02:32.360","Text":"This is what in fact we\u0027re going to do here."},{"Start":"02:32.360 ","End":"02:35.885","Text":"I\u0027m going to just write it in words this time and say,"},{"Start":"02:35.885 ","End":"02:40.845","Text":"I\u0027m going to subtract 2 from both sides."},{"Start":"02:40.845 ","End":"02:43.580","Text":"If I subtract 2 from both sides on the left,"},{"Start":"02:43.580 ","End":"02:45.560","Text":"I\u0027ll get x plus 2."},{"Start":"02:45.560 ","End":"02:49.100","Text":"Here\u0027s what I subtract 2, but remember,"},{"Start":"02:49.100 ","End":"02:52.775","Text":"I\u0027ve to do everything to both sides to keep it equal,"},{"Start":"02:52.775 ","End":"02:56.195","Text":"is equal to 5 and I\u0027ll subtract 2 from here."},{"Start":"02:56.195 ","End":"02:59.795","Text":"Now we know how to simplify algebraic expressions,"},{"Start":"02:59.795 ","End":"03:03.070","Text":"x plus 2 minus 2 is just x,"},{"Start":"03:03.070 ","End":"03:05.595","Text":"5 minus 2 is 3,"},{"Start":"03:05.595 ","End":"03:09.190","Text":"and so the solution is x=3,"},{"Start":"03:09.190 ","End":"03:11.300","Text":"and we did it not by guesswork."},{"Start":"03:11.300 ","End":"03:14.105","Text":"Now let me take another example."},{"Start":"03:14.105 ","End":"03:16.800","Text":"Let\u0027s say this was example 1,"},{"Start":"03:16.800 ","End":"03:18.990","Text":"and now I\u0027ll go for example 2,"},{"Start":"03:18.990 ","End":"03:20.760","Text":"try a different color."},{"Start":"03:20.760 ","End":"03:25.750","Text":"Let\u0027s say x minus 3=8."},{"Start":"03:27.890 ","End":"03:33.500","Text":"If we do it mentally, you can easily see that we\u0027re asking what minus 3=8,"},{"Start":"03:33.500 ","End":"03:35.570","Text":"and we know straight away that it\u0027s going to be 11,"},{"Start":"03:35.570 ","End":"03:37.115","Text":"but how would I do this?"},{"Start":"03:37.115 ","End":"03:40.670","Text":"Well, just like I said, if I do the same thing to both sides of an equation,"},{"Start":"03:40.670 ","End":"03:42.155","Text":"it\u0027ll still be an equation."},{"Start":"03:42.155 ","End":"03:46.820","Text":"In this case, what I really want to do is to add 3 to both sides."},{"Start":"03:46.820 ","End":"03:50.505","Text":"I\u0027m going to now add 3,"},{"Start":"03:50.505 ","End":"03:56.760","Text":"and then I will get that x minus 3 wherever was,"},{"Start":"03:56.760 ","End":"03:59.175","Text":"now I\u0027m adding 3 to the left side."},{"Start":"03:59.175 ","End":"04:00.780","Text":"This is the left hand side of the equation."},{"Start":"04:00.780 ","End":"04:02.010","Text":"This is the right hand side."},{"Start":"04:02.010 ","End":"04:08.940","Text":"Right hand side, I will also add 3 and then simplification,"},{"Start":"04:08.940 ","End":"04:12.270","Text":"x minus 3 plus 3 is just x,"},{"Start":"04:12.270 ","End":"04:15.465","Text":"will give me that x is equal to 11."},{"Start":"04:15.465 ","End":"04:19.440","Text":"Continuing, I need a bit more space here."},{"Start":"04:19.440 ","End":"04:26.220","Text":"This time my equation will be 4x=12."},{"Start":"04:26.220 ","End":"04:27.840","Text":"Remember in algebra,"},{"Start":"04:27.840 ","End":"04:30.215","Text":"if we don\u0027t have any operation written"},{"Start":"04:30.215 ","End":"04:32.930","Text":"2 things next to each other, it means multiplication."},{"Start":"04:32.930 ","End":"04:38.750","Text":"I\u0027ll just put the dot in to remind us that 4x means 4 times x."},{"Start":"04:38.750 ","End":"04:41.630","Text":"If I was to say to you 4 times what is 12,"},{"Start":"04:41.630 ","End":"04:44.915","Text":"you\u0027d have no trouble in telling me that that something is 3."},{"Start":"04:44.915 ","End":"04:47.260","Text":"I know the answer is going to be 3,"},{"Start":"04:47.260 ","End":"04:52.819","Text":"but I want to do it using this technique of doing the same thing to both sides."},{"Start":"04:52.819 ","End":"04:57.375","Text":"Clearly we want to somehow divide 12 by 4,."},{"Start":"04:57.375 ","End":"05:03.100","Text":"So what I\u0027m going to do is say, divide by 4."},{"Start":"05:03.500 ","End":"05:05.700","Text":"This is not the way we\u0027re going to do it."},{"Start":"05:05.700 ","End":"05:07.560","Text":"I\u0027m going to show you shortcuts."},{"Start":"05:07.560 ","End":"05:10.260","Text":"You wouldn\u0027t actually write subtract 2."},{"Start":"05:10.260 ","End":"05:12.185","Text":"I\u0027m going to show you afterwards how we would do it."},{"Start":"05:12.185 ","End":"05:13.370","Text":"I\u0027m doing it the long way,"},{"Start":"05:13.370 ","End":"05:15.980","Text":"just for educational purposes."},{"Start":"05:15.980 ","End":"05:17.420","Text":"So divide by 4,"},{"Start":"05:17.420 ","End":"05:19.310","Text":"I mean divide both sides by 4,"},{"Start":"05:19.310 ","End":"05:24.195","Text":"do the same thing to both sides of an equation and it\u0027ll still be an equation."},{"Start":"05:24.195 ","End":"05:26.330","Text":"I\u0027ve got 4x."},{"Start":"05:26.330 ","End":"05:29.475","Text":"Let\u0027s do our division like this,"},{"Start":"05:29.475 ","End":"05:35.310","Text":"divide it by 4 is equal to 12 divided by 4."},{"Start":"05:35.310 ","End":"05:38.510","Text":"Now we know how to cancel."},{"Start":"05:38.510 ","End":"05:42.110","Text":"We just say this cancels with this and this is 1,"},{"Start":"05:42.110 ","End":"05:44.630","Text":"but divided by 1 it\u0027s just x,"},{"Start":"05:44.630 ","End":"05:48.660","Text":"and so we end up with just x here,"},{"Start":"05:48.660 ","End":"05:53.360","Text":"and 12 divided by 4 with or without a calculator is 3."},{"Start":"05:53.360 ","End":"05:57.275","Text":"That\u0027s the solution to this equation and,"},{"Start":"05:57.275 ","End":"06:00.350","Text":"what do these all have in common and not in common?"},{"Start":"06:00.350 ","End":"06:01.670","Text":"I\u0027ve used addition."},{"Start":"06:01.670 ","End":"06:02.840","Text":"I\u0027ve used subtraction."},{"Start":"06:02.840 ","End":"06:04.044","Text":"I\u0027ve used multiplication."},{"Start":"06:04.044 ","End":"06:06.965","Text":"Let\u0027s give one more example with a division,"},{"Start":"06:06.965 ","End":"06:10.050","Text":"a bit more space, there I go."},{"Start":"06:10.050 ","End":"06:12.015","Text":"Example number 4,"},{"Start":"06:12.015 ","End":"06:19.470","Text":"let\u0027s take x divided by 5=6."},{"Start":"06:19.470 ","End":"06:23.565","Text":"What number divided by 5 is 6?"},{"Start":"06:23.565 ","End":"06:27.290","Text":"You could probably mentally do this and you\u0027d probably somehow end up mentally"},{"Start":"06:27.290 ","End":"06:31.085","Text":"doing 5 times 6 is 30 and indeed 30 over 5 is 6."},{"Start":"06:31.085 ","End":"06:32.839","Text":"But how do we do this formally?"},{"Start":"06:32.839 ","End":"06:35.930","Text":"In this case, we do something to both sides of the equation."},{"Start":"06:35.930 ","End":"06:41.280","Text":"In this case, we want to multiply by 5."},{"Start":"06:41.280 ","End":"06:46.525","Text":"That will do the trick for us because then we will get x over"},{"Start":"06:46.525 ","End":"06:54.615","Text":"5 times 5=6 times 5."},{"Start":"06:54.615 ","End":"06:57.225","Text":"Once again, we know how to cancel,"},{"Start":"06:57.225 ","End":"07:01.035","Text":"5 in the numerator cancels with 5 in the denominator,"},{"Start":"07:01.035 ","End":"07:10.030","Text":"and we\u0027re left here with just x is equal to 6 times 5 is 30 as we expected and suspected."},{"Start":"07:10.030 ","End":"07:14.190","Text":"That\u0027s 4 examples of 4 basic operations."},{"Start":"07:14.190 ","End":"07:18.525","Text":"Let me show you the short way of doing all these."},{"Start":"07:18.525 ","End":"07:21.175","Text":"I\u0027ll go back to the top,"},{"Start":"07:21.175 ","End":"07:24.800","Text":"and let me just get rid of this,"},{"Start":"07:24.800 ","End":"07:26.255","Text":"get some space here."},{"Start":"07:26.255 ","End":"07:33.620","Text":"In practice, if I was given x plus 2=5 what we do is,"},{"Start":"07:33.620 ","End":"07:40.715","Text":"we say, bring the 2 over to the other side and change its sign."},{"Start":"07:40.715 ","End":"07:46.455","Text":"What we say is that x=5 minus 2."},{"Start":"07:46.455 ","End":"07:51.185","Text":"If you have a plus something and you put it on the other side and make it a minus,"},{"Start":"07:51.185 ","End":"07:54.840","Text":"that\u0027s the practical way of what we do,"},{"Start":"07:54.840 ","End":"07:57.990","Text":"and then we just say that 5 minus 2 is 3."},{"Start":"07:57.990 ","End":"08:00.510","Text":"Addition, changes to subtraction."},{"Start":"08:00.510 ","End":"08:03.480","Text":"If you want the reasoning, this is the reasoning."},{"Start":"08:03.480 ","End":"08:07.770","Text":"Next example, x minus 3=8,"},{"Start":"08:07.770 ","End":"08:13.200","Text":"what we say is, we have a subtraction here of minus 3."},{"Start":"08:13.200 ","End":"08:17.765","Text":"We bring the 3 over to the other side and we change its sign,"},{"Start":"08:17.765 ","End":"08:19.985","Text":"and then we make it a plus 3."},{"Start":"08:19.985 ","End":"08:25.300","Text":"We say that x is equal to 8 plus 3."},{"Start":"08:25.300 ","End":"08:28.060","Text":"Just like a plus goes over to become a minus,"},{"Start":"08:28.060 ","End":"08:30.320","Text":"a minus goes over to become a plus."},{"Start":"08:30.320 ","End":"08:32.645","Text":"Then we say that x=11,"},{"Start":"08:32.645 ","End":"08:34.385","Text":"of course the same answer."},{"Start":"08:34.385 ","End":"08:39.050","Text":"Next example, here we had a multiplication,"},{"Start":"08:39.050 ","End":"08:43.100","Text":"4 times x or x times 4=12,"},{"Start":"08:43.100 ","End":"08:45.305","Text":"and the multiplication,"},{"Start":"08:45.305 ","End":"08:48.185","Text":"when I bring it to the other side becomes a division."},{"Start":"08:48.185 ","End":"08:49.580","Text":"Bring the 4 over,"},{"Start":"08:49.580 ","End":"08:56.105","Text":"so I get x=12 divided by 4 and that\u0027s equal to 3."},{"Start":"08:56.105 ","End":"08:58.175","Text":"This is how we do it in practice,"},{"Start":"08:58.175 ","End":"09:00.200","Text":"and as for the last one,"},{"Start":"09:00.200 ","End":"09:03.690","Text":"this time we had a division x divided by 5=6,"},{"Start":"09:03.690 ","End":"09:08.420","Text":"and just like multiplication became division, division becomes multiplication."},{"Start":"09:08.420 ","End":"09:12.799","Text":"We bring the 5 over to the other side and make it a multiplication,"},{"Start":"09:12.799 ","End":"09:14.795","Text":"6 times 5,"},{"Start":"09:14.795 ","End":"09:17.585","Text":"and that is equal to 30."},{"Start":"09:17.585 ","End":"09:19.621","Text":"That\u0027s how we do it in practice."},{"Start":"09:19.621 ","End":"09:22.200","Text":"That\u0027s enough for this clip."}],"ID":4931}],"Thumbnail":null,"ID":3861},{"Name":"Linear Equations- One Variable","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Linear in One Unknown - Part 1","Duration":"7m 46s","ChapterTopicVideoID":4937,"CourseChapterTopicPlaylistID":3862,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.370","Text":"In this clip, I\u0027m going to show you how to solve linear equations in one variable."},{"Start":"00:05.370 ","End":"00:09.000","Text":"The one variable means we just have x."},{"Start":"00:09.000 ","End":"00:11.760","Text":"As I mentioned, equations can have more than one variable;"},{"Start":"00:11.760 ","End":"00:14.475","Text":"x and y or any other letters,"},{"Start":"00:14.475 ","End":"00:17.550","Text":"here we\u0027re going to have just one variable and I\u0027ll just use"},{"Start":"00:17.550 ","End":"00:22.155","Text":"x not to confuse you with a or t or any other letter."},{"Start":"00:22.155 ","End":"00:24.990","Text":"One variable x. Linear,"},{"Start":"00:24.990 ","End":"00:26.190","Text":"I could define it,"},{"Start":"00:26.190 ","End":"00:27.645","Text":"but I don\u0027t want to get too technical."},{"Start":"00:27.645 ","End":"00:29.745","Text":"Let me just say what is not linear."},{"Start":"00:29.745 ","End":"00:33.150","Text":"Not linear, I\u0027m talking about the word linear here,"},{"Start":"00:33.150 ","End":"00:39.420","Text":"is not exponents, meaning stuff like x squared or in general,"},{"Start":"00:39.420 ","End":"00:43.205","Text":"x^n, not square roots of x,"},{"Start":"00:43.205 ","End":"00:45.785","Text":"not the nth root of x,"},{"Start":"00:45.785 ","End":"00:47.180","Text":"or anything like that,"},{"Start":"00:47.180 ","End":"00:51.350","Text":"and not 1/x or x in a denominator."},{"Start":"00:51.350 ","End":"00:52.985","Text":"These are the things that are not."},{"Start":"00:52.985 ","End":"00:56.650","Text":"Let\u0027s just get started and I think I\u0027ll do about 4 examples."},{"Start":"00:56.650 ","End":"01:03.260","Text":"The first example, 6x+2=8."},{"Start":"01:03.260 ","End":"01:04.670","Text":"Certainly, there\u0027s no exponents,"},{"Start":"01:04.670 ","End":"01:07.370","Text":"roots, denominators, anything like that."},{"Start":"01:07.370 ","End":"01:11.285","Text":"We\u0027re going to use the techniques that we learned in the last clip,"},{"Start":"01:11.285 ","End":"01:16.400","Text":"which is mainly whatever you do to one side you can do to the other."},{"Start":"01:16.400 ","End":"01:20.420","Text":"Also, we learned the technique of how to bring things"},{"Start":"01:20.420 ","End":"01:24.680","Text":"from one side to the other by taking the opposite."},{"Start":"01:24.680 ","End":"01:28.750","Text":"In this case, what I want to do first is bring the 2"},{"Start":"01:28.750 ","End":"01:32.900","Text":"to the other side and change it from a plus to a minus."},{"Start":"01:32.900 ","End":"01:37.610","Text":"6x=8-2."},{"Start":"01:37.610 ","End":"01:40.385","Text":"Sometimes I like to draw a little arrow,"},{"Start":"01:40.385 ","End":"01:42.470","Text":"something like this, to show what I\u0027m doing,"},{"Start":"01:42.470 ","End":"01:43.940","Text":"is I\u0027m taking the 2 to the other side,"},{"Start":"01:43.940 ","End":"01:45.710","Text":"but it changes sign."},{"Start":"01:45.710 ","End":"01:48.270","Text":"This is equal to 6,"},{"Start":"01:48.270 ","End":"01:50.415","Text":"and I\u0027ll just copy the 6x here."},{"Start":"01:50.415 ","End":"01:53.750","Text":"Remember 6x means 6*x."},{"Start":"01:53.750 ","End":"01:55.670","Text":"This gives me a hint to what to do next."},{"Start":"01:55.670 ","End":"01:57.020","Text":"I want to get rid of that 6,"},{"Start":"01:57.020 ","End":"02:00.095","Text":"so I\u0027m going to bring it over to the other side."},{"Start":"02:00.095 ","End":"02:03.020","Text":"This means that if it\u0027s multiplication here,"},{"Start":"02:03.020 ","End":"02:04.790","Text":"it\u0027s going to be division here."},{"Start":"02:04.790 ","End":"02:11.230","Text":"X is equal to this 6/6."},{"Start":"02:11.230 ","End":"02:14.175","Text":"We\u0027re left with x=1,"},{"Start":"02:14.175 ","End":"02:16.620","Text":"and that\u0027s the answer to this."},{"Start":"02:16.620 ","End":"02:18.630","Text":"Onto Number 2."},{"Start":"02:18.630 ","End":"02:23.505","Text":"7-2x=7."},{"Start":"02:23.505 ","End":"02:27.310","Text":"I want to get rid of this 7 here,"},{"Start":"02:27.310 ","End":"02:29.360","Text":"and I\u0027ll throw it to the other side."},{"Start":"02:29.360 ","End":"02:31.607","Text":"I\u0027ll just put an arrow here."},{"Start":"02:31.607 ","End":"02:35.040","Text":"When I throw it over to the other side, it changes sign."},{"Start":"02:35.040 ","End":"02:37.810","Text":"Here it\u0027s a plus. If here\u0027s nothing in front of it\u0027s a plus,"},{"Start":"02:37.810 ","End":"02:40.055","Text":"so here it becomes a minus."},{"Start":"02:40.055 ","End":"02:44.970","Text":"I get -2x=7 that was here,"},{"Start":"02:44.970 ","End":"02:50.325","Text":"and this 7 over here becomes minus.-2x,"},{"Start":"02:50.325 ","End":"02:53.410","Text":"just copying, equals 7-7 is 0."},{"Start":"02:53.410 ","End":"02:57.485","Text":"Now I want to get rid of this -2."},{"Start":"02:57.485 ","End":"03:00.290","Text":"I\u0027m going to take this-2,"},{"Start":"03:00.290 ","End":"03:04.060","Text":"the whole of it, and move it to the other side."},{"Start":"03:04.060 ","End":"03:06.890","Text":"But what do we do with it on the other side?"},{"Start":"03:06.890 ","End":"03:08.930","Text":"Well, here there\u0027s a multiplication."},{"Start":"03:08.930 ","End":"03:11.000","Text":"If there\u0027s nothing written, it\u0027s a multiplication."},{"Start":"03:11.000 ","End":"03:12.470","Text":"When I put it on the other side,"},{"Start":"03:12.470 ","End":"03:14.045","Text":"it becomes a division."},{"Start":"03:14.045 ","End":"03:19.515","Text":"x is 0 divided by-2."},{"Start":"03:19.515 ","End":"03:21.960","Text":"Well, 0 divided by anything is 0,"},{"Start":"03:21.960 ","End":"03:25.980","Text":"so x=0 and that\u0027s the answer to this."},{"Start":"03:25.980 ","End":"03:30.350","Text":"I want to remind you or tell you that you can always check"},{"Start":"03:30.350 ","End":"03:35.660","Text":"your answer by substituting at the end what you get in the original equation."},{"Start":"03:35.660 ","End":"03:39.890","Text":"For example, here if I put x=1, I get 6*1+2."},{"Start":"03:39.890 ","End":"03:41.360","Text":"Does that equal 8?"},{"Start":"03:41.360 ","End":"03:42.815","Text":"Well, yes it does."},{"Start":"03:42.815 ","End":"03:44.925","Text":"Or here, x=0."},{"Start":"03:44.925 ","End":"03:48.030","Text":"Is 7-2*0=7?"},{"Start":"03:48.030 ","End":"03:49.680","Text":"The answer is yes."},{"Start":"03:49.680 ","End":"03:51.300","Text":"This is very important."},{"Start":"03:51.300 ","End":"03:52.850","Text":"I\u0027m not going to say it each time,"},{"Start":"03:52.850 ","End":"03:55.505","Text":"but really, if you have the time,"},{"Start":"03:55.505 ","End":"03:58.130","Text":"you want to be sure when you get your answer"},{"Start":"03:58.130 ","End":"04:02.615","Text":"substituted in the original and see if it gives you something true."},{"Start":"04:02.615 ","End":"04:05.065","Text":"Let\u0027s get onto Number 3."},{"Start":"04:05.065 ","End":"04:14.270","Text":"Number 3 will be 2x+x=24."},{"Start":"04:14.270 ","End":"04:17.585","Text":"This is a bit different than the ones we\u0027ve had so far."},{"Start":"04:17.585 ","End":"04:24.130","Text":"It won\u0027t really help us to move an x to the other side or even the 2x,"},{"Start":"04:24.130 ","End":"04:27.290","Text":"but still I\u0027m going to have x on both sides and that\u0027s not good."},{"Start":"04:27.290 ","End":"04:29.885","Text":"What is needed here is a different technique."},{"Start":"04:29.885 ","End":"04:32.700","Text":"We need to factorize;"},{"Start":"04:32.700 ","End":"04:34.850","Text":"to take something outside the brackets."},{"Start":"04:34.850 ","End":"04:37.700","Text":"Now, remembering that x is 1x,"},{"Start":"04:37.700 ","End":"04:40.745","Text":"and I\u0027ll just emphasize it by putting 1x here."},{"Start":"04:40.745 ","End":"04:44.030","Text":"Now, you remember how to collect like terms."},{"Start":"04:44.030 ","End":"04:45.665","Text":"We have 2x and 1x,"},{"Start":"04:45.665 ","End":"04:48.095","Text":"altogether we have 3x."},{"Start":"04:48.095 ","End":"04:51.155","Text":"Now we have 3x is 24."},{"Start":"04:51.155 ","End":"04:55.685","Text":"Don\u0027t forget that 3x means 3*x."},{"Start":"04:55.685 ","End":"04:59.225","Text":"Now we want to move the 3 to the other side,"},{"Start":"04:59.225 ","End":"05:02.170","Text":"but here it\u0027s a multiplication and here it\u0027s a division."},{"Start":"05:02.170 ","End":"05:09.240","Text":"x=24/3, in other words,"},{"Start":"05:09.240 ","End":"05:11.235","Text":"x is equal to 8."},{"Start":"05:11.235 ","End":"05:12.915","Text":"That solved the third one."},{"Start":"05:12.915 ","End":"05:14.970","Text":"Now one last one;"},{"Start":"05:14.970 ","End":"05:25.190","Text":"Number 4, 2-5x+7=-3x+8."},{"Start":"05:25.190 ","End":"05:27.800","Text":"Here\u0027s a little bit more work and it brings me to"},{"Start":"05:27.800 ","End":"05:31.370","Text":"a general principle which I won\u0027t write but I\u0027ll repeat often,"},{"Start":"05:31.370 ","End":"05:34.490","Text":"that the idea in this equation is to"},{"Start":"05:34.490 ","End":"05:37.490","Text":"bring the x\u0027s to the left and the numbers to the right."},{"Start":"05:37.490 ","End":"05:40.220","Text":"I\u0027ll say that again, x\u0027s on the left, numbers on the right."},{"Start":"05:40.220 ","End":"05:42.620","Text":"If you look at the ones we\u0027ve done so far,"},{"Start":"05:42.620 ","End":"05:44.060","Text":"this is what we did."},{"Start":"05:44.060 ","End":"05:48.245","Text":"We left the x\u0027s on the left and we put the numbers on the right."},{"Start":"05:48.245 ","End":"05:50.600","Text":"In Number 2, I\u0027ve put the number on the right,"},{"Start":"05:50.600 ","End":"05:52.835","Text":"so we had x\u0027s on the left, numbers on the right."},{"Start":"05:52.835 ","End":"05:57.395","Text":"Here, we already had x\u0027s on the left and numbers on the right."},{"Start":"05:57.395 ","End":"05:59.270","Text":"Same thing here."},{"Start":"05:59.270 ","End":"06:02.876","Text":"What I want to do is keep the 5x or"},{"Start":"06:02.876 ","End":"06:07.605","Text":"-5x here and bring this over to this side. I\u0027ll indicate that."},{"Start":"06:07.605 ","End":"06:11.983","Text":"I want the 3x moved over to the left because it\u0027s an x,"},{"Start":"06:11.983 ","End":"06:15.080","Text":"I want the 7 to go over to the right,"},{"Start":"06:15.080 ","End":"06:18.125","Text":"and I want the 2 to go over to the right."},{"Start":"06:18.125 ","End":"06:22.325","Text":"Remember, with addition and subtraction we reverse the sign."},{"Start":"06:22.325 ","End":"06:23.780","Text":"Let\u0027s see what we get."},{"Start":"06:23.780 ","End":"06:26.895","Text":"On the left we already had a -5x,"},{"Start":"06:26.895 ","End":"06:28.825","Text":"so it stays -5x."},{"Start":"06:28.825 ","End":"06:36.385","Text":"But we brought this one over or over about to bring it over and it becomes +3x."},{"Start":"06:36.385 ","End":"06:38.370","Text":"That\u0027s all the x\u0027s on the left."},{"Start":"06:38.370 ","End":"06:41.344","Text":"Now the numbers on the right is,"},{"Start":"06:41.344 ","End":"06:42.960","Text":"we already had an 8 here,"},{"Start":"06:42.960 ","End":"06:45.495","Text":"so let\u0027s leave that 8, it\u0027s +8."},{"Start":"06:45.495 ","End":"06:53.110","Text":"The 2 comes over as -2 and the +7 comes over as -7."},{"Start":"06:53.110 ","End":"06:56.855","Text":"Now we\u0027ve separated like terms."},{"Start":"06:56.855 ","End":"06:59.075","Text":"The x\u0027s on the left, numbers on the right."},{"Start":"06:59.075 ","End":"07:01.310","Text":"Now we just have to collect them."},{"Start":"07:01.310 ","End":"07:03.380","Text":"Let\u0027s see, how many x do we have?"},{"Start":"07:03.380 ","End":"07:09.555","Text":"Well, we have-5+3, which is-2x."},{"Start":"07:09.555 ","End":"07:13.620","Text":"On the right we have 8-2-7,"},{"Start":"07:13.620 ","End":"07:17.265","Text":"and that comes out to be-1."},{"Start":"07:17.265 ","End":"07:20.150","Text":"Now we have this multiplication here,"},{"Start":"07:20.150 ","End":"07:24.860","Text":"we want to take this-2 and move it to the other side."},{"Start":"07:24.860 ","End":"07:28.180","Text":"This time a multiplication turns into a division."},{"Start":"07:28.180 ","End":"07:33.520","Text":"What we get is that x=-1/-2."},{"Start":"07:33.520 ","End":"07:36.290","Text":"Remember, negative over negative is positive,"},{"Start":"07:36.290 ","End":"07:40.525","Text":"so this just gives us that x is 1.5."},{"Start":"07:40.525 ","End":"07:42.960","Text":"A fraction. That\u0027s okay."},{"Start":"07:42.960 ","End":"07:46.690","Text":"That\u0027s these 4 and we\u0027re done for now."}],"ID":4932},{"Watched":false,"Name":"Linear in One Unknown - Part 2","Duration":"9m 21s","ChapterTopicVideoID":4936,"CourseChapterTopicPlaylistID":3862,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"Now this might look familiar to you because the previous clip"},{"Start":"00:03.180 ","End":"00:06.120","Text":"was solving linear equations in one variable"},{"Start":"00:06.120 ","End":"00:12.300","Text":"so is this so I\u0027ll just add the words continued because it\u0027s a big topic."},{"Start":"00:12.300 ","End":"00:17.190","Text":"One of the main things that we\u0027ll see in this clip will be the use"},{"Start":"00:17.190 ","End":"00:22.525","Text":"of the expansion of brackets when we\u0027re solving linear equations."},{"Start":"00:22.525 ","End":"00:24.345","Text":"I\u0027ll just write that down."},{"Start":"00:24.345 ","End":"00:29.415","Text":"We\u0027re going to be using brackets expansions which you learned in algebraic expression."},{"Start":"00:29.415 ","End":"00:31.665","Text":"Here\u0027s our first example."},{"Start":"00:31.665 ","End":"00:38.440","Text":"Twice x minus 2 minus 3 equals 1."},{"Start":"00:38.440 ","End":"00:40.430","Text":"Now, in the previous lesson,"},{"Start":"00:40.430 ","End":"00:43.130","Text":"I kept telling you always that we want to get it"},{"Start":"00:43.130 ","End":"00:46.625","Text":"with x\u0027s on the left, numbers on the right."},{"Start":"00:46.625 ","End":"00:48.545","Text":"Again, I\u0027m going to repeat this."},{"Start":"00:48.545 ","End":"00:51.620","Text":"In fact, I even brought a little cartoon to illustrate it,"},{"Start":"00:51.620 ","End":"00:55.310","Text":"it\u0027s taken from the New Testament, I believe."},{"Start":"00:55.310 ","End":"00:57.530","Text":"We\u0027re going to use it here only in our case,"},{"Start":"00:57.530 ","End":"01:00.875","Text":"we don\u0027t have sheep and goats we\u0027re going to have x\u0027s and numbers."},{"Start":"01:00.875 ","End":"01:03.140","Text":"Before we can bring stuff over,"},{"Start":"01:03.140 ","End":"01:08.615","Text":"we need to expand the brackets and that\u0027s the topic of this section so let\u0027s do that."},{"Start":"01:08.615 ","End":"01:11.000","Text":"Remember how we do this."},{"Start":"01:11.000 ","End":"01:14.420","Text":"We just multiply this by this and this by this."},{"Start":"01:14.420 ","End":"01:18.590","Text":"We get 2x minus 2 times 2 is 4,"},{"Start":"01:18.590 ","End":"01:21.185","Text":"minus 3 is 1."},{"Start":"01:21.185 ","End":"01:23.510","Text":"Now we do the separation."},{"Start":"01:23.510 ","End":"01:26.570","Text":"Terms containing x will go on the left,"},{"Start":"01:26.570 ","End":"01:28.490","Text":"just numbers will go on the right."},{"Start":"01:28.490 ","End":"01:34.085","Text":"What I have to do is move this over to the right and this over to the right."},{"Start":"01:34.085 ","End":"01:35.240","Text":"But this will stay on the right."},{"Start":"01:35.240 ","End":"01:37.585","Text":"It is a number and this will stay on the left."},{"Start":"01:37.585 ","End":"01:41.400","Text":"We get 2x equals the 1 that was here,"},{"Start":"01:41.400 ","End":"01:47.510","Text":"plus 4, plus 3 because when it deserts to the other side, it changes sign."},{"Start":"01:47.510 ","End":"01:50.575","Text":"We get 2x equals 8."},{"Start":"01:50.575 ","End":"01:53.660","Text":"Now again, we bring 2 over to the other side,"},{"Start":"01:53.660 ","End":"01:55.730","Text":"but this time what we have here is"},{"Start":"01:55.730 ","End":"01:58.820","Text":"the multiplication so on the other side it becomes a division."},{"Start":"01:58.820 ","End":"02:00.890","Text":"We\u0027re going to divide by 2."},{"Start":"02:00.890 ","End":"02:07.790","Text":"What we get is that x is equal to 8 over 2."},{"Start":"02:07.790 ","End":"02:11.060","Text":"In other words, x is equal to 4."},{"Start":"02:11.060 ","End":"02:12.950","Text":"That\u0027s our first example."},{"Start":"02:12.950 ","End":"02:14.844","Text":"Let\u0027s go for the second."},{"Start":"02:14.844 ","End":"02:16.850","Text":"Number 2,"},{"Start":"02:16.850 ","End":"02:17.390","Text":"see"},{"Start":"02:17.390 ","End":"02:18.960","Text":"4x"},{"Start":"02:26.020 ","End":"02:28.190","Text":"minus"},{"Start":"02:28.190 ","End":"02:28.880","Text":"5(2-3x)=-x."},{"Start":"02:28.880 ","End":"02:32.990","Text":"Once again, the name of this clip is expansion of"},{"Start":"02:32.990 ","End":"02:38.101","Text":"brackets so that\u0027s what we\u0027re going to do first, 4x there."},{"Start":"02:38.101 ","End":"02:41.090","Text":"Now, minus 5 times this and this."},{"Start":"02:41.090 ","End":"02:42.620","Text":"I have multiply by each one,"},{"Start":"02:42.620 ","End":"02:44.945","Text":"I\u0027m not going to keep putting these red arrows."},{"Start":"02:44.945 ","End":"02:47.930","Text":"Minus 5 times 2 is minus 10,"},{"Start":"02:47.930 ","End":"02:52.665","Text":"minus 5 times minus 3x is plus 15x."},{"Start":"02:52.665 ","End":"02:54.590","Text":"Watch out for the minus minus,"},{"Start":"02:54.590 ","End":"02:58.730","Text":"which is plus and equals minus x."},{"Start":"02:58.730 ","End":"03:02.600","Text":"Once again, the picture will remind me something goes to the left,"},{"Start":"03:02.600 ","End":"03:06.185","Text":"something to the right, the x\u0027s on the left numbers on the right."},{"Start":"03:06.185 ","End":"03:12.270","Text":"Let\u0027s see, x\u0027s; 4x plus 15x."},{"Start":"03:12.440 ","End":"03:15.415","Text":"Then what\u0027s going to change sign?"},{"Start":"03:15.415 ","End":"03:17.570","Text":"This has to go over to the left,"},{"Start":"03:17.570 ","End":"03:21.865","Text":"doesn\u0027t belong and this has to go over to the right it doesn\u0027t belong."},{"Start":"03:21.865 ","End":"03:25.670","Text":"This becomes over here plus x,"},{"Start":"03:25.670 ","End":"03:27.980","Text":"and then we have nothing left meanwhile,"},{"Start":"03:27.980 ","End":"03:30.920","Text":"but the 10 comes over as plus 10."},{"Start":"03:30.920 ","End":"03:32.600","Text":"Now let\u0027s do the addition."},{"Start":"03:32.600 ","End":"03:34.220","Text":"How many x\u0027s all together?"},{"Start":"03:34.220 ","End":"03:36.320","Text":"4 plus 15 plus,"},{"Start":"03:36.320 ","End":"03:44.075","Text":"if it doesn\u0027t say anything it\u0027s 1x so that\u0027s 4 plus 15 plus 1 is 20x equals 10."},{"Start":"03:44.075 ","End":"03:45.980","Text":"This of course, is a multiplication,"},{"Start":"03:45.980 ","End":"03:50.333","Text":"so when I bring it to the other side, it\u0027s a division."},{"Start":"03:50.333 ","End":"03:55.790","Text":"I get that x equals 10 divided by 20."},{"Start":"03:55.790 ","End":"03:59.435","Text":"In other words, x equals 1/2."},{"Start":"03:59.435 ","End":"04:02.120","Text":"That\u0027s that one. Next one,"},{"Start":"04:02.120 ","End":"04:03.925","Text":"that would be number 3,"},{"Start":"04:03.925 ","End":"04:07.980","Text":"5 times 5 minus x,"},{"Start":"04:07.980 ","End":"04:15.540","Text":"minus 5 minus 2x out of 5 there, equals 4x."},{"Start":"04:15.540 ","End":"04:17.895","Text":"Better scroll a bit,"},{"Start":"04:17.895 ","End":"04:20.760","Text":"get some extra room there."},{"Start":"04:20.760 ","End":"04:29.555","Text":"Again, expanding 5 times 5 is 25, then minus 5x."},{"Start":"04:29.555 ","End":"04:31.280","Text":"A minus is like a minus 1,"},{"Start":"04:31.280 ","End":"04:34.685","Text":"or you can just remember that a minus reverses all the signs."},{"Start":"04:34.685 ","End":"04:39.220","Text":"It\u0027s a minus 5 and a plus 2x."},{"Start":"04:39.220 ","End":"04:42.995","Text":"I\u0027m going to bring the 4x over at the same time."},{"Start":"04:42.995 ","End":"04:48.070","Text":"Minus 4x equals 0."},{"Start":"04:48.070 ","End":"04:51.405","Text":"I already brought over this."},{"Start":"04:51.405 ","End":"04:52.850","Text":"Everything\u0027s on the left,"},{"Start":"04:52.850 ","End":"04:55.670","Text":"but the numbers are going to have to go to the right."},{"Start":"04:55.670 ","End":"04:58.400","Text":"This will have to go to the other side and this will have to"},{"Start":"04:58.400 ","End":"05:01.190","Text":"go to the other side. What do we have?"},{"Start":"05:01.190 ","End":"05:04.880","Text":"We have minus 5x from here,"},{"Start":"05:04.880 ","End":"05:08.180","Text":"plus 2x minus 4x."},{"Start":"05:08.180 ","End":"05:10.415","Text":"That\u0027s the x terms."},{"Start":"05:10.415 ","End":"05:13.220","Text":"Then we get on the other side,"},{"Start":"05:13.220 ","End":"05:18.340","Text":"minus 25 plus 5."},{"Start":"05:18.340 ","End":"05:21.135","Text":"If we collect here, let\u0027s see."},{"Start":"05:21.135 ","End":"05:26.100","Text":"Minus 5, minus 4 is minus 9,"},{"Start":"05:26.100 ","End":"05:30.480","Text":"plus 2 is minus 7,"},{"Start":"05:30.480 ","End":"05:38.140","Text":"and on the other side we have minus 25 plus 5 is minus 20."},{"Start":"05:38.140 ","End":"05:40.510","Text":"We have a multiplication here."},{"Start":"05:40.510 ","End":"05:44.725","Text":"We want to bring the minus 7 to the other side in the denominator."},{"Start":"05:44.725 ","End":"05:50.830","Text":"Then we get that x is equal to minus 20 over minus"},{"Start":"05:50.830 ","End":"05:57.595","Text":"7 and that is equal to minuses will cancel."},{"Start":"05:57.595 ","End":"06:00.445","Text":"Minus over minus is plus 20 over 7,"},{"Start":"06:00.445 ","End":"06:02.440","Text":"an improper fraction, let\u0027s see,"},{"Start":"06:02.440 ","End":"06:07.690","Text":"7 goes into 20 twice and 6 leftover 2 and 6/7."},{"Start":"06:07.690 ","End":"06:10.540","Text":"I was supposed to get a whole number I must have missed copied somewhere,"},{"Start":"06:10.540 ","End":"06:13.420","Text":"but never mind this is an exercise in its own right."},{"Start":"06:13.420 ","End":"06:18.440","Text":"This would be the answer for the way I wrote the exercise,"},{"Start":"06:18.440 ","End":"06:23.525","Text":"x equals 2 and 6/7 underline it and that\u0027s the third one."},{"Start":"06:23.525 ","End":"06:26.610","Text":"We have one more. Let\u0027s see."},{"Start":"06:26.610 ","End":"06:37.565","Text":"This one will be 7x minus 4x minus 6 times 3 equals 13."},{"Start":"06:37.565 ","End":"06:41.060","Text":"There\u0027s a slight difference here in that we usually"},{"Start":"06:41.060 ","End":"06:44.540","Text":"put the number on the left of the brackets here it\u0027s on the right."},{"Start":"06:44.540 ","End":"06:48.560","Text":"So I\u0027m just going to imagine that this 3 is not here,"},{"Start":"06:48.560 ","End":"06:50.059","Text":"but it\u0027s here instead,"},{"Start":"06:50.059 ","End":"06:51.230","Text":"it doesn\u0027t make any difference."},{"Start":"06:51.230 ","End":"06:53.680","Text":"I\u0027m just used to having the number on the left."},{"Start":"06:53.680 ","End":"06:57.600","Text":"Expand brackets 7x minus 3 times"},{"Start":"06:57.600 ","End":"07:05.085","Text":"4x minus 12x minus 3 times minus 6, plus 18."},{"Start":"07:05.085 ","End":"07:07.635","Text":"Then this is equal to 13."},{"Start":"07:07.635 ","End":"07:10.215","Text":"Let\u0027s see which has to switch sides,"},{"Start":"07:10.215 ","End":"07:11.840","Text":"x\u0027s are good on the left,"},{"Start":"07:11.840 ","End":"07:17.960","Text":"just the 18 has to go over so 7x minus 12x I can already do that."},{"Start":"07:17.960 ","End":"07:20.160","Text":"That\u0027s minus 5x."},{"Start":"07:21.460 ","End":"07:26.540","Text":"Here I have 13 minus 18,"},{"Start":"07:26.540 ","End":"07:30.455","Text":"which is minus 5."},{"Start":"07:30.455 ","End":"07:34.555","Text":"Now I can divide by minus 5."},{"Start":"07:34.555 ","End":"07:36.750","Text":"Let me copy it again, minus 5x."},{"Start":"07:36.750 ","End":"07:43.160","Text":"This is a multiplication so this can go over to the other side in the denominator."},{"Start":"07:43.160 ","End":"07:48.875","Text":"We get that x is equal to minus 5 over minus 5."},{"Start":"07:48.875 ","End":"07:51.425","Text":"Anything over itself is 1,"},{"Start":"07:51.425 ","End":"07:53.900","Text":"x is equal to 1."},{"Start":"07:53.900 ","End":"07:56.300","Text":"We\u0027re basically done with this set."},{"Start":"07:56.300 ","End":"08:01.190","Text":"But I\u0027ll tell you sometimes you should really check your answer by substituting."},{"Start":"08:01.190 ","End":"08:04.040","Text":"Let\u0027s do one by substitution here,"},{"Start":"08:04.040 ","End":"08:06.305","Text":"just to see that it really comes out."},{"Start":"08:06.305 ","End":"08:08.240","Text":"If you want to skip this bit, you may,"},{"Start":"08:08.240 ","End":"08:10.655","Text":"but I recommend not skipping."},{"Start":"08:10.655 ","End":"08:17.140","Text":"What I\u0027m about to do is check that x equals 1 works, 7 times 1."},{"Start":"08:17.140 ","End":"08:19.010","Text":"I\u0027m ignoring the orange bit."},{"Start":"08:19.010 ","End":"08:20.510","Text":"In fact, not to confuse."},{"Start":"08:20.510 ","End":"08:22.989","Text":"Here it\u0027s multiplied, you don\u0027t see it very well,"},{"Start":"08:22.989 ","End":"08:26.100","Text":"the dot there, minus, let\u0027s see,"},{"Start":"08:26.100 ","End":"08:30.690","Text":"4 times 1 minus 6 times 3,"},{"Start":"08:30.690 ","End":"08:31.926","Text":"is it equal to,"},{"Start":"08:31.926 ","End":"08:34.320","Text":"have to check, to 13?"},{"Start":"08:34.320 ","End":"08:43.200","Text":"Well, let\u0027s see. This is 7 and 4 minus 6 is minus 2."},{"Start":"08:43.200 ","End":"08:45.520","Text":"Just this bit."},{"Start":"08:45.530 ","End":"08:48.090","Text":"We have a minus,"},{"Start":"08:48.090 ","End":"08:51.030","Text":"minus 2 times 3."},{"Start":"08:51.030 ","End":"08:55.695","Text":"So we end up with 7 plus 2 times 3,"},{"Start":"08:55.695 ","End":"08:58.100","Text":"which is 7 plus 6,"},{"Start":"08:58.100 ","End":"09:05.225","Text":"I\u0027m going in the opposite direction somehow, which is 13."},{"Start":"09:05.225 ","End":"09:07.699","Text":"That is correct."},{"Start":"09:07.699 ","End":"09:11.525","Text":"We\u0027ve even checked and you should do this from time to time."},{"Start":"09:11.525 ","End":"09:13.429","Text":"You could do it on every exercise,"},{"Start":"09:13.429 ","End":"09:17.270","Text":"substitute what you get in the original equation and see if it works."},{"Start":"09:17.270 ","End":"09:21.720","Text":"See if you get a true statement. Now I\u0027m done."}],"ID":4933},{"Watched":false,"Name":"Linear in One Unknown - Part 3","Duration":"5m 35s","ChapterTopicVideoID":4938,"CourseChapterTopicPlaylistID":3862,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.070","Text":"Is a continuation of the previous one,"},{"Start":"00:02.070 ","End":"00:03.450","Text":"that was already continued."},{"Start":"00:03.450 ","End":"00:06.405","Text":"I mean we\u0027re continuing again."},{"Start":"00:06.405 ","End":"00:10.560","Text":"I remember we did exercises 1-4 in the previous clip,"},{"Start":"00:10.560 ","End":"00:13.110","Text":"which means that we\u0027re going to continue with number 5."},{"Start":"00:13.110 ","End":"00:17.565","Text":"Number 5 says that x times x minus"},{"Start":"00:17.565 ","End":"00:24.315","Text":"5=x^2 minus 7x plus 8."},{"Start":"00:24.315 ","End":"00:26.580","Text":"Once again, we\u0027re going be expanding"},{"Start":"00:26.580 ","End":"00:30.075","Text":"brackets which is on the left where we have brackets."},{"Start":"00:30.075 ","End":"00:34.155","Text":"X times x is x^2."},{"Start":"00:34.155 ","End":"00:41.205","Text":"x times minus 5 is minus 5x and this just copied just as it is."},{"Start":"00:41.205 ","End":"00:46.085","Text":"Now, here something funny because I remember telling you that"},{"Start":"00:46.085 ","End":"00:51.395","Text":"linear equations should not have x^2 in it.. Well,"},{"Start":"00:51.395 ","End":"00:55.175","Text":"you\u0027re right, and in a way this does have x^2 in it,"},{"Start":"00:55.175 ","End":"00:56.660","Text":"but if you look,"},{"Start":"00:56.660 ","End":"01:00.020","Text":"I can subtract x^2 from both sides."},{"Start":"01:00.020 ","End":"01:04.505","Text":"That I\u0027m subtracting x^2 from left and from right."},{"Start":"01:04.505 ","End":"01:05.915","Text":"If I do that,"},{"Start":"01:05.915 ","End":"01:08.885","Text":"this will just disappear and this will just disappear."},{"Start":"01:08.885 ","End":"01:11.720","Text":"It\u0027s quite common to see a thing like this and just put"},{"Start":"01:11.720 ","End":"01:15.410","Text":"a line through and a line through to say this cancels with this."},{"Start":"01:15.410 ","End":"01:20.090","Text":"What we get is just minus 5x equals"},{"Start":"01:20.090 ","End":"01:26.570","Text":"minus 7x plus 8 and this is a linear equation in 1 variable."},{"Start":"01:26.570 ","End":"01:29.120","Text":"There\u0027s no x^2 in it or anything."},{"Start":"01:29.120 ","End":"01:33.410","Text":"All I have to do now is collect like terms,"},{"Start":"01:33.410 ","End":"01:37.220","Text":"and I\u0027ll move this to the other side and it becomes a plus,"},{"Start":"01:37.220 ","End":"01:42.300","Text":"so minus 5x plus 7x is equal to 8."},{"Start":"01:42.300 ","End":"01:45.570","Text":"Collect the x\u0027s minus 5 plus 7 is 2,"},{"Start":"01:45.570 ","End":"01:47.520","Text":"2x is 8,"},{"Start":"01:47.520 ","End":"01:49.950","Text":"and dividing by 2,"},{"Start":"01:49.950 ","End":"01:54.075","Text":"x is 8/2,"},{"Start":"01:54.075 ","End":"01:57.340","Text":"so x is 4."},{"Start":"01:57.340 ","End":"02:00.980","Text":"What we learned here is that sometimes it might"},{"Start":"02:00.980 ","End":"02:04.705","Text":"appear that it\u0027s not a linear equation and we have an x^2 in it,"},{"Start":"02:04.705 ","End":"02:07.020","Text":"but an actual fact that canceled."},{"Start":"02:07.020 ","End":"02:11.150","Text":"Equations that do have x^2 in them are quadratic equations,"},{"Start":"02:11.150 ","End":"02:13.565","Text":"which we will learn but not now."},{"Start":"02:13.565 ","End":"02:16.750","Text":"I\u0027ll just erase this and then go on to the next."},{"Start":"02:16.750 ","End":"02:18.995","Text":"It\u0027ll be number 6."},{"Start":"02:18.995 ","End":"02:24.165","Text":"7 minus x times 1 minus"},{"Start":"02:24.165 ","End":"02:29.700","Text":"x minus x minus 3^2,"},{"Start":"02:29.700 ","End":"02:32.075","Text":"and this is equal to 0."},{"Start":"02:32.075 ","End":"02:34.330","Text":"We need to expand brackets here,"},{"Start":"02:34.330 ","End":"02:36.800","Text":"so remember that each one goes with each one,"},{"Start":"02:36.800 ","End":"02:38.330","Text":"this with this, this with this,"},{"Start":"02:38.330 ","End":"02:40.205","Text":"this with this, and this with this."},{"Start":"02:40.205 ","End":"02:43.010","Text":"Also, I want to remind you that we did have"},{"Start":"02:43.010 ","End":"02:47.210","Text":"this special binomial products or expansions,"},{"Start":"02:47.210 ","End":"02:53.295","Text":"that we have the formula that a minus b^2 is"},{"Start":"02:53.295 ","End":"02:59.900","Text":"a^2 minus 2ab plus b^2 and I\u0027m going to apply this formula to the second bit."},{"Start":"02:59.900 ","End":"03:01.405","Text":"But let\u0027s start with this one,"},{"Start":"03:01.405 ","End":"03:03.855","Text":"7 times 1 is 7."},{"Start":"03:03.855 ","End":"03:07.305","Text":"7 times minus x minus 7x."},{"Start":"03:07.305 ","End":"03:10.710","Text":"You know me and my 7\u0027s are the European style,"},{"Start":"03:10.710 ","End":"03:13.620","Text":"so that looks odd, that\u0027s a 7."},{"Start":"03:13.620 ","End":"03:17.985","Text":"Then minus x times 1 is minus x."},{"Start":"03:17.985 ","End":"03:22.695","Text":"Minus x times minus x is plus x^2."},{"Start":"03:22.695 ","End":"03:25.110","Text":"Once again, we note an x^2 here."},{"Start":"03:25.110 ","End":"03:27.500","Text":"It better cancel, otherwise we can\u0027t do it."},{"Start":"03:27.500 ","End":"03:30.043","Text":"Let\u0027s continue. Minus,"},{"Start":"03:30.043 ","End":"03:32.870","Text":"now the x minus 3^2 I\u0027ll use with this formula,"},{"Start":"03:32.870 ","End":"03:35.650","Text":"but I\u0027ll need to put everything in the brackets."},{"Start":"03:35.650 ","End":"03:38.040","Text":"X is in the role of a,"},{"Start":"03:38.040 ","End":"03:40.080","Text":"3 in the role of b,"},{"Start":"03:40.080 ","End":"03:48.570","Text":"a^2 means x^2 minus 2ab is minus 2 times x times 3,"},{"Start":"03:48.570 ","End":"03:52.650","Text":"and then plus b^2 is plus 3^2."},{"Start":"03:52.650 ","End":"03:56.065","Text":"All this, close brackets,"},{"Start":"03:56.065 ","End":"03:58.370","Text":"is equal to 0."},{"Start":"03:58.370 ","End":"04:00.995","Text":"Now I have to open this bracket,"},{"Start":"04:00.995 ","End":"04:04.030","Text":"there\u0027s a minus in front or minus 1."},{"Start":"04:04.030 ","End":"04:06.030","Text":"This, we just copy the first bit,"},{"Start":"04:06.030 ","End":"04:11.700","Text":"7 minus 7x minus x plus x^2 and minus 1 times,"},{"Start":"04:11.700 ","End":"04:16.010","Text":"or you could just simply say a minus in front of a bracket reverses all the signs."},{"Start":"04:16.010 ","End":"04:22.310","Text":"Either way, it\u0027s minus x^2 plus 2 times x times 3 is 6x,"},{"Start":"04:22.310 ","End":"04:26.325","Text":"and then minus 3^2 is 9."},{"Start":"04:26.325 ","End":"04:29.750","Text":"This is equal to 0. Everything\u0027s on the left."},{"Start":"04:29.750 ","End":"04:32.725","Text":"I want to put the numbers on the right."},{"Start":"04:32.725 ","End":"04:34.680","Text":"I\u0027ll just copy the x\u0027s,"},{"Start":"04:34.680 ","End":"04:36.410","Text":"here I have an x."},{"Start":"04:36.410 ","End":"04:38.480","Text":"Make sure not to miss anything."},{"Start":"04:38.480 ","End":"04:40.885","Text":"This is a minus x,"},{"Start":"04:40.885 ","End":"04:42.795","Text":"and I forgot to say of course,"},{"Start":"04:42.795 ","End":"04:48.990","Text":"that plus x^2 and minus x^2 cancel each other because that\u0027s 0."},{"Start":"04:48.990 ","End":"04:54.140","Text":"Back to this, we have the minus x and then plus 6x,"},{"Start":"04:54.140 ","End":"04:58.610","Text":"that\u0027s all the x\u0027s on the left and our numbers on the right, what do I have?"},{"Start":"04:58.610 ","End":"05:00.830","Text":"I have a 7, so it\u0027s minus 7."},{"Start":"05:00.830 ","End":"05:05.879","Text":"We have here a plus 9 because minus 9 goes over as plus."},{"Start":"05:05.879 ","End":"05:10.005","Text":"Let\u0027s see how many x\u0027s minus 7 minus 1 plus 6,"},{"Start":"05:10.005 ","End":"05:18.185","Text":"I make it minus 2x and minus 7 plus 9 is positive 2."},{"Start":"05:18.185 ","End":"05:22.840","Text":"Now all I have to do is divide both sides by this minus 2,"},{"Start":"05:22.840 ","End":"05:26.955","Text":"and I\u0027ve got x=2 over minus 2."},{"Start":"05:26.955 ","End":"05:31.050","Text":"Plus over minus is minus and 2 over 2 is 1,"},{"Start":"05:31.050 ","End":"05:36.310","Text":"so x is equal to minus 1 and we\u0027re done."}],"ID":4934},{"Watched":false,"Name":"Exercise 1","Duration":"6m 28s","ChapterTopicVideoID":4932,"CourseChapterTopicPlaylistID":3862,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"In this exercise, we have to solve the following equations,"},{"Start":"00:04.260 ","End":"00:08.475","Text":"a, b, c and d. Let\u0027s take them one at a time."},{"Start":"00:08.475 ","End":"00:11.835","Text":"First one, 6x plus 2 equals 8."},{"Start":"00:11.835 ","End":"00:16.335","Text":"Now I\u0027m going to do the long method and then we\u0027ll do the shortcut method."},{"Start":"00:16.335 ","End":"00:18.195","Text":"If 2 things are equal,"},{"Start":"00:18.195 ","End":"00:20.910","Text":"then I can do the same thing to both sides."},{"Start":"00:20.910 ","End":"00:25.650","Text":"I\u0027m going to subtract 2 from both sides and so I get"},{"Start":"00:25.650 ","End":"00:31.500","Text":"6x plus 2 minus 2 equals 8 minus 2."},{"Start":"00:31.500 ","End":"00:35.370","Text":"You might ask, why did I know to subtract 2 from both sides?"},{"Start":"00:35.370 ","End":"00:38.760","Text":"Well, I wanted to leave just x\u0027s here and just numbers"},{"Start":"00:38.760 ","End":"00:42.810","Text":"here and I saw that this 2 is getting in the way so I subtracted 2."},{"Start":"00:42.810 ","End":"00:47.340","Text":"After I subtract 2 the 2 minus 2 is nothing so I just"},{"Start":"00:47.340 ","End":"00:53.410","Text":"get that 6x is equal to 8 minus 2 is 6."},{"Start":"00:53.410 ","End":"00:56.300","Text":"Now, here\u0027s an equation,"},{"Start":"00:56.300 ","End":"01:01.925","Text":"so I can do the same thing to both sides and I can divide both sides by 6."},{"Start":"01:01.925 ","End":"01:08.400","Text":"If I divide both sides by 6 and I\u0027ll just slightly skip a step,"},{"Start":"01:08.400 ","End":"01:10.635","Text":"but I won\u0027t say 6x over 6."},{"Start":"01:10.635 ","End":"01:14.070","Text":"You can already see that 6x over 6 is going to be"},{"Start":"01:14.070 ","End":"01:18.820","Text":"just x and 6 divided by 6 we can do in our heads is equal to 1."},{"Start":"01:18.820 ","End":"01:26.165","Text":"But if we were doing it long handed we\u0027d write 6x over 6 equals 6 over 6 and so on."},{"Start":"01:26.165 ","End":"01:32.420","Text":"Now, the quicker way of doing things is there are some rules of thumb that you can use."},{"Start":"01:32.420 ","End":"01:35.300","Text":"For example, I start from here again,"},{"Start":"01:35.300 ","End":"01:40.895","Text":"6x plus 2 equals 8 and normally what we would do is say, okay,"},{"Start":"01:40.895 ","End":"01:42.890","Text":"the opposite of addition is subtraction,"},{"Start":"01:42.890 ","End":"01:48.005","Text":"I can move the 2 over to the other side if I do the opposite, which is subtract,"},{"Start":"01:48.005 ","End":"01:53.990","Text":"so I would say straightaway that 6x equals 8 minus 2 and I could even write here,"},{"Start":"01:53.990 ","End":"01:55.565","Text":"say that equals 6."},{"Start":"01:55.565 ","End":"01:59.155","Text":"Maybe I\u0027ll write it on another line, 6x equals 6."},{"Start":"01:59.155 ","End":"02:02.510","Text":"Now here when I say 6x it\u0027s really a multiplication."},{"Start":"02:02.510 ","End":"02:05.030","Text":"It\u0027s 6 times x, so again,"},{"Start":"02:05.030 ","End":"02:09.770","Text":"we can use the trick of moving something to the other side and doing the opposite."},{"Start":"02:09.770 ","End":"02:12.170","Text":"Instead of multiplying on the other side,"},{"Start":"02:12.170 ","End":"02:13.580","Text":"it will be dividing."},{"Start":"02:13.580 ","End":"02:18.200","Text":"X will equal 6 divided by 6,"},{"Start":"02:18.200 ","End":"02:20.970","Text":"so I can write it 6 colon 6."},{"Start":"02:20.970 ","End":"02:24.180","Text":"I\u0027ll write the division this way and 6 over 6 is 1"},{"Start":"02:24.180 ","End":"02:29.375","Text":"so x is equal to 1 because it comes out the same answer both ways."},{"Start":"02:29.375 ","End":"02:33.910","Text":"Next one,7 minus 2x equals 7."},{"Start":"02:33.910 ","End":"02:36.870","Text":"What I\u0027d like is x\u0027s on the left and numbers on the right,"},{"Start":"02:36.870 ","End":"02:40.514","Text":"that\u0027s my idea so I was going to subtract 7 from both sides."},{"Start":"02:40.514 ","End":"02:46.650","Text":"I say at the side that I\u0027m going to subtract 7 and so now I get 7"},{"Start":"02:46.650 ","End":"02:54.660","Text":"minus 2x minus 7 equals 7 minus 7 and notice this is the style of 7."},{"Start":"02:54.660 ","End":"02:58.995","Text":"I use its European no confusion."},{"Start":"02:58.995 ","End":"03:07.380","Text":"7 minus 7 is nothing so I have just minus 2x and 7 minus 7 is 0."},{"Start":"03:07.380 ","End":"03:11.360","Text":"I\u0027m going to proceed mechanically even though there\u0027s a shortcut here,"},{"Start":"03:11.360 ","End":"03:12.605","Text":"but I\u0027m just going to say,"},{"Start":"03:12.605 ","End":"03:15.350","Text":"okay, I want to have just x on the left."},{"Start":"03:15.350 ","End":"03:20.345","Text":"I\u0027m going to divide both sides by minus 2,"},{"Start":"03:20.345 ","End":"03:29.080","Text":"so I get minus 2x over minus 2 is just x and 0 divided by minus 2,"},{"Start":"03:29.080 ","End":"03:33.500","Text":"remember 0 divided by anything is 0 except 0 by 0,"},{"Start":"03:33.500 ","End":"03:34.925","Text":"which is not here."},{"Start":"03:34.925 ","End":"03:39.110","Text":"The quicker way of doing it is to say as follows."},{"Start":"03:39.110 ","End":"03:43.490","Text":"I\u0027ll start with 7 minus 2x equals 7 just the same and then what I"},{"Start":"03:43.490 ","End":"03:48.030","Text":"say is I bring this over to the other side and because it\u0027s a positive 7,"},{"Start":"03:48.030 ","End":"03:50.385","Text":"it becomes a negative 7 over here."},{"Start":"03:50.385 ","End":"03:55.910","Text":"Minus 2x equals 7 minus 7 which equals 0 and I would"},{"Start":"03:55.910 ","End":"04:02.115","Text":"usually do this all in 1 step and say minus 2x is 0."},{"Start":"04:02.115 ","End":"04:05.100","Text":"Then I would say bring the minus 2 over to"},{"Start":"04:05.100 ","End":"04:09.155","Text":"the other side because it\u0027s a multiplication on the other side it will be a division."},{"Start":"04:09.155 ","End":"04:16.855","Text":"So x will equal 0 divided by minus 2 and then that is equal to 0."},{"Start":"04:16.855 ","End":"04:23.390","Text":"Continuing with c and d. I think you get the idea,"},{"Start":"04:23.390 ","End":"04:26.345","Text":"you know what, I\u0027m going to move to the quick method already."},{"Start":"04:26.345 ","End":"04:30.605","Text":"2x plus x equals 24."},{"Start":"04:30.605 ","End":"04:34.490","Text":"First of all, I have to collect like terms this is something we"},{"Start":"04:34.490 ","End":"04:38.965","Text":"didn\u0027t have in these 2 and so collecting like terms,"},{"Start":"04:38.965 ","End":"04:42.825","Text":"the 2x and the x like 2x plus 1x,"},{"Start":"04:42.825 ","End":"04:46.440","Text":"2 apples plus 1 apple is 3 apples."},{"Start":"04:46.440 ","End":"04:51.225","Text":"2x plus 1x is 3x is 24."},{"Start":"04:51.225 ","End":"04:54.095","Text":"Then I say bring the 3 over to the other side,"},{"Start":"04:54.095 ","End":"04:56.914","Text":"because here it\u0027s multiplication on the other side,"},{"Start":"04:56.914 ","End":"05:00.975","Text":"it will be division, divided by 3."},{"Start":"05:00.975 ","End":"05:06.965","Text":"Then I would say either on the same line or on the next line, x=8."},{"Start":"05:06.965 ","End":"05:12.605","Text":"I should really be coloring the solutions so let\u0027s do that."},{"Start":"05:12.605 ","End":"05:14.990","Text":"So x is equal to 1."},{"Start":"05:14.990 ","End":"05:20.210","Text":"Same thing here x equals 0 confirmed."},{"Start":"05:20.210 ","End":"05:23.600","Text":"Here just did it one way, x equals 8."},{"Start":"05:23.600 ","End":"05:27.769","Text":"Now let\u0027s go and do number d. Gradually"},{"Start":"05:27.769 ","End":"05:31.580","Text":"things get a little bit more complicated as you get used to them."},{"Start":"05:31.580 ","End":"05:35.915","Text":"In this case, we have x\u0027s and numbers on both sides,"},{"Start":"05:35.915 ","End":"05:41.295","Text":"so what we do is we can do 2 things at once."},{"Start":"05:41.295 ","End":"05:46.910","Text":"We can throw the 6 over to this side and at the same time,"},{"Start":"05:46.910 ","End":"05:50.030","Text":"I can throw the x over to this side."},{"Start":"05:50.030 ","End":"05:52.895","Text":"I prefer the x\u0027s on the left and the numbers on the right,"},{"Start":"05:52.895 ","End":"05:55.205","Text":"but it could be the other way."},{"Start":"05:55.205 ","End":"05:59.330","Text":"What I get is because this was an addition,"},{"Start":"05:59.330 ","End":"06:01.250","Text":"when it goes over to the left,"},{"Start":"06:01.250 ","End":"06:03.110","Text":"it becomes a subtraction."},{"Start":"06:03.110 ","End":"06:06.580","Text":"The same thing on the right here I had the 8,"},{"Start":"06:06.580 ","End":"06:10.570","Text":"but the 6 when I throw it over becomes minus 6."},{"Start":"06:10.570 ","End":"06:13.815","Text":"Now I collect like terms 2x minus x,"},{"Start":"06:13.815 ","End":"06:20.595","Text":"2 apples minus 1 apple is just 1 apple and 8 minus 6 is equal to"},{"Start":"06:20.595 ","End":"06:28.720","Text":"2 and so that is the answer and that\u0027s done."}],"ID":4935},{"Watched":false,"Name":"Exercise 2","Duration":"3m 56s","ChapterTopicVideoID":4933,"CourseChapterTopicPlaylistID":3862,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.090","Text":"In this exercise, we have to solve the equations."},{"Start":"00:03.090 ","End":"00:06.180","Text":"There\u0027s 3 of them. I have copied them over here."},{"Start":"00:06.180 ","End":"00:08.445","Text":"Start with a,"},{"Start":"00:08.445 ","End":"00:10.260","Text":"and just to make things easier,"},{"Start":"00:10.260 ","End":"00:12.675","Text":"let me mark like terms."},{"Start":"00:12.675 ","End":"00:17.310","Text":"I\u0027ll let the terms with x be the apples in green,"},{"Start":"00:17.310 ","End":"00:22.725","Text":"and I\u0027ll let the numbers be oranges,"},{"Start":"00:22.725 ","End":"00:25.215","Text":"and that will be in orange of course."},{"Start":"00:25.215 ","End":"00:27.953","Text":"Now, what we want is to keep the apples on the left,"},{"Start":"00:27.953 ","End":"00:30.285","Text":"and the oranges on the right."},{"Start":"00:30.285 ","End":"00:34.013","Text":"We already have minus 7x,"},{"Start":"00:34.013 ","End":"00:36.555","Text":"and the plus 2x on the left."},{"Start":"00:36.555 ","End":"00:38.780","Text":"But I want the 4x also on the left."},{"Start":"00:38.780 ","End":"00:40.775","Text":"Since it switches sides,"},{"Start":"00:40.775 ","End":"00:45.175","Text":"it changes to a negative 4x minus 4x."},{"Start":"00:45.175 ","End":"00:50.210","Text":"Similarly, I have on the right a minus 13,"},{"Start":"00:50.210 ","End":"00:52.550","Text":"but I also want to throw the 5 over to the right,"},{"Start":"00:52.550 ","End":"00:53.570","Text":"so instead of plus 5,"},{"Start":"00:53.570 ","End":"00:55.550","Text":"it becomes minus 5."},{"Start":"00:55.550 ","End":"00:57.275","Text":"Now we do the arithmetic."},{"Start":"00:57.275 ","End":"00:59.325","Text":"We have x\u0027s only."},{"Start":"00:59.325 ","End":"01:00.770","Text":"How many x\u0027s do we have?"},{"Start":"01:00.770 ","End":"01:02.690","Text":"We have minus 7 plus 2,"},{"Start":"01:02.690 ","End":"01:04.100","Text":"minus 4 of them."},{"Start":"01:04.100 ","End":"01:05.630","Text":"Minus 7, plus 4,"},{"Start":"01:05.630 ","End":"01:06.920","Text":"minus 2, we can collect."},{"Start":"01:06.920 ","End":"01:08.555","Text":"We\u0027ll do the negatives first."},{"Start":"01:08.555 ","End":"01:11.570","Text":"It\u0027s minus 11 and the positive is 2."},{"Start":"01:11.570 ","End":"01:14.135","Text":"The difference between 11 and 2 is 9,"},{"Start":"01:14.135 ","End":"01:17.480","Text":"but it\u0027s minus 9 because the negatives win."},{"Start":"01:17.480 ","End":"01:22.560","Text":"Minus 9x is equal to,"},{"Start":"01:22.560 ","End":"01:24.060","Text":"now here we have 2 negatives,"},{"Start":"01:24.060 ","End":"01:26.479","Text":"so we add 13 and 5 is 18,"},{"Start":"01:26.479 ","End":"01:29.435","Text":"but because it\u0027s negative, it\u0027s minus 18."},{"Start":"01:29.435 ","End":"01:34.690","Text":"Now, divide both sides by minus 9."},{"Start":"01:34.690 ","End":"01:38.585","Text":"If I divide this by minus 9, I\u0027m just left with x."},{"Start":"01:38.585 ","End":"01:42.560","Text":"Here, I could do it at the side, or we can do it in our heads."},{"Start":"01:42.560 ","End":"01:44.840","Text":"Minus 18 over minus 9,"},{"Start":"01:44.840 ","End":"01:46.775","Text":"minus over minus is plus,"},{"Start":"01:46.775 ","End":"01:49.880","Text":"18 over 9 is 2."},{"Start":"01:49.880 ","End":"01:52.430","Text":"This is the answer."},{"Start":"01:52.430 ","End":"01:58.340","Text":"Next one. Better scroll up a bit. There we are."},{"Start":"01:58.340 ","End":"02:00.950","Text":"I\u0027m not going to use colors this time."},{"Start":"02:00.950 ","End":"02:04.265","Text":"We just take the x\u0027s to the left and numbers to the right."},{"Start":"02:04.265 ","End":"02:09.190","Text":"What already was on the left is 6x minus 7x, this and this."},{"Start":"02:09.190 ","End":"02:11.925","Text":"I got to throw these two over to the left."},{"Start":"02:11.925 ","End":"02:15.870","Text":"This becomes minus x and plus 5,"},{"Start":"02:15.870 ","End":"02:17.340","Text":"we switch sign."},{"Start":"02:17.340 ","End":"02:18.720","Text":"Now, numbers."},{"Start":"02:18.720 ","End":"02:20.790","Text":"We already had minus 7 here."},{"Start":"02:20.790 ","End":"02:23.130","Text":"You want to throw the 3 and the 5 over,"},{"Start":"02:23.130 ","End":"02:26.755","Text":"so it\u0027s plus 3 minus 5."},{"Start":"02:26.755 ","End":"02:29.060","Text":"Now, we just collect like terms."},{"Start":"02:29.060 ","End":"02:30.830","Text":"Here are all the x\u0027s,"},{"Start":"02:30.830 ","End":"02:33.860","Text":"so what we have to do is 6 minus 7,"},{"Start":"02:33.860 ","End":"02:35.615","Text":"minus 1, plus 5."},{"Start":"02:35.615 ","End":"02:38.480","Text":"I\u0027ll use the technique of taking the positives first,"},{"Start":"02:38.480 ","End":"02:40.085","Text":"6 and 5 is 11,"},{"Start":"02:40.085 ","End":"02:42.750","Text":"less 7 and 1 is 8,"},{"Start":"02:42.750 ","End":"02:44.970","Text":"11 minus 8 is 3."},{"Start":"02:44.970 ","End":"02:46.815","Text":"We\u0027re left with 3x."},{"Start":"02:46.815 ","End":"02:52.040","Text":"Here, we have negative 12 and plus 3,"},{"Start":"02:52.040 ","End":"02:56.640","Text":"so all together negative minus 9."},{"Start":"02:56.640 ","End":"02:59.840","Text":"Then we divide by 3 both sides."},{"Start":"02:59.840 ","End":"03:06.415","Text":"We\u0027re left with x equals minus 9 over 3, is minus 3."},{"Start":"03:06.415 ","End":"03:10.005","Text":"Next, x\u0027s on the left,"},{"Start":"03:10.005 ","End":"03:14.475","Text":"that\u0027s minus 5x, and here, plus 3x."},{"Start":"03:14.475 ","End":"03:16.050","Text":"Numbers on the right,"},{"Start":"03:16.050 ","End":"03:17.595","Text":"we already had 8."},{"Start":"03:17.595 ","End":"03:19.020","Text":"Let me throw over this."},{"Start":"03:19.020 ","End":"03:21.855","Text":"Minus 2, minus 7."},{"Start":"03:21.855 ","End":"03:24.885","Text":"Collect like terms. How many x\u0027s?"},{"Start":"03:24.885 ","End":"03:27.030","Text":"Minus 5 plus 3 of them."},{"Start":"03:27.030 ","End":"03:29.025","Text":"That\u0027s minus 2 of them."},{"Start":"03:29.025 ","End":"03:31.270","Text":"Numbers, 8 minus 2,"},{"Start":"03:31.270 ","End":"03:34.080","Text":"minus 7 is 6."},{"Start":"03:34.080 ","End":"03:36.985","Text":"6 minus 7 is minus 1."},{"Start":"03:36.985 ","End":"03:40.355","Text":"Divide both sides by minus 2,"},{"Start":"03:40.355 ","End":"03:46.245","Text":"and we\u0027ve got x equals minus 1 over minus 2 is plus,"},{"Start":"03:46.245 ","End":"03:47.900","Text":"but it\u0027s 1/2,"},{"Start":"03:47.900 ","End":"03:49.115","Text":"which is a half."},{"Start":"03:49.115 ","End":"03:51.878","Text":"X is equal to a half,"},{"Start":"03:51.878 ","End":"03:56.710","Text":"and I\u0027ll highlight it and we are done."}],"ID":4936},{"Watched":false,"Name":"Exercise 3","Duration":"5m ","ChapterTopicVideoID":4934,"CourseChapterTopicPlaylistID":3862,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"We have to solve these equations here."},{"Start":"00:02.460 ","End":"00:05.880","Text":"There\u0027s 4 of them and we\u0027ll scroll down."},{"Start":"00:05.880 ","End":"00:08.205","Text":"That should do it for now."},{"Start":"00:08.205 ","End":"00:10.590","Text":"Let\u0027s start with the first 1."},{"Start":"00:10.590 ","End":"00:12.795","Text":"Let\u0027s open the brackets first,"},{"Start":"00:12.795 ","End":"00:16.320","Text":"3 times x and then 3 times minus 1."},{"Start":"00:16.320 ","End":"00:19.605","Text":"It\u0027s 3x minus 3,"},{"Start":"00:19.605 ","End":"00:22.314","Text":"minus 4 equals 2,"},{"Start":"00:22.314 ","End":"00:25.710","Text":"x\u0027s on the left numbers on the right."},{"Start":"00:25.710 ","End":"00:29.115","Text":"3x is all alone with the x\u0027s."},{"Start":"00:29.115 ","End":"00:31.020","Text":"Everything else goes to the right,"},{"Start":"00:31.020 ","End":"00:35.805","Text":"so it\u0027s 2 plus 3 plus 4."},{"Start":"00:35.805 ","End":"00:38.149","Text":"I\u0027ll continue on the same line,"},{"Start":"00:38.149 ","End":"00:39.840","Text":"it\u0027s equal to 9."},{"Start":"00:39.840 ","End":"00:42.255","Text":"Then we divide by 3."},{"Start":"00:42.255 ","End":"00:46.995","Text":"Then we get x equals 9 divided by 3 is 3."},{"Start":"00:46.995 ","End":"00:50.225","Text":"That does a. Now onto b."},{"Start":"00:50.225 ","End":"00:52.421","Text":"Once again, we have to open brackets,"},{"Start":"00:52.421 ","End":"00:55.595","Text":"so 7x is here and I can either"},{"Start":"00:55.595 ","End":"00:59.450","Text":"put a minus and then the brackets and just multiply it by 4,"},{"Start":"00:59.450 ","End":"01:05.030","Text":"or I can just straightaway multiply everything by negative 4 by minus 4,"},{"Start":"01:05.030 ","End":"01:07.455","Text":"which is probably shorter."},{"Start":"01:07.455 ","End":"01:12.165","Text":"Minus 4 times 3, is minus 12."},{"Start":"01:12.165 ","End":"01:15.570","Text":"Minus 4 times minus 4x,"},{"Start":"01:15.570 ","End":"01:17.115","Text":"4 times 4 is 16,"},{"Start":"01:17.115 ","End":"01:24.320","Text":"it\u0027s plus 16x because minus times minus is plus is equal to minus x."},{"Start":"01:24.320 ","End":"01:27.290","Text":"Now, we do the collecting like terms."},{"Start":"01:27.290 ","End":"01:28.940","Text":"x\u0027s on the left,"},{"Start":"01:28.940 ","End":"01:30.740","Text":"numbers on the right."},{"Start":"01:30.740 ","End":"01:35.930","Text":"I have 7x plus 16 x and then plus"},{"Start":"01:35.930 ","End":"01:42.200","Text":"x and the minus 12 on the right becomes plus 12."},{"Start":"01:42.200 ","End":"01:45.950","Text":"Continue over here. How many x\u0027s do we have?"},{"Start":"01:45.950 ","End":"01:55.940","Text":"7 plus 16 plus 1 is 24. x equals 12 then dividing by 24,"},{"Start":"01:55.940 ","End":"02:00.620","Text":"I just get x is 12 over 24."},{"Start":"02:00.620 ","End":"02:05.510","Text":"But we can cancel because 12 goes into 24 twice,"},{"Start":"02:05.510 ","End":"02:10.955","Text":"you end up with x equals 1.5. There we are."},{"Start":"02:10.955 ","End":"02:16.265","Text":"The next, distributive law, expand brackets."},{"Start":"02:16.265 ","End":"02:19.205","Text":"6 times 4 is 24."},{"Start":"02:19.205 ","End":"02:22.990","Text":"6 times minus x minus 6x."},{"Start":"02:22.990 ","End":"02:25.335","Text":"Minus is like minus 1."},{"Start":"02:25.335 ","End":"02:28.170","Text":"Minus 1 times 6 is minus 6."},{"Start":"02:28.170 ","End":"02:31.500","Text":"Minus 1 times minus x is plus x."},{"Start":"02:31.500 ","End":"02:35.850","Text":"2 minuses multiplied give a plus equals 3x."},{"Start":"02:35.850 ","End":"02:37.620","Text":"Collecting like terms,"},{"Start":"02:37.620 ","End":"02:40.250","Text":"x\u0027s on the left, numbers on the right."},{"Start":"02:40.250 ","End":"02:46.993","Text":"Minus 6x plus x as this and this and this 1 with a reverse sign."},{"Start":"02:46.993 ","End":"02:48.810","Text":"Then numbers on the right,"},{"Start":"02:48.810 ","End":"02:52.960","Text":"so it\u0027s minus 24 plus 6."},{"Start":"02:52.960 ","End":"02:54.935","Text":"How many x\u0027s do we have?"},{"Start":"02:54.935 ","End":"03:03.015","Text":"Minus 6 plus 1 minus 3 is minus 8x."},{"Start":"03:03.015 ","End":"03:10.200","Text":"Minus 24 plus 6 is minus 18."},{"Start":"03:10.200 ","End":"03:12.345","Text":"Continuing here."},{"Start":"03:12.345 ","End":"03:16.755","Text":"If I divide by minus 8,"},{"Start":"03:16.755 ","End":"03:24.475","Text":"I get x equals minus 18 over minus 8,"},{"Start":"03:24.475 ","End":"03:29.950","Text":"so it\u0027s plus, but also I can divide by 2."},{"Start":"03:29.950 ","End":"03:38.638","Text":"It\u0027s 9 over 2 because I\u0027m canceling and I change the improper fraction to a mixed number."},{"Start":"03:38.638 ","End":"03:42.770","Text":"2 into 9 gives us 4 and 1 leftover, so it\u0027s 4.5."},{"Start":"03:42.770 ","End":"03:45.090","Text":"That\u0027s the answer,"},{"Start":"03:45.090 ","End":"03:49.910","Text":"x equals 4.5 although we could also write it as 9 over 2,"},{"Start":"03:49.910 ","End":"03:54.770","Text":"but it\u0027s considered better to write it as a mixed number."},{"Start":"03:54.770 ","End":"03:56.540","Text":"Now the last 1,"},{"Start":"03:56.540 ","End":"03:57.980","Text":"let\u0027s scroll a bit."},{"Start":"03:57.980 ","End":"04:02.585","Text":"Some room here. 5x, expand the brackets."},{"Start":"04:02.585 ","End":"04:07.580","Text":"I\u0027m going to imagine that the 4 is in front of the brackets."},{"Start":"04:07.580 ","End":"04:08.960","Text":"It doesn\u0027t matter that it\u0027s the right."},{"Start":"04:08.960 ","End":"04:10.310","Text":"The distributive law still works."},{"Start":"04:10.310 ","End":"04:12.170","Text":"You can just say it works this way,"},{"Start":"04:12.170 ","End":"04:14.375","Text":"this times this and this times this."},{"Start":"04:14.375 ","End":"04:20.105","Text":"It\u0027s minus 3x times 4 is minus 12x."},{"Start":"04:20.105 ","End":"04:24.200","Text":"Either way, if you imagine the 4 in front or not."},{"Start":"04:24.200 ","End":"04:27.305","Text":"Then let\u0027s imagine the 4 is here."},{"Start":"04:27.305 ","End":"04:36.290","Text":"Minus 4 times minus 7 is plus 28 is equal to 21."},{"Start":"04:36.290 ","End":"04:42.050","Text":"5 x minus 12x is equal to 21 minus 28."},{"Start":"04:42.050 ","End":"04:47.310","Text":"Here we have 5 minus 12 is minus is 7x."},{"Start":"04:47.310 ","End":"04:55.945","Text":"21 minus 28 is minus 7 divide by minus 7 and we just get that x is equal to 1."},{"Start":"04:55.945 ","End":"05:00.220","Text":"That concludes Part D. We are done."}],"ID":4937},{"Watched":false,"Name":"Exercise 4","Duration":"6m 8s","ChapterTopicVideoID":4935,"CourseChapterTopicPlaylistID":3862,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.825","Text":"In this exercise, we have to solve the following equation."},{"Start":"00:03.825 ","End":"00:05.475","Text":"There\u0027s only 2 of them."},{"Start":"00:05.475 ","End":"00:09.570","Text":"Let\u0027s get started with the first which I copied over here."},{"Start":"00:09.570 ","End":"00:16.110","Text":"Already there is something fishy because this is in the chapter on linear equations,"},{"Start":"00:16.110 ","End":"00:20.610","Text":"meaning equations where there is no x^2 yet here I see x^2."},{"Start":"00:20.610 ","End":"00:25.499","Text":"Well, the mystery will be solved during the solution of the question."},{"Start":"00:25.499 ","End":"00:27.600","Text":"First thing we have to do is open the bracket."},{"Start":"00:27.600 ","End":"00:30.945","Text":"Remember we multiply this by this and this by this."},{"Start":"00:30.945 ","End":"00:35.715","Text":"We get x times x is x^2."},{"Start":"00:35.715 ","End":"00:45.550","Text":"x times minus 5 is minus 5x is equal to x^2 minus 7x plus 8."},{"Start":"00:45.550 ","End":"00:49.510","Text":"Now, remember what I said about there shouldn\u0027t be an x^2?"},{"Start":"00:49.510 ","End":"00:51.320","Text":"In fact, it cancels."},{"Start":"00:51.320 ","End":"00:53.270","Text":"When you have something on both sides,"},{"Start":"00:53.270 ","End":"00:55.250","Text":"you can just cancel it."},{"Start":"00:55.250 ","End":"01:00.065","Text":"Really what I did was I subtracted x^2 from both sides of the equation."},{"Start":"01:00.065 ","End":"01:03.110","Text":"Or if you like moving something to the other side,"},{"Start":"01:03.110 ","End":"01:06.080","Text":"I brought it to this side and it became minus x^2,"},{"Start":"01:06.080 ","End":"01:07.729","Text":"whichever way you look at it at cancels."},{"Start":"01:07.729 ","End":"01:12.580","Text":"Now we are in familiar territory with just x\u0027s and numbers."},{"Start":"01:12.580 ","End":"01:17.250","Text":"Let\u0027s separate the sheep from the goats,"},{"Start":"01:17.250 ","End":"01:18.595","Text":"as the expression goes."},{"Start":"01:18.595 ","End":"01:22.790","Text":"Let\u0027s put the terms containing x on the left and the numbers on the right."},{"Start":"01:22.790 ","End":"01:25.725","Text":"This is an x term, it stays."},{"Start":"01:25.725 ","End":"01:27.230","Text":"This is an x term,"},{"Start":"01:27.230 ","End":"01:29.240","Text":"but it goes over to the other side."},{"Start":"01:29.240 ","End":"01:30.650","Text":"It comes out as plus,"},{"Start":"01:30.650 ","End":"01:33.680","Text":"and the 8 was here, it stays here."},{"Start":"01:33.680 ","End":"01:38.770","Text":"Next, collect like terms minus 5x plus 7x."},{"Start":"01:38.770 ","End":"01:44.310","Text":"How many x altogether minus 5 plus 7 is plus 2."},{"Start":"01:44.310 ","End":"01:48.150","Text":"We have 2x is equal to 8,"},{"Start":"01:48.150 ","End":"01:51.020","Text":"and now we bring the 2 over to"},{"Start":"01:51.020 ","End":"01:54.665","Text":"the other side because it was multiplied here, it\u0027s divided here."},{"Start":"01:54.665 ","End":"01:58.640","Text":"I\u0027m just going to write here, divide by 2."},{"Start":"01:58.640 ","End":"02:02.285","Text":"Just like a should\u0027ve written here, subtract x^2."},{"Start":"02:02.285 ","End":"02:04.190","Text":"When you do some of the same thing to both sides,"},{"Start":"02:04.190 ","End":"02:06.455","Text":"you write down what it is that you did to both sides."},{"Start":"02:06.455 ","End":"02:09.320","Text":"Anyway, we get x equals 8 divided by 2,"},{"Start":"02:09.320 ","End":"02:11.975","Text":"which I immediately write as 4."},{"Start":"02:11.975 ","End":"02:14.090","Text":"That is the answer."},{"Start":"02:14.090 ","End":"02:17.540","Text":"Now, let\u0027s go to the second one."},{"Start":"02:17.540 ","End":"02:19.655","Text":"Let\u0027s see what we have here."},{"Start":"02:19.655 ","End":"02:23.450","Text":"Looks like we have brackets here and here we also have one of"},{"Start":"02:23.450 ","End":"02:27.500","Text":"those special binomial expansions products."},{"Start":"02:27.500 ","End":"02:29.690","Text":"Everything is on the left, only 0 is on the right."},{"Start":"02:29.690 ","End":"02:32.330","Text":"Well, let\u0027s just open the brackets."},{"Start":"02:32.330 ","End":"02:34.460","Text":"We call expansion."},{"Start":"02:34.460 ","End":"02:37.300","Text":"I\u0027ll use the red arc."},{"Start":"02:37.300 ","End":"02:39.450","Text":"This with this, this with this,"},{"Start":"02:39.450 ","End":"02:41.365","Text":"this with this, this with this."},{"Start":"02:41.365 ","End":"02:43.615","Text":"Some people even put little arrows at the end."},{"Start":"02:43.615 ","End":"02:47.885","Text":"I\u0027ll do it for ones. Let\u0027s get to it."},{"Start":"02:47.885 ","End":"02:51.090","Text":"7 times 1 is 7."},{"Start":"02:51.090 ","End":"02:54.900","Text":"7 times minus x is minus 7x."},{"Start":"02:54.900 ","End":"02:59.385","Text":"Minus x times 1 is minus x."},{"Start":"02:59.385 ","End":"03:01.635","Text":"Minus x times minus x,"},{"Start":"03:01.635 ","End":"03:04.075","Text":"negative times negative is positive,"},{"Start":"03:04.075 ","End":"03:05.800","Text":"so it\u0027s like x times x,"},{"Start":"03:05.800 ","End":"03:08.770","Text":"which is x^2, minus."},{"Start":"03:08.770 ","End":"03:10.855","Text":"Now we come across this."},{"Start":"03:10.855 ","End":"03:13.120","Text":"This is one of those special binomial."},{"Start":"03:13.120 ","End":"03:14.784","Text":"Let me just write the equation."},{"Start":"03:14.784 ","End":"03:16.885","Text":"I believe this is the equation,"},{"Start":"03:16.885 ","End":"03:21.755","Text":"the identity, a formula that we\u0027re looking for."},{"Start":"03:21.755 ","End":"03:24.170","Text":"But since there\u0027s a minus here,"},{"Start":"03:24.170 ","End":"03:27.950","Text":"I have to put this in brackets, so I\u0027ll expand."},{"Start":"03:27.950 ","End":"03:31.535","Text":"But I\u0027ll put the answer to the expansion in brackets."},{"Start":"03:31.535 ","End":"03:35.480","Text":"Let\u0027s see, we need a^2 minus 2ab plus b^2,"},{"Start":"03:35.480 ","End":"03:40.040","Text":"where x is my a and 3 is the b."},{"Start":"03:40.040 ","End":"03:43.660","Text":"It\u0027s x^2 minus 2ab,"},{"Start":"03:43.660 ","End":"03:48.720","Text":"minus 2 times x times 3 is minus 6x."},{"Start":"03:48.720 ","End":"03:50.865","Text":"Then plus 3^2."},{"Start":"03:50.865 ","End":"03:52.825","Text":"3^2 is 9."},{"Start":"03:52.825 ","End":"03:54.680","Text":"Now I have to open the minus."},{"Start":"03:54.680 ","End":"03:58.430","Text":"Remember the minus in front of a bracket just reverses all the signs."},{"Start":"03:58.430 ","End":"04:02.945","Text":"This bit is the same 7 minus 7x minus x plus x^2."},{"Start":"04:02.945 ","End":"04:08.460","Text":"Then minus x^2 plus 6x minus 9."},{"Start":"04:08.460 ","End":"04:10.335","Text":"I reverse all the signs."},{"Start":"04:10.335 ","End":"04:12.840","Text":"I forgot the is equal to 0,"},{"Start":"04:12.840 ","End":"04:14.120","Text":"otherwise, it\u0027s not an equation."},{"Start":"04:14.120 ","End":"04:15.368","Text":"We don\u0027t have an equality."},{"Start":"04:15.368 ","End":"04:18.334","Text":"That\u0027s fixed. Collecting like terms."},{"Start":"04:18.334 ","End":"04:22.430","Text":"We have really 3 terms apples,"},{"Start":"04:22.430 ","End":"04:25.380","Text":"oranges, bananas. I\u0027ll mark them."},{"Start":"04:25.380 ","End":"04:29.850","Text":"These are the apples and the oranges are the x\u0027s."},{"Start":"04:29.850 ","End":"04:32.550","Text":"That\u0027s these and the third fruit,"},{"Start":"04:32.550 ","End":"04:35.055","Text":"bananas or lemons, these."},{"Start":"04:35.055 ","End":"04:37.745","Text":"Let\u0027s collect them. Let\u0027s go with the green first."},{"Start":"04:37.745 ","End":"04:41.090","Text":"Plus x^2 minus x^2 that cancels out."},{"Start":"04:41.090 ","End":"04:42.680","Text":"That\u0027s just nothing."},{"Start":"04:42.680 ","End":"04:45.500","Text":"Just don\u0027t write anything at all."},{"Start":"04:45.500 ","End":"04:48.230","Text":"I\u0027ll even just indicate that I cancel them."},{"Start":"04:48.230 ","End":"04:52.295","Text":"I expected that to happen because this is in the chapter on linear equations,"},{"Start":"04:52.295 ","End":"04:54.720","Text":"x^2 isn\u0027t supposed to come into this."},{"Start":"04:54.720 ","End":"04:57.830","Text":"Next, do the x\u0027s, the orange ones."},{"Start":"04:57.830 ","End":"04:59.360","Text":"How many of them do we have?"},{"Start":"04:59.360 ","End":"05:02.285","Text":"Minus 7 minus 1 plus 6."},{"Start":"05:02.285 ","End":"05:06.380","Text":"Minus 7 minus 1 is minus 8,"},{"Start":"05:06.380 ","End":"05:08.435","Text":"plus 6 is minus 2."},{"Start":"05:08.435 ","End":"05:11.840","Text":"We have minus 2 oranges, x\u0027s,"},{"Start":"05:11.840 ","End":"05:15.320","Text":"I mean, and let\u0027s see how many bananas."},{"Start":"05:15.320 ","End":"05:19.760","Text":"We have 7 minus 9 is minus 2."},{"Start":"05:19.760 ","End":"05:23.770","Text":"Minus 2 is equal to 0."},{"Start":"05:23.770 ","End":"05:28.970","Text":"Now as usual, I like the x\u0027s on the left and the numbers on the right."},{"Start":"05:28.970 ","End":"05:31.190","Text":"Well, the x is already is on the left."},{"Start":"05:31.190 ","End":"05:33.875","Text":"This I throw over to the other side."},{"Start":"05:33.875 ","End":"05:36.635","Text":"But I\u0027ve changed it from a minus to a plus,"},{"Start":"05:36.635 ","End":"05:39.005","Text":"minus 2x equals 2."},{"Start":"05:39.005 ","End":"05:44.410","Text":"Finally, I need to divide it by minus 2."},{"Start":"05:44.410 ","End":"05:47.190","Text":"If I\u0027m going to just be left with x,"},{"Start":"05:47.190 ","End":"05:49.630","Text":"so, I divide it by minus 2."},{"Start":"05:49.630 ","End":"05:51.335","Text":"I\u0027ll continue over here."},{"Start":"05:51.335 ","End":"05:53.090","Text":"If I divide it by minus 2,"},{"Start":"05:53.090 ","End":"05:58.225","Text":"this is just x and this is 2 divided by minus 2,"},{"Start":"05:58.225 ","End":"06:05.130","Text":"plus over minus is minus and 2 over 2 is 1. x is equal to minus 1."},{"Start":"06:05.130 ","End":"06:09.040","Text":"I\u0027ll highlight it and we\u0027re done."}],"ID":4938}],"Thumbnail":null,"ID":3862},{"Name":"Linear One Unknown, 0 Or Infinite Solutions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Linear in One Unknown - Degenerate","Duration":"5m ","ChapterTopicVideoID":4941,"CourseChapterTopicPlaylistID":3863,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.605","Text":"I\u0027m now going to talk about something I call unusual linear equations."},{"Start":"00:04.605 ","End":"00:09.510","Text":"Still in 1 variable that would be x. I use the informal term unusual."},{"Start":"00:09.510 ","End":"00:13.260","Text":"A technical term might be degenerate or singular,"},{"Start":"00:13.260 ","End":"00:17.610","Text":"but I don\u0027t want to use fancy words, just say unusual."},{"Start":"00:17.610 ","End":"00:19.665","Text":"Now I\u0027ll tell you in what sense unusual."},{"Start":"00:19.665 ","End":"00:24.165","Text":"Up to now, we\u0027ve had the linear equation and we\u0027ve solved it and we found what x equals."},{"Start":"00:24.165 ","End":"00:25.815","Text":"We\u0027ve got 1 solution."},{"Start":"00:25.815 ","End":"00:28.050","Text":"I mentioned in the introduction that sometimes"},{"Start":"00:28.050 ","End":"00:31.050","Text":"equations have more than 1 solution when they\u0027re not linear."},{"Start":"00:31.050 ","End":"00:35.950","Text":"I gave an example of x^2 equals 9 can have 2 solutions."},{"Start":"00:35.950 ","End":"00:37.640","Text":"Now with linear equations,"},{"Start":"00:37.640 ","End":"00:39.730","Text":"they almost always just have 1 solution,"},{"Start":"00:39.730 ","End":"00:42.085","Text":"but there are 2 other possibilities."},{"Start":"00:42.085 ","End":"00:43.810","Text":"The possibilities, I\u0027ll just write the of"},{"Start":"00:43.810 ","End":"00:49.400","Text":"the regular 1 solution is what we regularly get 1."},{"Start":"00:49.400 ","End":"00:52.115","Text":"That\u0027s the normal situation."},{"Start":"00:52.115 ","End":"00:53.300","Text":"That\u0027s not the unusual."},{"Start":"00:53.300 ","End":"00:54.995","Text":"That\u0027s the usual."},{"Start":"00:54.995 ","End":"00:59.665","Text":"What I mean by unusual is it could be no solution."},{"Start":"00:59.665 ","End":"01:03.370","Text":"That\u0027s 1 possibility."},{"Start":"01:03.370 ","End":"01:11.450","Text":"The other unusual possibility is there are an infinite number of solutions."},{"Start":"01:11.450 ","End":"01:14.930","Text":"It\u0027s not just that there are infinite number of solutions."},{"Start":"01:14.930 ","End":"01:17.960","Text":"In this case, it\u0027s that every number is a solution."},{"Start":"01:17.960 ","End":"01:22.175","Text":"All x are solutions in here,"},{"Start":"01:22.175 ","End":"01:26.975","Text":"no x whereas we\u0027re used to exactly just 1x."},{"Start":"01:26.975 ","End":"01:31.430","Text":"Let me give an example of each and they don\u0027t happen that often,"},{"Start":"01:31.430 ","End":"01:33.920","Text":"but you have to be aware that it could be."},{"Start":"01:33.920 ","End":"01:37.745","Text":"Well, I just went ahead and wrote them up to save time."},{"Start":"01:37.745 ","End":"01:39.815","Text":"Which one should we go for first?"},{"Start":"01:39.815 ","End":"01:44.135","Text":"Think I\u0027ll go for this one first and see what happens here."},{"Start":"01:44.135 ","End":"01:47.524","Text":"Here we want to collect like terms."},{"Start":"01:47.524 ","End":"01:50.855","Text":"x is on the left and numbers on the right."},{"Start":"01:50.855 ","End":"01:52.340","Text":"How many x do we have?"},{"Start":"01:52.340 ","End":"01:54.035","Text":"We have 2x from here."},{"Start":"01:54.035 ","End":"01:56.845","Text":"We have 8x from here,"},{"Start":"01:56.845 ","End":"02:00.555","Text":"minus 3x, it\u0027s a lot."},{"Start":"02:00.555 ","End":"02:04.610","Text":"7x that comes over becomes minus 7x."},{"Start":"02:04.610 ","End":"02:06.950","Text":"That\u0027s x\u0027s. Now numbers,"},{"Start":"02:06.950 ","End":"02:08.975","Text":"not so many, but yeah,"},{"Start":"02:08.975 ","End":"02:12.645","Text":"this 4 comes across as minus 4,"},{"Start":"02:12.645 ","End":"02:17.680","Text":"and then I have plus 9 from here and minus 5."},{"Start":"02:17.680 ","End":"02:20.674","Text":"Let\u0027s put them together."},{"Start":"02:20.674 ","End":"02:22.505","Text":"How many x\u0027s do I have?"},{"Start":"02:22.505 ","End":"02:24.095","Text":"2 plus 8,"},{"Start":"02:24.095 ","End":"02:25.930","Text":"minus 3 minus 7."},{"Start":"02:25.930 ","End":"02:28.395","Text":"Well, the pluses of 2 and 8 is 10,"},{"Start":"02:28.395 ","End":"02:30.915","Text":"the minuses of 3 and 7 is 10."},{"Start":"02:30.915 ","End":"02:33.405","Text":"We end up with 0x."},{"Start":"02:33.405 ","End":"02:37.100","Text":"Here I get 9 minus 4 minus 5."},{"Start":"02:37.100 ","End":"02:39.455","Text":"If I change the order, I also get 0."},{"Start":"02:39.455 ","End":"02:42.990","Text":"In other words, 0 equals 0."},{"Start":"02:43.010 ","End":"02:46.595","Text":"That looks very strange, that\u0027s always true."},{"Start":"02:46.595 ","End":"02:48.215","Text":"It turns out that, yeah,"},{"Start":"02:48.215 ","End":"02:52.340","Text":"x is a solution to this equation if and only if 0 equals 0."},{"Start":"02:52.340 ","End":"02:53.881","Text":"Now, this is always true,"},{"Start":"02:53.881 ","End":"02:55.760","Text":"so this is always true."},{"Start":"02:55.760 ","End":"02:58.310","Text":"In fact, if you throw me any number,"},{"Start":"02:58.310 ","End":"03:00.225","Text":"suppose you threw, I don\u0027t know,"},{"Start":"03:00.225 ","End":"03:05.355","Text":"x equals 1, I would do 2 plus 2 plus 8."},{"Start":"03:05.355 ","End":"03:07.770","Text":"Anyway, you can check. Substitute x equals 1,"},{"Start":"03:07.770 ","End":"03:09.435","Text":"it will give you equality."},{"Start":"03:09.435 ","End":"03:12.180","Text":"Maybe x equals 0, it\u0027ll be easier to compute."},{"Start":"03:12.180 ","End":"03:16.965","Text":"If x is 0, I just go 4 equals 9 minus 5,"},{"Start":"03:16.965 ","End":"03:19.350","Text":"but every number you put in."},{"Start":"03:19.350 ","End":"03:24.985","Text":"All x is a solutions which also implies that there\u0027s an infinite number of solutions."},{"Start":"03:24.985 ","End":"03:31.460","Text":"Typically, when you get to something like 0 equals 0 or 5 equals 5 at the end,"},{"Start":"03:31.460 ","End":"03:35.750","Text":"that indicates to you that every x is a solution."},{"Start":"03:35.750 ","End":"03:39.705","Text":"We write down solution,"},{"Start":"03:39.705 ","End":"03:44.765","Text":"all x or every x, something like that."},{"Start":"03:44.765 ","End":"03:46.940","Text":"Every x is a solution."},{"Start":"03:46.940 ","End":"03:49.175","Text":"Now let\u0027s go to this one."},{"Start":"03:49.175 ","End":"03:51.620","Text":"Here we have to also expand brackets."},{"Start":"03:51.620 ","End":"03:52.940","Text":"Let\u0027s see 5 times x,"},{"Start":"03:52.940 ","End":"03:54.320","Text":"5 times minus 2."},{"Start":"03:54.320 ","End":"04:00.780","Text":"5x minus 10 equals 3x plus again it brackets 2 with this 2 with this,"},{"Start":"04:00.780 ","End":"04:05.340","Text":"so 2x minus 6. x is to the left."},{"Start":"04:05.340 ","End":"04:07.200","Text":"I\u0027ll start with the one that is on the left,"},{"Start":"04:07.200 ","End":"04:13.350","Text":"and then I bring over the 3x is minus, minus 2x."},{"Start":"04:13.350 ","End":"04:15.555","Text":"On this side, let\u0027s see,"},{"Start":"04:15.555 ","End":"04:23.295","Text":"10 and plus and minus 6."},{"Start":"04:23.295 ","End":"04:27.020","Text":"If I do this computation, how many x\u0027s?"},{"Start":"04:27.020 ","End":"04:28.430","Text":"5 minus 3, minus 2,"},{"Start":"04:28.430 ","End":"04:31.505","Text":"I\u0027ve got 0x which is 0 again,"},{"Start":"04:31.505 ","End":"04:33.520","Text":"but here I have 4."},{"Start":"04:33.520 ","End":"04:35.625","Text":"In contrast to this,"},{"Start":"04:35.625 ","End":"04:39.650","Text":"where this is always true wherever x is 0 equals 0."},{"Start":"04:39.650 ","End":"04:41.525","Text":"But here whatever x is,"},{"Start":"04:41.525 ","End":"04:44.090","Text":"it will not be true that 0 equals 4."},{"Start":"04:44.090 ","End":"04:47.140","Text":"Here the solution is no x or no solution."},{"Start":"04:47.140 ","End":"04:50.599","Text":"We will just write in parallel solution,"},{"Start":"04:50.599 ","End":"04:54.485","Text":"no such x or no x or no solution."},{"Start":"04:54.485 ","End":"04:57.230","Text":"Just write it in your own words."},{"Start":"04:57.230 ","End":"05:00.090","Text":"That\u0027s it for this clip."}],"ID":4939},{"Watched":false,"Name":"Exercise 1","Duration":"1m 58s","ChapterTopicVideoID":4939,"CourseChapterTopicPlaylistID":3863,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"Here we have to solve this equation which looks innocent enough."},{"Start":"00:04.770 ","End":"00:07.110","Text":"Let\u0027s see what happens and why this is here,"},{"Start":"00:07.110 ","End":"00:09.135","Text":"why we\u0027re back to linear equations."},{"Start":"00:09.135 ","End":"00:11.070","Text":"Well, let\u0027s simplify,"},{"Start":"00:11.070 ","End":"00:15.165","Text":"expand brackets 6 times x and then 6 times minus 2."},{"Start":"00:15.165 ","End":"00:23.445","Text":"We have 6x minus 12 equals 2x plus 5 plus 4x."},{"Start":"00:23.445 ","End":"00:27.525","Text":"Now we bring the x\u0027s to the left and the numbers to the right."},{"Start":"00:27.525 ","End":"00:37.900","Text":"X\u0027s we have here and here and here and number here and here."},{"Start":"00:38.120 ","End":"00:42.860","Text":"On the left we have 6x minus 2x."},{"Start":"00:42.860 ","End":"00:47.195","Text":"Remember everything that crosses side gets reversed."},{"Start":"00:47.195 ","End":"00:55.350","Text":"Minus 4x and on the right we get a plus 12 and a plus 5."},{"Start":"00:55.600 ","End":"00:58.205","Text":"What do we get now?"},{"Start":"00:58.205 ","End":"01:02.060","Text":"6 minus 2 minus 4, that\u0027s 0."},{"Start":"01:02.060 ","End":"01:06.815","Text":"I get 0x equals 17."},{"Start":"01:06.815 ","End":"01:08.615","Text":"0 times anything is 0."},{"Start":"01:08.615 ","End":"01:11.720","Text":"I get 0 equals 17."},{"Start":"01:11.720 ","End":"01:14.400","Text":"What\u0027s going on here?"},{"Start":"01:14.400 ","End":"01:18.070","Text":"The answer is that if while you\u0027re trying to solve it,"},{"Start":"01:18.070 ","End":"01:22.495","Text":"you get something impossible, a definite falsehood."},{"Start":"01:22.495 ","End":"01:26.035","Text":"It means that there is no solution to this equation."},{"Start":"01:26.035 ","End":"01:30.910","Text":"You simply write no solution."},{"Start":"01:32.720 ","End":"01:35.080","Text":"That\u0027s all there is to it."},{"Start":"01:35.080 ","End":"01:39.220","Text":"Any value of x you substitute in here will not work."},{"Start":"01:39.220 ","End":"01:45.160","Text":"Because any value of x that satisfies this equation implies that 0 equals 17,"},{"Start":"01:45.160 ","End":"01:47.020","Text":"and this could never going to happen."},{"Start":"01:47.020 ","End":"01:49.690","Text":"No solution, and that\u0027s it."},{"Start":"01:49.690 ","End":"01:57.320","Text":"Or no x satisfies the equation or something like that you can write. That\u0027s It."}],"ID":4940},{"Watched":false,"Name":"Exercise 2","Duration":"2m 36s","ChapterTopicVideoID":4940,"CourseChapterTopicPlaylistID":3863,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"Here we have to solve this equation and one variable,"},{"Start":"00:05.250 ","End":"00:07.950","Text":"x, looks innocent enough,"},{"Start":"00:07.950 ","End":"00:09.435","Text":"let\u0027s try and solve it."},{"Start":"00:09.435 ","End":"00:12.408","Text":"The usual technique is to bring the x\u0027s to the left,"},{"Start":"00:12.408 ","End":"00:14.340","Text":"numbers to the right just to make it easier,"},{"Start":"00:14.340 ","End":"00:16.050","Text":"I\u0027ll mark where the x\u0027s are."},{"Start":"00:16.050 ","End":"00:17.805","Text":"X marks the spot."},{"Start":"00:17.805 ","End":"00:20.969","Text":"These are the x\u0027s and I\u0027ll bring them to the left."},{"Start":"00:20.969 ","End":"00:28.530","Text":"I\u0027ve got 5x plus x minus 4x because it\u0027s switched sides and minus 2x."},{"Start":"00:28.530 ","End":"00:32.160","Text":"Now the numbers, number I have here and here."},{"Start":"00:32.160 ","End":"00:35.975","Text":"Minus 3 was on the right already,"},{"Start":"00:35.975 ","End":"00:37.580","Text":"but this one crossed sides,"},{"Start":"00:37.580 ","End":"00:39.800","Text":"so it becomes plus 3."},{"Start":"00:39.800 ","End":"00:41.884","Text":"Let\u0027s see, let\u0027s do the arithmetic."},{"Start":"00:41.884 ","End":"00:48.440","Text":"5 and 1 minus 4 minus 2 is 6 minus 6 is 0,"},{"Start":"00:48.440 ","End":"00:51.160","Text":"0x that\u0027s just 0."},{"Start":"00:51.160 ","End":"00:57.140","Text":"Minus 3 plus 3 is also 0 and 0 equals 0 is always true."},{"Start":"00:57.140 ","End":"01:00.335","Text":"It\u0027s called tautology, self-evident."},{"Start":"01:00.335 ","End":"01:05.135","Text":"Now, what does this mean in terms of the solution to this equation?"},{"Start":"01:05.135 ","End":"01:09.230","Text":"Well, if you get something that\u0027s a tautology,"},{"Start":"01:09.230 ","End":"01:12.890","Text":"that\u0027s always true like 5 equals 5 or something like that,"},{"Start":"01:12.890 ","End":"01:18.305","Text":"or 0 equals 0, it means that there\u0027s an infinite number of solutions."},{"Start":"01:18.305 ","End":"01:21.605","Text":"More than that, it means that every x is a solution."},{"Start":"01:21.605 ","End":"01:25.460","Text":"This is satisfied by all x."},{"Start":"01:25.460 ","End":"01:27.275","Text":"That\u0027s the solution."},{"Start":"01:27.275 ","End":"01:29.060","Text":"Satisfy this equation."},{"Start":"01:29.060 ","End":"01:31.550","Text":"Let\u0027s just illustrate it by, I don\u0027t know,"},{"Start":"01:31.550 ","End":"01:37.425","Text":"give me a value of x, say 3."},{"Start":"01:37.425 ","End":"01:43.710","Text":"We got 15 minus 3 is 12,"},{"Start":"01:43.710 ","End":"01:52.040","Text":"plus 3 is 15 and here we get 4 times 3 is 12 plus 6 minus 3 is also 15."},{"Start":"01:52.040 ","End":"01:54.005","Text":"Try x equals 0,"},{"Start":"01:54.005 ","End":"01:57.215","Text":"we just get minus 3 equals minus 3, it works."},{"Start":"01:57.215 ","End":"01:58.970","Text":"Every x you plug in,"},{"Start":"01:58.970 ","End":"02:01.985","Text":"you\u0027ll get an equality to one where x equals 1."},{"Start":"02:01.985 ","End":"02:08.995","Text":"5 minus 3 plus 1 is 6,"},{"Start":"02:08.995 ","End":"02:15.710","Text":"and 4 plus 2 minus 3 is 3."},{"Start":"02:15.710 ","End":"02:17.210","Text":"Wait, what did I do wrong here?"},{"Start":"02:17.210 ","End":"02:19.955","Text":"When x is 1, it\u0027s 5 plus 1 minus 3, yeah,"},{"Start":"02:19.955 ","End":"02:22.805","Text":"it\u0027s 3 and 4 plus 2 minus 3 is also 3."},{"Start":"02:22.805 ","End":"02:26.820","Text":"All x satisfy the equation."},{"Start":"02:27.250 ","End":"02:34.920","Text":"You can just write all x. That\u0027s it."}],"ID":4941}],"Thumbnail":null,"ID":3863},{"Name":"Linear One Unknown Denom","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"1m 31s","ChapterTopicVideoID":4942,"CourseChapterTopicPlaylistID":3864,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"This is one of these equations with denominators."},{"Start":"00:03.900 ","End":"00:06.000","Text":"The first thing we want to do is get rid of"},{"Start":"00:06.000 ","End":"00:09.330","Text":"the denominators and then we can see what we have,"},{"Start":"00:09.330 ","End":"00:11.965","Text":"a linear or quadratic or whatever."},{"Start":"00:11.965 ","End":"00:14.950","Text":"Let\u0027s see, 3 and 9."},{"Start":"00:14.950 ","End":"00:17.310","Text":"What\u0027s a good common denominator?"},{"Start":"00:17.310 ","End":"00:20.639","Text":"Of course, you can multiply 3 by 9 and get 27,"},{"Start":"00:20.639 ","End":"00:22.739","Text":"but we prefer something smaller."},{"Start":"00:22.739 ","End":"00:25.990","Text":"You can see that 9 will work for both."},{"Start":"00:27.320 ","End":"00:33.690","Text":"I\u0027m going to multiply both sides of the equation by 9."},{"Start":"00:34.480 ","End":"00:39.055","Text":"That means that every term gets multiplied by 9,"},{"Start":"00:39.055 ","End":"00:42.620","Text":"and 9 will cancel with the denominator in each case."},{"Start":"00:42.620 ","End":"00:46.335","Text":"Here, we\u0027ll be left with 3, 9/3."},{"Start":"00:46.335 ","End":"00:48.960","Text":"Here, 9/9 is 1."},{"Start":"00:48.960 ","End":"00:51.210","Text":"Here, the denominator is 1,"},{"Start":"00:51.210 ","End":"00:53.840","Text":"we\u0027ll get the full 9 here."},{"Start":"00:53.840 ","End":"00:55.970","Text":"After we multiply out,"},{"Start":"00:55.970 ","End":"01:01.710","Text":"we get 3 times x is 3x minus 1 times x is"},{"Start":"01:01.710 ","End":"01:09.375","Text":"x equals 9 times minus 4 plus times minus is minus 36."},{"Start":"01:09.375 ","End":"01:11.805","Text":"Collecting the like terms,"},{"Start":"01:11.805 ","End":"01:17.160","Text":"3x-x is 2x is minus 36."},{"Start":"01:17.160 ","End":"01:20.415","Text":"Then we divide both sides by 2."},{"Start":"01:20.415 ","End":"01:22.775","Text":"Take the 2 to the denominator."},{"Start":"01:22.775 ","End":"01:27.990","Text":"Either way, x=-18."},{"Start":"01:28.000 ","End":"01:31.140","Text":"That\u0027s the answer."}],"ID":4942},{"Watched":false,"Name":"Exercise 2","Duration":"1m 13s","ChapterTopicVideoID":4943,"CourseChapterTopicPlaylistID":3864,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"Here\u0027s another one of those equations where we"},{"Start":"00:03.000 ","End":"00:06.055","Text":"have a denominator that we should get rid of."},{"Start":"00:06.055 ","End":"00:07.865","Text":"Let\u0027s do it over here."},{"Start":"00:07.865 ","End":"00:10.675","Text":"Let\u0027s see. Now we have 15 and 10."},{"Start":"00:10.675 ","End":"00:13.800","Text":"I don\u0027t want to just say 15 times 10 it\u0027s 150."},{"Start":"00:13.800 ","End":"00:15.225","Text":"That\u0027s way too big."},{"Start":"00:15.225 ","End":"00:16.860","Text":"Just think about it a moment,"},{"Start":"00:16.860 ","End":"00:18.960","Text":"you\u0027ll see that 30 is good enough."},{"Start":"00:18.960 ","End":"00:22.170","Text":"If I multiply both sides of the equation by 30,"},{"Start":"00:22.170 ","End":"00:24.735","Text":"that should get rid of the fractions."},{"Start":"00:24.735 ","End":"00:28.125","Text":"Multiply each term by 30."},{"Start":"00:28.125 ","End":"00:31.950","Text":"Now 30 with 15 is going to cancel,"},{"Start":"00:31.950 ","End":"00:34.470","Text":"so we\u0027re going to get 2 in the numerator,"},{"Start":"00:34.470 ","End":"00:37.965","Text":"30 over 10 is 3,"},{"Start":"00:37.965 ","End":"00:41.100","Text":"and here there\u0027s no fraction or it\u0027s 1 in the denominator,"},{"Start":"00:41.100 ","End":"00:44.295","Text":"so we multiply by the whole 30."},{"Start":"00:44.295 ","End":"00:48.585","Text":"What we get is 2 times 4x is 8x,"},{"Start":"00:48.585 ","End":"00:51.060","Text":"3 times 3x is 9x,"},{"Start":"00:51.060 ","End":"00:53.025","Text":"but it\u0027s with a minus,"},{"Start":"00:53.025 ","End":"00:56.025","Text":"and 30 times 1 is 30."},{"Start":"00:56.025 ","End":"01:01.505","Text":"8x minus 9x is just minus x, which is 30."},{"Start":"01:01.505 ","End":"01:05.060","Text":"From here, it\u0027s easy to see that x is minus 30."},{"Start":"01:05.060 ","End":"01:07.900","Text":"You either multiply or divide by minus 1,"},{"Start":"01:07.900 ","End":"01:10.025","Text":"and that\u0027s what you get,"},{"Start":"01:10.025 ","End":"01:14.040","Text":"and this is the answer and we\u0027re done."}],"ID":4943},{"Watched":false,"Name":"Exercise 3","Duration":"2m 1s","ChapterTopicVideoID":4944,"CourseChapterTopicPlaylistID":3864,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.545","Text":"Here we have an equation with a lot of denominators."},{"Start":"00:04.545 ","End":"00:06.675","Text":"I want to get rid of those denominators."},{"Start":"00:06.675 ","End":"00:08.040","Text":"Let\u0027s do it over here."},{"Start":"00:08.040 ","End":"00:10.530","Text":"Look, we have here 3,"},{"Start":"00:10.530 ","End":"00:13.680","Text":"5, and 15."},{"Start":"00:13.680 ","End":"00:17.744","Text":"I have to think, what\u0027s a good common denominator for all of them?"},{"Start":"00:17.744 ","End":"00:19.140","Text":"If you reflect a moment,"},{"Start":"00:19.140 ","End":"00:21.720","Text":"you\u0027ll see that 15 is already 3 times 5."},{"Start":"00:21.720 ","End":"00:24.930","Text":"So 15 will do for all three of these."},{"Start":"00:24.930 ","End":"00:30.510","Text":"Of course, here, in front of the x is like 1x or even 1 over 1x,"},{"Start":"00:30.510 ","End":"00:32.340","Text":"if you want it as a fraction."},{"Start":"00:32.340 ","End":"00:38.670","Text":"What I\u0027m saying is we get to multiply both sides of this equation by 15,"},{"Start":"00:38.670 ","End":"00:42.075","Text":"which means multiplying every term by 15."},{"Start":"00:42.075 ","End":"00:45.540","Text":"What\u0027s going to happen is 15 is going to cancel with each of"},{"Start":"00:45.540 ","End":"00:49.225","Text":"the denominators and we\u0027re going to have something left in the numerator."},{"Start":"00:49.225 ","End":"00:51.670","Text":"15 over 3 is 5,"},{"Start":"00:51.670 ","End":"00:54.040","Text":"which means we multiply this by 5."},{"Start":"00:54.040 ","End":"00:56.155","Text":"15 over 5 is 3,"},{"Start":"00:56.155 ","End":"01:00.595","Text":"so we multiply this by 3."},{"Start":"01:00.595 ","End":"01:02.590","Text":"15, there is no denominator here,"},{"Start":"01:02.590 ","End":"01:05.845","Text":"denominator of 1, it means it gets the full 15."},{"Start":"01:05.845 ","End":"01:09.440","Text":"Here 15 over 15 is 1."},{"Start":"01:09.650 ","End":"01:14.180","Text":"We get rid of the denominators,"},{"Start":"01:14.180 ","End":"01:15.975","Text":"then we just do these multiplications."},{"Start":"01:15.975 ","End":"01:19.020","Text":"5 times 2 is 10,"},{"Start":"01:19.020 ","End":"01:20.490","Text":"but there\u0027s an x there,"},{"Start":"01:20.490 ","End":"01:26.925","Text":"plus 3 times 4 is 12x equals 15 times 1 is"},{"Start":"01:26.925 ","End":"01:34.800","Text":"15x and minus 1 times 7 is 7, no x."},{"Start":"01:34.800 ","End":"01:37.845","Text":"Bring the 15 to the other side."},{"Start":"01:37.845 ","End":"01:45.570","Text":"We\u0027ve got 10x+12x-15x=-7."},{"Start":"01:45.570 ","End":"01:52.785","Text":"10+12-15, 22-15=7x, equals minus 7."},{"Start":"01:52.785 ","End":"01:55.200","Text":"Then we divide by 7."},{"Start":"01:55.200 ","End":"01:58.248","Text":"So x is minus 1,"},{"Start":"01:58.248 ","End":"02:01.960","Text":"and that is our answer."}],"ID":4944},{"Watched":false,"Name":"Exercise 4","Duration":"3m 39s","ChapterTopicVideoID":4945,"CourseChapterTopicPlaylistID":3864,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.700","Text":"Here we have to solve this equation,"},{"Start":"00:01.700 ","End":"00:04.590","Text":"and the first thing I see is there\u0027s plenty of denominators,"},{"Start":"00:04.590 ","End":"00:06.000","Text":"I want to get rid of those."},{"Start":"00:06.000 ","End":"00:07.575","Text":"Let\u0027s do that over here."},{"Start":"00:07.575 ","End":"00:09.885","Text":"I look, we have here a 6,"},{"Start":"00:09.885 ","End":"00:12.240","Text":"we have here a 5, and a 4."},{"Start":"00:12.240 ","End":"00:16.350","Text":"Now I have to think of a number that all these three go into."},{"Start":"00:16.350 ","End":"00:22.830","Text":"You could just multiply them all together and get 6 times 5 is 30 times 4 is 120,"},{"Start":"00:22.830 ","End":"00:25.365","Text":"or we could try something smaller."},{"Start":"00:25.365 ","End":"00:27.885","Text":"In fact, if I think about it,"},{"Start":"00:27.885 ","End":"00:29.250","Text":"60 will be good enough,"},{"Start":"00:29.250 ","End":"00:30.930","Text":"we don\u0027t need the full 120."},{"Start":"00:30.930 ","End":"00:35.310","Text":"So I multiply by 60 on both sides,"},{"Start":"00:35.310 ","End":"00:39.675","Text":"and that means everything gets multiplied by 60 as usual."},{"Start":"00:39.675 ","End":"00:44.715","Text":"60 helps cancel all the denominators,"},{"Start":"00:44.715 ","End":"00:47.145","Text":"60/6 is 10,"},{"Start":"00:47.145 ","End":"00:51.750","Text":"I\u0027ll write those quotients here, division,"},{"Start":"00:51.750 ","End":"00:57.540","Text":"60/5 is 12,"},{"Start":"00:57.540 ","End":"00:59.880","Text":"60/4 is 15,"},{"Start":"00:59.880 ","End":"01:02.210","Text":"and here, the denominator is 1,"},{"Start":"01:02.210 ","End":"01:05.045","Text":"so it\u0027s just 60."},{"Start":"01:05.045 ","End":"01:11.020","Text":"Now we just multiply out throughout the denominators and do the multiplications."},{"Start":"01:11.020 ","End":"01:18.660","Text":"Thus, 10(5x+1), we need the brackets."},{"Start":"01:18.660 ","End":"01:23.120","Text":"An actual fact, the dividing line is considered to be a bracket."},{"Start":"01:23.120 ","End":"01:25.415","Text":"It\u0027s one of the forms of brackets."},{"Start":"01:25.415 ","End":"01:28.800","Text":"Minus, because this is a minus,"},{"Start":"01:28.800 ","End":"01:38.430","Text":"12(6x-1)=15(3x+1),"},{"Start":"01:38.430 ","End":"01:41.820","Text":"and minus 60."},{"Start":"01:41.820 ","End":"01:44.150","Text":"That\u0027s gotten rid of the fractions."},{"Start":"01:44.150 ","End":"01:46.610","Text":"Now we need to organize this."},{"Start":"01:46.610 ","End":"01:48.695","Text":"Let\u0027s expand brackets,"},{"Start":"01:48.695 ","End":"01:51.845","Text":"10 times 5x and then 10 times 1 and so on."},{"Start":"01:51.845 ","End":"01:55.895","Text":"So 50x, 10 times 1 is 10,"},{"Start":"01:55.895 ","End":"02:00.940","Text":"12 times 6 is 72x,"},{"Start":"02:00.940 ","End":"02:05.040","Text":"but minus 12 times minus 1 is plus 12,"},{"Start":"02:05.040 ","End":"02:08.935","Text":"and then 15 times 3x is 45x,"},{"Start":"02:08.935 ","End":"02:15.005","Text":"plus 15 minus 60. x is to the left,"},{"Start":"02:15.005 ","End":"02:16.660","Text":"numbers to the right,"},{"Start":"02:16.660 ","End":"02:21.495","Text":"50x, we\u0027ve got from here, minus 72x."},{"Start":"02:21.495 ","End":"02:26.400","Text":"From here, I get minus 45x because I brought it over,"},{"Start":"02:26.400 ","End":"02:29.640","Text":"and now, numbers, let\u0027s see."},{"Start":"02:29.640 ","End":"02:31.530","Text":"Let\u0027s start with this one,"},{"Start":"02:31.530 ","End":"02:33.570","Text":"10 becomes minus 10,"},{"Start":"02:33.570 ","End":"02:35.805","Text":"12 becomes minus 12,"},{"Start":"02:35.805 ","End":"02:39.760","Text":"and plus 15,"},{"Start":"02:40.370 ","End":"02:44.255","Text":"and then minus the 60."},{"Start":"02:44.255 ","End":"02:46.295","Text":"Let\u0027s see what we have left."},{"Start":"02:46.295 ","End":"02:51.780","Text":"50 minus 72 minus 45,"},{"Start":"02:52.210 ","End":"02:55.735","Text":"I get minus 67x."},{"Start":"02:55.735 ","End":"02:59.100","Text":"Let me just check that 50 minus 45 is 5,"},{"Start":"02:59.100 ","End":"03:01.330","Text":"5 minus 72, yeah."},{"Start":"03:01.330 ","End":"03:03.380","Text":"Then this is equal to,"},{"Start":"03:03.380 ","End":"03:10.180","Text":"let\u0027s see, minus 10 minus 12, minus 22."},{"Start":"03:10.180 ","End":"03:13.800","Text":"Let\u0027s do all the minuses, minus 22,"},{"Start":"03:13.800 ","End":"03:19.035","Text":"and then minus 60 is minus 82 plus 15."},{"Start":"03:19.035 ","End":"03:26.140","Text":"So it\u0027s minus, we need a difference of 82 and 15, is 67."},{"Start":"03:26.170 ","End":"03:33.390","Text":"So just divide by minus 67 on both sides and we get x=1."},{"Start":"03:33.560 ","End":"03:39.760","Text":"This is our answer and we are done."}],"ID":4945},{"Watched":false,"Name":"Exercise 5","Duration":"2m 26s","ChapterTopicVideoID":4946,"CourseChapterTopicPlaylistID":3864,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.075","Text":"Here we have to solve this equation"},{"Start":"00:03.075 ","End":"00:05.970","Text":"and the first thing I notice is that there are denominators,"},{"Start":"00:05.970 ","End":"00:09.120","Text":"fractions, and I\u0027d like to get rid of the denominators."},{"Start":"00:09.120 ","End":"00:10.800","Text":"Let\u0027s do that over here."},{"Start":"00:10.800 ","End":"00:20.190","Text":"The denominators are 5 and 15 and I need to find something that both of them go into."},{"Start":"00:20.190 ","End":"00:21.900","Text":"I don\u0027t have to look very far,"},{"Start":"00:21.900 ","End":"00:24.030","Text":"15 will be good for both of them."},{"Start":"00:24.030 ","End":"00:29.520","Text":"Let\u0027s multiply both sides of this equation by 15."},{"Start":"00:29.520 ","End":"00:38.415","Text":"With the usual way we do things,15/5 is 3 so I multiply this by 3,"},{"Start":"00:38.415 ","End":"00:42.120","Text":"15/15 is 1 so this gets multiplied by 1,"},{"Start":"00:42.120 ","End":"00:50.555","Text":"and this is like over 1 so I have to multiply it by the full 15."},{"Start":"00:50.555 ","End":"00:58.530","Text":"What we get is 3 times 2 is 6 but we still have the x minus 3, minus,"},{"Start":"00:58.530 ","End":"01:01.340","Text":"1 times 3 is 3,"},{"Start":"01:01.340 ","End":"01:04.035","Text":"and we still have the 4 minus x,"},{"Start":"01:04.035 ","End":"01:10.990","Text":"and here 15 times all of x plus 2 so it needs the brackets."},{"Start":"01:11.240 ","End":"01:13.935","Text":"Let\u0027s expand the brackets,"},{"Start":"01:13.935 ","End":"01:18.305","Text":"6 times x is 6x minus 6 times 3,"},{"Start":"01:18.305 ","End":"01:21.270","Text":"18 minus 3,"},{"Start":"01:21.270 ","End":"01:22.845","Text":"that\u0027s minus 12,"},{"Start":"01:22.845 ","End":"01:26.460","Text":"and minus minus is plus 3x."},{"Start":"01:26.460 ","End":"01:27.750","Text":"On the other side, 15x,"},{"Start":"01:27.750 ","End":"01:31.450","Text":"15 times 2 is 30."},{"Start":"01:31.450 ","End":"01:35.405","Text":"As usual, I want the x\u0027s on the left and the numbers on the right."},{"Start":"01:35.405 ","End":"01:43.477","Text":"On the left, I have 6x+3x-15x."},{"Start":"01:43.477 ","End":"01:52.965","Text":"On the right, let\u0027s start with the 30 here that we had and then plus 18 and plus 12."},{"Start":"01:52.965 ","End":"01:54.855","Text":"Where does that leave us?"},{"Start":"01:54.855 ","End":"01:56.970","Text":"6 plus 3 is 9,"},{"Start":"01:56.970 ","End":"02:07.100","Text":"9 minus 15 is minus 6x,18 plus 12 is 30 so I\u0027ve got 60 altogether."},{"Start":"02:07.100 ","End":"02:12.425","Text":"Now I want to divide both sides by minus 6,"},{"Start":"02:12.425 ","End":"02:17.640","Text":"so that\u0027ll leave me here with just x and 60 over minus 6,"},{"Start":"02:17.640 ","End":"02:21.880","Text":"plus over minus is minus 60/6 is 10."},{"Start":"02:21.980 ","End":"02:24.060","Text":"This is the answer,"},{"Start":"02:24.060 ","End":"02:26.920","Text":"x=-10 and we\u0027re done."}],"ID":4946},{"Watched":false,"Name":"Exercise 6","Duration":"2m 36s","ChapterTopicVideoID":4947,"CourseChapterTopicPlaylistID":3864,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.860","Text":"Here we have an equation with fractions."},{"Start":"00:03.860 ","End":"00:05.249","Text":"I want to get rid of the fractions."},{"Start":"00:05.249 ","End":"00:08.040","Text":"It\u0027s a little bit different than the usual."},{"Start":"00:08.040 ","End":"00:10.395","Text":"Let me solve it down here."},{"Start":"00:10.395 ","End":"00:15.225","Text":"Because the fractions are all located in this one bit."},{"Start":"00:15.225 ","End":"00:17.130","Text":"Let me show you how we handle this."},{"Start":"00:17.130 ","End":"00:22.230","Text":"We still think of say 3 and 7 and look for a common denominator."},{"Start":"00:22.230 ","End":"00:25.710","Text":"The least you can get is 21."},{"Start":"00:25.710 ","End":"00:28.965","Text":"I want to multiply both sides by 21."},{"Start":"00:28.965 ","End":"00:31.545","Text":"The way I\u0027ll do it,"},{"Start":"00:31.545 ","End":"00:34.120","Text":"is I\u0027ll write this as 5,"},{"Start":"00:34.120 ","End":"00:35.960","Text":"and the 21,"},{"Start":"00:35.960 ","End":"00:43.300","Text":"I\u0027ll put right next to these brackets here, x/3 minus x/7."},{"Start":"00:43.300 ","End":"00:45.980","Text":"I still have to multiply everything by 21,"},{"Start":"00:45.980 ","End":"00:49.040","Text":"so this also gets multiplied by 21."},{"Start":"00:49.040 ","End":"00:52.900","Text":"The right-hand side also gets multiplied by 21."},{"Start":"00:52.900 ","End":"00:56.820","Text":"What I want to concentrate on is this bit here."},{"Start":"00:56.820 ","End":"01:00.700","Text":"In fact, why don\u0027t I do that at the side?"},{"Start":"01:00.820 ","End":"01:04.460","Text":"I\u0027m just taking this bit in orange,"},{"Start":"01:04.460 ","End":"01:12.665","Text":"21(x/3-x/7)."},{"Start":"01:12.665 ","End":"01:19.120","Text":"In this case, I do just like we usually do,"},{"Start":"01:19.120 ","End":"01:24.480","Text":"I say 21/3 is 7 and I write a little 7 here,"},{"Start":"01:24.480 ","End":"01:26.985","Text":"and 21/7 is 3."},{"Start":"01:26.985 ","End":"01:32.570","Text":"What you get then is you get rid of the 21 as well as the denominators"},{"Start":"01:32.570 ","End":"01:41.670","Text":"and we get 7 times x is 7x and minus 3x."},{"Start":"01:43.210 ","End":"01:46.380","Text":"This is just 4x."},{"Start":"01:46.810 ","End":"01:49.895","Text":"This is the orange bit."},{"Start":"01:49.895 ","End":"01:52.939","Text":"I\u0027m going to put it back here."},{"Start":"01:52.939 ","End":"01:56.675","Text":"What we get now is 5 times"},{"Start":"01:56.675 ","End":"02:04.070","Text":"4x minus 21x = 21."},{"Start":"02:04.070 ","End":"02:06.020","Text":"5 times 4x is 20x,"},{"Start":"02:06.020 ","End":"02:12.620","Text":"so you\u0027ve got 20x-21x is 21."},{"Start":"02:12.620 ","End":"02:16.905","Text":"20x-21x is just minus x,"},{"Start":"02:16.905 ","End":"02:21.065","Text":"minus 1x, which is minus x, is 21."},{"Start":"02:21.065 ","End":"02:23.705","Text":"Then if minus x is 21,"},{"Start":"02:23.705 ","End":"02:26.970","Text":"just divide by minus 1 or multiply by minus 1,"},{"Start":"02:26.970 ","End":"02:36.520","Text":"and we get that x=-21. That is it."}],"ID":4947},{"Watched":false,"Name":"Exercise 7","Duration":"1m 8s","ChapterTopicVideoID":4948,"CourseChapterTopicPlaylistID":3864,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.400","Text":"We have to solve this equation."},{"Start":"00:02.400 ","End":"00:04.860","Text":"We notice there are denominators,"},{"Start":"00:04.860 ","End":"00:06.600","Text":"that is. I\u0027ll do it over here."},{"Start":"00:06.600 ","End":"00:13.380","Text":"I\u0027ll look for a common denominator of 4 and x."},{"Start":"00:13.380 ","End":"00:18.090","Text":"The best I can do is 4x because that will be"},{"Start":"00:18.090 ","End":"00:23.670","Text":"divisible by 4 and by x. I\u0027ll multiply both sides by 4x,"},{"Start":"00:23.670 ","End":"00:26.010","Text":"which means multiplying every term."},{"Start":"00:26.010 ","End":"00:32.620","Text":"Why we usually do this because we say 4 into 4x goes x times,"},{"Start":"00:32.780 ","End":"00:38.725","Text":"x into 4x goes 4 times."},{"Start":"00:38.725 ","End":"00:43.710","Text":"The right-hand side gets multiplied by 4x."},{"Start":"00:44.660 ","End":"00:52.530","Text":"What it comes down to is x times 1 is x minus 4 times 2 is 8,"},{"Start":"00:52.530 ","End":"00:58.205","Text":"and 4x times 0 is 0 because anything times 0 is 0."},{"Start":"00:58.205 ","End":"01:00.050","Text":"X minus 8 is 0."},{"Start":"01:00.050 ","End":"01:02.525","Text":"Let\u0027s bring the 8 over to the other side."},{"Start":"01:02.525 ","End":"01:05.730","Text":"We\u0027ve got x equals 8."},{"Start":"01:05.730 ","End":"01:08.890","Text":"That\u0027s all there is to it."}],"ID":4948},{"Watched":false,"Name":"Exercise 8","Duration":"1m 30s","ChapterTopicVideoID":4949,"CourseChapterTopicPlaylistID":3864,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"We have to solve the following equation."},{"Start":"00:02.760 ","End":"00:05.370","Text":"First thing I see is denominators."},{"Start":"00:05.370 ","End":"00:07.950","Text":"Don\u0027t like denominators. I want to get rid of them."},{"Start":"00:07.950 ","End":"00:10.275","Text":"Let\u0027s do it over here."},{"Start":"00:10.275 ","End":"00:12.585","Text":"I\u0027ll look for something in common."},{"Start":"00:12.585 ","End":"00:18.670","Text":"Both of these will go into and the obvious choice is just 2(x-1)."},{"Start":"00:19.910 ","End":"00:24.675","Text":"I ask 2 into this goes how many times?"},{"Start":"00:24.675 ","End":"00:28.360","Text":"X minus 1 times. Better put it in brackets."},{"Start":"00:28.520 ","End":"00:37.830","Text":"X minus 1 into 2(x-1) goes in twice and multiply the left minus hand side."},{"Start":"00:37.830 ","End":"00:42.200","Text":"I also have to multiply the right minus hand side by 2(x-1)."},{"Start":"00:43.550 ","End":"00:50.385","Text":"Multiplying it out, I get x minus 1 times 1 is x minus 1."},{"Start":"00:50.385 ","End":"00:53.830","Text":"I don\u0027t think I\u0027ll need the brackets now."},{"Start":"00:54.170 ","End":"00:57.570","Text":"Minus 2 times x is 2x,"},{"Start":"00:57.570 ","End":"01:00.170","Text":"and equals, doesn\u0027t matter what this is,"},{"Start":"01:00.170 ","End":"01:03.449","Text":"because times 0 it\u0027s just 0."},{"Start":"01:03.460 ","End":"01:07.273","Text":"Now, collect like terms,"},{"Start":"01:07.273 ","End":"01:11.995","Text":"x minus 2x is minus x,"},{"Start":"01:11.995 ","End":"01:16.720","Text":"and the minus 1 goes to the other side and becomes 1."},{"Start":"01:16.720 ","End":"01:19.830","Text":"So minus x is 1,"},{"Start":"01:19.830 ","End":"01:24.355","Text":"so x is minus 1."},{"Start":"01:24.355 ","End":"01:27.815","Text":"That was just from dividing both sides by minus 1,"},{"Start":"01:27.815 ","End":"01:31.020","Text":"and this is our answer, and we\u0027re done."}],"ID":4949},{"Watched":false,"Name":"Exercise 9","Duration":"1m 16s","ChapterTopicVideoID":4950,"CourseChapterTopicPlaylistID":3864,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.389","Text":"We have here an equation to solve."},{"Start":"00:03.389 ","End":"00:04.680","Text":"This is the equation."},{"Start":"00:04.680 ","End":"00:06.150","Text":"Let\u0027s solve it down here."},{"Start":"00:06.150 ","End":"00:10.455","Text":"Now, I look at this and I see it\u0027s just a fraction equals a fraction,"},{"Start":"00:10.455 ","End":"00:11.880","Text":"and when this happens,"},{"Start":"00:11.880 ","End":"00:13.590","Text":"there\u0027s little shortcut you can use."},{"Start":"00:13.590 ","End":"00:17.640","Text":"You can say that the product of this diagonal equals the product of this diagonal."},{"Start":"00:17.640 ","End":"00:22.100","Text":"I\u0027ll write that down even formula."},{"Start":"00:22.100 ","End":"00:28.290","Text":"If a/b=c/d, then what I get is that"},{"Start":"00:28.290 ","End":"00:38.115","Text":"this tag and little ad=bc and that always works."},{"Start":"00:38.115 ","End":"00:40.885","Text":"In our case, what we get is,"},{"Start":"00:40.885 ","End":"00:50.880","Text":"this diagonal is 3(x+2x)=x,"},{"Start":"00:50.880 ","End":"00:55.900","Text":"if I expand here, I get 3x+6=x."},{"Start":"00:56.300 ","End":"01:01.575","Text":"Bring the x over and I\u0027ve got 2x."},{"Start":"01:01.575 ","End":"01:04.760","Text":"But I also want to bring the 6 over to the other side,"},{"Start":"01:04.760 ","End":"01:06.530","Text":"so that\u0027s minus 6,"},{"Start":"01:06.530 ","End":"01:08.585","Text":"and if I divide by 2,"},{"Start":"01:08.585 ","End":"01:13.205","Text":"I get x equals minus 6/2 is -3,"},{"Start":"01:13.205 ","End":"01:16.440","Text":"and this is our answer."}],"ID":4950},{"Watched":false,"Name":"Exercise 10","Duration":"1m 39s","ChapterTopicVideoID":4951,"CourseChapterTopicPlaylistID":3864,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"Here we have an equation to solve, I\u0027ll do it over here."},{"Start":"00:04.830 ","End":"00:09.300","Text":"I want to remind you that when you have something like this,"},{"Start":"00:09.300 ","End":"00:11.055","Text":"a fraction equals a fraction,"},{"Start":"00:11.055 ","End":"00:14.085","Text":"we can do what is called cross multiplication."},{"Start":"00:14.085 ","End":"00:18.930","Text":"In general, if a over b is equal to c over d,"},{"Start":"00:18.930 ","End":"00:26.255","Text":"then a times d is equal to b times c. If we apply that here,"},{"Start":"00:26.255 ","End":"00:30.950","Text":"we get that 5 times 3x plus 2,"},{"Start":"00:30.950 ","End":"00:32.800","Text":"I need the brackets,"},{"Start":"00:32.800 ","End":"00:37.600","Text":"equals 4 times 2x minus 1."},{"Start":"00:37.790 ","End":"00:41.400","Text":"Now just it\u0027s a matter of tidying up."},{"Start":"00:41.400 ","End":"00:45.570","Text":"Let\u0027s see, 5 times 3x is 15x,"},{"Start":"00:45.570 ","End":"00:51.180","Text":"5 times 2 is 10 and here 4 times 2x"},{"Start":"00:51.180 ","End":"00:56.955","Text":"is 8x minus 4 times 1, minus 4."},{"Start":"00:56.955 ","End":"01:00.840","Text":"Bring the exes to the left and I\u0027ve"},{"Start":"01:00.840 ","End":"01:11.510","Text":"got 15x"},{"Start":"01:11.510 ","End":"01:12.680","Text":"minus 8x."},{"Start":"01:12.680 ","End":"01:14.420","Text":"Remember every time you cross the equals,"},{"Start":"01:14.420 ","End":"01:15.920","Text":"you change sign,"},{"Start":"01:15.920 ","End":"01:18.605","Text":"so we got minus 10,"},{"Start":"01:18.605 ","End":"01:24.255","Text":"minus 4,15 minus 8 is 7x equals,"},{"Start":"01:24.255 ","End":"01:27.240","Text":"this and this is minus 14,"},{"Start":"01:27.240 ","End":"01:31.500","Text":"and then just divide by 7 and you\u0027ve got that x is equal"},{"Start":"01:31.500 ","End":"01:38.230","Text":"to minus 14 over 7 minus 2 and this is the answer."}],"ID":4951},{"Watched":false,"Name":"Exercise 11","Duration":"2m 8s","ChapterTopicVideoID":4952,"CourseChapterTopicPlaylistID":3864,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.920","Text":"We have to solve this equation here,"},{"Start":"00:01.920 ","End":"00:03.900","Text":"and there\u0027s lots of stuff in the denominator."},{"Start":"00:03.900 ","End":"00:05.250","Text":"I want to get rid of fractions."},{"Start":"00:05.250 ","End":"00:06.855","Text":"Let\u0027s do it over here."},{"Start":"00:06.855 ","End":"00:13.680","Text":"What I want to do is find the common denominator for all the 3 separate denominators."},{"Start":"00:13.680 ","End":"00:18.105","Text":"I\u0027ve got x and x and x^2, so I need x^2."},{"Start":"00:18.105 ","End":"00:22.845","Text":"I\u0027ve got, as far as the numbers go 1 and 6 and 3 so 6 will do."},{"Start":"00:22.845 ","End":"00:26.520","Text":"If I multiply both sides by 6x^2,"},{"Start":"00:26.520 ","End":"00:29.130","Text":"that should get rid of all the fractions."},{"Start":"00:29.130 ","End":"00:34.080","Text":"Let\u0027s see, 6x^2 over 3x^2 is 2,"},{"Start":"00:34.080 ","End":"00:37.740","Text":"so you\u0027ll have to multiply this by 2 and here 6x"},{"Start":"00:37.740 ","End":"00:41.520","Text":"into 6x^2 goes x times because this, after all,"},{"Start":"00:41.520 ","End":"00:46.845","Text":"it\u0027s just 6 times x times x and x goes into 6x^2,"},{"Start":"00:46.845 ","End":"00:48.900","Text":"6x times same reason,"},{"Start":"00:48.900 ","End":"00:54.340","Text":"6x x and the x goes in 6x times."},{"Start":"00:54.670 ","End":"01:01.815","Text":"Now I actually do the multiplication 2x plus 5"},{"Start":"01:01.815 ","End":"01:11.070","Text":"minus x times 1 is just x and 6x times 1 is 6x."},{"Start":"01:11.070 ","End":"01:13.425","Text":"Expand the brackets,"},{"Start":"01:13.425 ","End":"01:21.360","Text":"2x plus 10 twice 5 is 10 minus x equals 6x."},{"Start":"01:21.360 ","End":"01:23.890","Text":"Now you know the rules."},{"Start":"01:23.890 ","End":"01:26.180","Text":"Variables to the left,"},{"Start":"01:26.180 ","End":"01:27.725","Text":"numbers to the right."},{"Start":"01:27.725 ","End":"01:36.695","Text":"I\u0027ve got 2x minus x minus 6x here,"},{"Start":"01:36.695 ","End":"01:39.815","Text":"and numbers to the right, minus 10."},{"Start":"01:39.815 ","End":"01:42.664","Text":"Let\u0027s see, collecting the like terms."},{"Start":"01:42.664 ","End":"01:47.810","Text":"2 minus 1 minus 6 is 2"},{"Start":"01:47.810 ","End":"01:53.590","Text":"minus 7 minus 5x is minus 10."},{"Start":"01:53.590 ","End":"01:56.850","Text":"Now if we divide by the minus 5,"},{"Start":"01:56.850 ","End":"02:05.520","Text":"we get that x equals minus 10 over minus 5 is like 10 over 5, which is 2."},{"Start":"02:05.520 ","End":"02:08.770","Text":"This is the answer."}],"ID":4952},{"Watched":false,"Name":"Exercise 12","Duration":"7m 52s","ChapterTopicVideoID":4953,"CourseChapterTopicPlaylistID":3864,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.105","Text":"Here we have an equation to solve,"},{"Start":"00:03.105 ","End":"00:06.750","Text":"and it\u0027s got a lot of stuff in the denominator."},{"Start":"00:06.750 ","End":"00:08.925","Text":"Let\u0027s work over here,"},{"Start":"00:08.925 ","End":"00:14.010","Text":"and I\u0027ll show you a couple of tricks and how to solve this kind of thing."},{"Start":"00:14.010 ","End":"00:18.510","Text":"When you see that the denominators are quite more complex,"},{"Start":"00:18.510 ","End":"00:24.165","Text":"I recommend factorizing each of the denominators as far as possible."},{"Start":"00:24.165 ","End":"00:31.560","Text":"Let\u0027s see. We have x^2 minus 1. I\u0027ll write all 3 of them."},{"Start":"00:31.560 ","End":"00:33.660","Text":"We have x plus 1,"},{"Start":"00:33.660 ","End":"00:37.905","Text":"and we have 2 minus 2x."},{"Start":"00:37.905 ","End":"00:42.175","Text":"Now this, using the difference of squares formula,"},{"Start":"00:42.175 ","End":"00:45.360","Text":"since it\u0027s x^2 minus 1^2,"},{"Start":"00:45.360 ","End":"00:48.155","Text":"we can write the 2 here, won\u0027t change anything."},{"Start":"00:48.155 ","End":"00:53.595","Text":"This is x minus 1 times x plus 1."},{"Start":"00:53.595 ","End":"00:56.420","Text":"X plus 1, I can\u0027t do anything with it."},{"Start":"00:56.420 ","End":"00:58.340","Text":"It\u0027s just x plus 1."},{"Start":"00:58.340 ","End":"01:01.580","Text":"We\u0027ll write it here as x plus 1, in brackets."},{"Start":"01:01.580 ","End":"01:04.220","Text":"Then it looks like this one."},{"Start":"01:04.220 ","End":"01:06.830","Text":"2 minus 2x, I can take 2 out,"},{"Start":"01:06.830 ","End":"01:12.955","Text":"and I\u0027ll write it as twice 1 minus x."},{"Start":"01:12.955 ","End":"01:16.595","Text":"Now here\u0027s the thing I wanted to show you."},{"Start":"01:16.595 ","End":"01:22.760","Text":"Notice that amongst all these factors on the right I have a 1 minus x,"},{"Start":"01:22.760 ","End":"01:25.415","Text":"and I have an x minus 1."},{"Start":"01:25.415 ","End":"01:29.660","Text":"Now I think that these are very close. Here\u0027s what I mean."},{"Start":"01:29.660 ","End":"01:31.685","Text":"In general in algebra,"},{"Start":"01:31.685 ","End":"01:39.290","Text":"if I have a minus b and b minus a,"},{"Start":"01:39.290 ","End":"01:42.245","Text":"I claim that each one is the negative of the other."},{"Start":"01:42.245 ","End":"01:48.410","Text":"For example, if I do minus of a minus b,"},{"Start":"01:48.410 ","End":"01:54.478","Text":"then this is equal to minus a plus b,"},{"Start":"01:54.478 ","End":"02:01.775","Text":"and minus a plus b with a slight rearrangement of order is the same as b minus a."},{"Start":"02:01.775 ","End":"02:06.935","Text":"What this means is if I put a minus in front of the subtraction,"},{"Start":"02:06.935 ","End":"02:12.305","Text":"it just changes the order."},{"Start":"02:12.305 ","End":"02:15.030","Text":"Just as for example,"},{"Start":"02:15.710 ","End":"02:20.400","Text":"5 minus 3 will equal 3 minus 5."},{"Start":"02:20.400 ","End":"02:22.315","Text":"You can check that out."},{"Start":"02:22.315 ","End":"02:31.610","Text":"What I\u0027d like to do here now is rewrite this as minus 2,"},{"Start":"02:31.610 ","End":"02:35.275","Text":"but then x minus 1."},{"Start":"02:35.275 ","End":"02:41.800","Text":"Once again, if I change the order of a subtraction and throw in a minus,"},{"Start":"02:41.800 ","End":"02:43.930","Text":"it will all work out okay."},{"Start":"02:43.930 ","End":"02:50.110","Text":"The reason I do this is because I\u0027d like to see if I can get some common factors here,"},{"Start":"02:50.110 ","End":"02:52.465","Text":"just like I have x plus 1 here and here,"},{"Start":"02:52.465 ","End":"02:55.645","Text":"I want an x minus 1 here and here."},{"Start":"02:55.645 ","End":"03:00.100","Text":"Now the situation is a lot clearer,"},{"Start":"03:00.100 ","End":"03:05.575","Text":"because all I have to do is collect the factors I need."},{"Start":"03:05.575 ","End":"03:07.675","Text":"Let me color, it\u0027ll be easier."},{"Start":"03:07.675 ","End":"03:11.540","Text":"X plus 1, I have here and here."},{"Start":"03:12.650 ","End":"03:15.820","Text":"I\u0027m going to erase the green,"},{"Start":"03:15.820 ","End":"03:17.310","Text":"it might be confusing."},{"Start":"03:17.310 ","End":"03:20.180","Text":"But I do want to leave the general formula that when I have"},{"Start":"03:20.180 ","End":"03:24.760","Text":"a difference I can put a minus and change the order."},{"Start":"03:24.760 ","End":"03:28.480","Text":"That\u0027s useful. I\u0027ll even put it in a little box."},{"Start":"03:28.480 ","End":"03:32.400","Text":"Reverse the order and throw in a minus."},{"Start":"03:32.400 ","End":"03:36.645","Text":"Back here I have x plus 1 here and here."},{"Start":"03:36.645 ","End":"03:43.995","Text":"I also have x minus 1 here and here."},{"Start":"03:43.995 ","End":"03:49.070","Text":"I also have a minus 2, so there."},{"Start":"03:49.070 ","End":"03:53.210","Text":"Now what I need for a common denominator is something that will"},{"Start":"03:53.210 ","End":"03:58.310","Text":"include everything that\u0027s in all of these 3, but no extra."},{"Start":"03:58.310 ","End":"03:59.975","Text":"Just write this down."},{"Start":"03:59.975 ","End":"04:04.379","Text":"My common denominator, denormal,"},{"Start":"04:04.379 ","End":"04:10.740","Text":"will be, I need x plus 1."},{"Start":"04:12.050 ","End":"04:16.905","Text":"That will cover this and this, and that\u0027s it."},{"Start":"04:16.905 ","End":"04:19.465","Text":"I\u0027ll need an x minus 1."},{"Start":"04:19.465 ","End":"04:22.924","Text":"Now I\u0027ve completely covered this and this. I\u0027ve got everything."},{"Start":"04:22.924 ","End":"04:24.560","Text":"I\u0027ve also got this covered."},{"Start":"04:24.560 ","End":"04:27.995","Text":"I don\u0027t have this covered because I still need a minus 2."},{"Start":"04:27.995 ","End":"04:29.675","Text":"I\u0027ve got the x minus 1 covered."},{"Start":"04:29.675 ","End":"04:31.850","Text":"If I put a minus 2 here,"},{"Start":"04:31.850 ","End":"04:34.910","Text":"this should be my common denominator."},{"Start":"04:34.910 ","End":"04:37.530","Text":"I\u0027ll just color it, there,"},{"Start":"04:37.530 ","End":"04:41.195","Text":"there, and there."},{"Start":"04:41.195 ","End":"04:43.825","Text":"I\u0027ve got these covered."},{"Start":"04:43.825 ","End":"04:47.390","Text":"Now I\u0027m going to put in the missing stuff."},{"Start":"04:47.390 ","End":"04:50.090","Text":"Let me just write lines here so you\u0027ll see where I\u0027m going."},{"Start":"04:50.090 ","End":"04:53.255","Text":"I want to take each one of them separately and look."},{"Start":"04:53.255 ","End":"04:56.270","Text":"Now x^2 minus 1 is here."},{"Start":"04:56.270 ","End":"04:59.940","Text":"I\u0027ve got x minus 1 times x plus 1."},{"Start":"04:59.940 ","End":"05:04.065","Text":"All I\u0027m missing to get this is the minus 2."},{"Start":"05:04.065 ","End":"05:06.080","Text":"See, this is here, this is here."},{"Start":"05:06.080 ","End":"05:07.520","Text":"I\u0027m missing a minus 2,"},{"Start":"05:07.520 ","End":"05:09.095","Text":"so that goes here."},{"Start":"05:09.095 ","End":"05:13.460","Text":"Next one. The denominator is x plus 1, that\u0027s here."},{"Start":"05:13.460 ","End":"05:15.215","Text":"I\u0027ve got the yellow covered,"},{"Start":"05:15.215 ","End":"05:17.180","Text":"I need this and this still."},{"Start":"05:17.180 ","End":"05:23.980","Text":"I put here minus 2 times x minus 1."},{"Start":"05:23.980 ","End":"05:27.350","Text":"Then the last one, 2 minus 2x, that\u0027s here."},{"Start":"05:27.350 ","End":"05:28.910","Text":"That\u0027s this one."},{"Start":"05:28.910 ","End":"05:30.350","Text":"I\u0027ve got one of these."},{"Start":"05:30.350 ","End":"05:31.640","Text":"I\u0027ve got one of these."},{"Start":"05:31.640 ","End":"05:33.590","Text":"What I\u0027m missing is this one,"},{"Start":"05:33.590 ","End":"05:35.945","Text":"is the x plus 1."},{"Start":"05:35.945 ","End":"05:38.960","Text":"That\u0027s basically the difficult part."},{"Start":"05:38.960 ","End":"05:40.730","Text":"The rest of it is now standard,"},{"Start":"05:40.730 ","End":"05:42.905","Text":"so let\u0027s just get to it."},{"Start":"05:42.905 ","End":"05:45.485","Text":"Let me just move this out the way."},{"Start":"05:45.485 ","End":"05:48.270","Text":"Let\u0027s put this over here."},{"Start":"05:49.670 ","End":"05:55.935","Text":"Minus 2 times 7 for this."},{"Start":"05:55.935 ","End":"06:01.720","Text":"Then minus 2 times x minus 1 times 2,"},{"Start":"06:03.530 ","End":"06:06.420","Text":"and plus x plus"},{"Start":"06:06.420 ","End":"06:12.765","Text":"1 times"},{"Start":"06:12.765 ","End":"06:17.350","Text":"3 equals something times 0."},{"Start":"06:17.350 ","End":"06:21.670","Text":"This is the common denominator."},{"Start":"06:21.670 ","End":"06:23.859","Text":"I\u0027ll just write it as common denominator."},{"Start":"06:23.859 ","End":"06:26.545","Text":"But it doesn\u0027t matter what it is is what I\u0027m saying."},{"Start":"06:26.545 ","End":"06:28.915","Text":"It should have been from here,"},{"Start":"06:28.915 ","End":"06:31.735","Text":"but because I\u0027m multiplying by 0,"},{"Start":"06:31.735 ","End":"06:35.280","Text":"this times 0, it\u0027s just 0."},{"Start":"06:35.280 ","End":"06:37.575","Text":"It doesn\u0027t matter what this is."},{"Start":"06:37.575 ","End":"06:42.645","Text":"Let\u0027s expand, minus 14."},{"Start":"06:42.645 ","End":"06:47.340","Text":"This is, minus 2 with the 2 is minus 4,"},{"Start":"06:47.340 ","End":"06:49.080","Text":"so it\u0027s minus 4x,"},{"Start":"06:49.080 ","End":"06:52.230","Text":"and then plus 4,"},{"Start":"06:52.230 ","End":"06:54.630","Text":"minus minus is plus."},{"Start":"06:54.630 ","End":"07:03.210","Text":"Then plus 3x plus 3 equals 0."},{"Start":"07:03.210 ","End":"07:11.550","Text":"X\u0027s on the left and numbers on the right."},{"Start":"07:11.550 ","End":"07:12.910","Text":"What have I got on the left?"},{"Start":"07:12.910 ","End":"07:19.015","Text":"I\u0027ve got minus 4x plus 3x,"},{"Start":"07:19.015 ","End":"07:21.460","Text":"and on the right, that\u0027s everything."},{"Start":"07:21.460 ","End":"07:30.120","Text":"It\u0027s 14 minus 4 minus 3."},{"Start":"07:30.120 ","End":"07:32.445","Text":"Let\u0027s collect and see."},{"Start":"07:32.445 ","End":"07:36.015","Text":"Minus 4x plus 3x is minus x,"},{"Start":"07:36.015 ","End":"07:40.800","Text":"14 minus 4 minus 3 is 7."},{"Start":"07:40.800 ","End":"07:47.400","Text":"If I divide by minus 1, I\u0027ve got x=-7."},{"Start":"07:47.400 ","End":"07:52.450","Text":"This is our answer. We are done."}],"ID":4953},{"Watched":false,"Name":"Exercise 13","Duration":"6m 41s","ChapterTopicVideoID":4954,"CourseChapterTopicPlaylistID":3864,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"In this exercise, we have to solve this equation."},{"Start":"00:03.600 ","End":"00:08.235","Text":"The main feature of this equation is the denominators that we want to get rid of."},{"Start":"00:08.235 ","End":"00:10.900","Text":"They\u0027re a little bit complicated."},{"Start":"00:11.480 ","End":"00:14.850","Text":"I\u0027ll work down here. Let\u0027s first of all find what is"},{"Start":"00:14.850 ","End":"00:17.760","Text":"a good common denominator, hopefully the least."},{"Start":"00:17.760 ","End":"00:19.950","Text":"The whole business with the denominators,"},{"Start":"00:19.950 ","End":"00:21.615","Text":"I\u0027m going to do it the side."},{"Start":"00:21.615 ","End":"00:25.410","Text":"First of all, let me write the denominators involved."},{"Start":"00:25.410 ","End":"00:29.100","Text":"One of them is 2 minus x^2."},{"Start":"00:29.100 ","End":"00:31.275","Text":"I\u0027ll see what this is in a moment."},{"Start":"00:31.275 ","End":"00:35.034","Text":"The other one is 12 minus 3x,"},{"Start":"00:35.034 ","End":"00:37.185","Text":"^2, and we\u0027ll see what this is."},{"Start":"00:37.185 ","End":"00:39.660","Text":"We did a similar thing in the previous exercise."},{"Start":"00:39.660 ","End":"00:43.970","Text":"What I\u0027d like to do is factorize the denominator and then I\u0027ll"},{"Start":"00:43.970 ","End":"00:48.875","Text":"take all handpick old and all the bits that I need to find the common denominator."},{"Start":"00:48.875 ","End":"00:50.705","Text":"Now if I factorize this,"},{"Start":"00:50.705 ","End":"00:54.695","Text":"this is just 2 minus x times 2 minus x."},{"Start":"00:54.695 ","End":"00:56.600","Text":"That\u0027s what it means to be squared."},{"Start":"00:56.600 ","End":"00:59.135","Text":"I\u0027m just writing it out."},{"Start":"00:59.135 ","End":"01:03.770","Text":"Here, the first thing I can do is take out a constant 3."},{"Start":"01:03.770 ","End":"01:05.360","Text":"If I do that,"},{"Start":"01:05.360 ","End":"01:07.290","Text":"I\u0027ll get, let\u0027s see,"},{"Start":"01:07.290 ","End":"01:13.900","Text":"3 out of 12 leaves me 4 and just 1 x^2."},{"Start":"01:13.900 ","End":"01:17.875","Text":"This 4 minus x^2, I want to do it at the side,"},{"Start":"01:17.875 ","End":"01:20.285","Text":"because 4 minus x^2,"},{"Start":"01:20.285 ","End":"01:23.945","Text":"4 I can write as 2^2 minus x^2."},{"Start":"01:23.945 ","End":"01:25.715","Text":"Does this remind you of anything?"},{"Start":"01:25.715 ","End":"01:28.445","Text":"It should remind you of"},{"Start":"01:28.445 ","End":"01:30.030","Text":"the"},{"Start":"01:40.840 ","End":"01:42.260","Text":"difference"},{"Start":"01:42.260 ","End":"01:42.830","Text":"of squares"},{"Start":"01:42.830 ","End":"01:48.755","Text":"formula that if we have a^2 minus b^2,"},{"Start":"01:48.755 ","End":"01:56.390","Text":"that this factorizes as a plus b, a minus b."},{"Start":"01:56.390 ","End":"01:58.745","Text":"Or sometimes we write it in the other order."},{"Start":"01:58.745 ","End":"02:00.980","Text":"A minus b, a plus b."},{"Start":"02:00.980 ","End":"02:02.600","Text":"It doesn\u0027t really matter,"},{"Start":"02:02.600 ","End":"02:05.060","Text":"as long as 1 is a plus and 1 is a minus."},{"Start":"02:05.060 ","End":"02:07.534","Text":"Now if I apply this here,"},{"Start":"02:07.534 ","End":"02:13.570","Text":"what I get is that this is equal to 2 plus x,"},{"Start":"02:13.570 ","End":"02:20.505","Text":"2 minus x. I mean the a is the 2 and the b is the x."},{"Start":"02:20.505 ","End":"02:23.175","Text":"When I go back over here,"},{"Start":"02:23.175 ","End":"02:30.855","Text":"I can now write it as 3 and here times 2 plus x,"},{"Start":"02:30.855 ","End":"02:32.870","Text":"and 2 minus x."},{"Start":"02:32.870 ","End":"02:35.645","Text":"That\u0027s writing this as this."},{"Start":"02:35.645 ","End":"02:40.160","Text":"Now I want to cover all possible factors here."},{"Start":"02:40.160 ","End":"02:43.415","Text":"When I get my common denominator,"},{"Start":"02:43.415 ","End":"02:46.965","Text":"I\u0027ll just call it cd for the moment."},{"Start":"02:46.965 ","End":"02:48.390","Text":"A common denominator,"},{"Start":"02:48.390 ","End":"02:50.570","Text":"what I have to do is take all possible factors."},{"Start":"02:50.570 ","End":"02:52.630","Text":"I\u0027ll use colors here."},{"Start":"02:52.630 ","End":"02:57.570","Text":"Here, I have a 2 minus x and another 2 minus x."},{"Start":"02:57.570 ","End":"03:00.470","Text":"Here, I have a 2 minus x,"},{"Start":"03:00.470 ","End":"03:04.225","Text":"but I also have a 2 plus x,"},{"Start":"03:04.225 ","End":"03:06.420","Text":"and I also have a 3."},{"Start":"03:06.420 ","End":"03:10.310","Text":"I need to make sure that all of these go into the common denominator."},{"Start":"03:10.310 ","End":"03:13.075","Text":"I have to have at least 2 minus x,"},{"Start":"03:13.075 ","End":"03:17.000","Text":"2 minus x in order to accommodate the first one."},{"Start":"03:17.000 ","End":"03:18.545","Text":"For the second one,"},{"Start":"03:18.545 ","End":"03:20.690","Text":"I already have a 2 minus x,"},{"Start":"03:20.690 ","End":"03:23.525","Text":"but I still need the other factors."},{"Start":"03:23.525 ","End":"03:26.060","Text":"I need a 3 times 2 plus x,"},{"Start":"03:26.060 ","End":"03:27.635","Text":"let me put the 3 here."},{"Start":"03:27.635 ","End":"03:29.720","Text":"I don\u0027t like to put numbers in the middle."},{"Start":"03:29.720 ","End":"03:36.585","Text":"Then 2 plus x. I colored this while you weren\u0027t looking."},{"Start":"03:36.585 ","End":"03:39.410","Text":"Now that we have the common denominator,"},{"Start":"03:39.410 ","End":"03:45.160","Text":"I have to see how many times each one goes into the common denominator."},{"Start":"03:45.160 ","End":"03:48.435","Text":"This one goes into this one."},{"Start":"03:48.435 ","End":"03:52.455","Text":"What\u0027s left is the 3 times 2 plus x."},{"Start":"03:52.455 ","End":"03:56.820","Text":"This bit I write over here,"},{"Start":"03:56.820 ","End":"04:03.320","Text":"I write it 3 times 2 plus x."},{"Start":"04:03.320 ","End":"04:08.180","Text":"This is the missing bits that I needed to complete this to this,"},{"Start":"04:08.180 ","End":"04:11.390","Text":"and I have to multiply it by this numerator."},{"Start":"04:11.390 ","End":"04:15.100","Text":"Now, for the second one,"},{"Start":"04:15.100 ","End":"04:17.030","Text":"I have the 3,"},{"Start":"04:17.030 ","End":"04:18.230","Text":"I have this and I have this."},{"Start":"04:18.230 ","End":"04:21.575","Text":"I\u0027m just missing one more yellow one."},{"Start":"04:21.575 ","End":"04:23.135","Text":"I\u0027m missing from here to here,"},{"Start":"04:23.135 ","End":"04:28.700","Text":"a 2 minus x. I have to multiply it by this."},{"Start":"04:28.700 ","End":"04:30.875","Text":"Here, because I have a 0,"},{"Start":"04:30.875 ","End":"04:32.600","Text":"doesn\u0027t matter what I multiply."},{"Start":"04:32.600 ","End":"04:34.940","Text":"I should multiply by the whole common denominator,"},{"Start":"04:34.940 ","End":"04:40.175","Text":"but because it\u0027s a 0, you can multiply it by anything and it will still be 0."},{"Start":"04:40.175 ","End":"04:41.870","Text":"Maybe I\u0027ll go over it once more."},{"Start":"04:41.870 ","End":"04:44.420","Text":"It\u0027s important. I looked at each of these."},{"Start":"04:44.420 ","End":"04:48.265","Text":"This one here, is this here."},{"Start":"04:48.265 ","End":"04:53.660","Text":"I see what factors are missing from it in order to make it the common denominator."},{"Start":"04:53.660 ","End":"04:58.040","Text":"I see that this and this are missing and that\u0027s what I put here,"},{"Start":"04:58.040 ","End":"04:59.405","Text":"what it needs to complete."},{"Start":"04:59.405 ","End":"05:02.870","Text":"If I look at this, the only thing that\u0027s missing from this to be"},{"Start":"05:02.870 ","End":"05:06.620","Text":"this is an extra 2 minus x, which I put here."},{"Start":"05:06.620 ","End":"05:15.535","Text":"Normally, I would put over here the whole of the common denominator."},{"Start":"05:15.535 ","End":"05:19.775","Text":"But I don\u0027t need to here because this is 0."},{"Start":"05:19.775 ","End":"05:23.630","Text":"Now comes the routine part of just multiplying out."},{"Start":"05:23.630 ","End":"05:31.845","Text":"What I get is 3 times 3 times 2 plus x. I\u0027ll just write it,3 times 3 times 2 plus x."},{"Start":"05:31.845 ","End":"05:33.135","Text":"That covers the first."},{"Start":"05:33.135 ","End":"05:35.715","Text":"The second is 2 minus x times 5."},{"Start":"05:35.715 ","End":"05:38.425","Text":"I prefer to write the 5 in front."},{"Start":"05:38.425 ","End":"05:42.125","Text":"Here, as we said, whatever it was times 0 it\u0027s 0."},{"Start":"05:42.125 ","End":"05:44.330","Text":"But if it wasn\u0027t 0, if it was 1 here,"},{"Start":"05:44.330 ","End":"05:46.070","Text":"or 7 or whatever,"},{"Start":"05:46.070 ","End":"05:49.165","Text":"you would multiply by the common denominator."},{"Start":"05:49.165 ","End":"05:52.950","Text":"Let\u0027s open brackets and collect the like terms."},{"Start":"05:52.950 ","End":"05:56.370","Text":"From here, 3 times 3 is 9."},{"Start":"05:56.370 ","End":"05:59.080","Text":"In fact, I\u0027ll immediately think of this as 9,"},{"Start":"05:59.080 ","End":"06:02.880","Text":"9 times 2 is 18,"},{"Start":"06:02.880 ","End":"06:06.840","Text":"9 times x is 9x,"},{"Start":"06:06.840 ","End":"06:09.585","Text":"5 times 2 is 10,"},{"Start":"06:09.585 ","End":"06:13.945","Text":"5 times minus x is minus 5x=0."},{"Start":"06:13.945 ","End":"06:17.105","Text":"Leave the x\u0027s on the left and move the numbers to the right."},{"Start":"06:17.105 ","End":"06:22.035","Text":"X\u0027s I have 9 minus 5 let\u0027s just skip a step."},{"Start":"06:22.035 ","End":"06:29.300","Text":"9 minus 5 is 4x and 18 and 10 is 28,"},{"Start":"06:29.300 ","End":"06:34.385","Text":"which goes to the other side as minus 28 and dividing by 4,"},{"Start":"06:34.385 ","End":"06:37.865","Text":"I get that x equals minus 7."},{"Start":"06:37.865 ","End":"06:41.730","Text":"This is the answer. So we are done."}],"ID":4954},{"Watched":false,"Name":"Exercise 14 parts 1-3","Duration":"13m 32s","ChapterTopicVideoID":4955,"CourseChapterTopicPlaylistID":3864,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.140 ","End":"00:04.880","Text":"In this exercise, there are 6 equations to solve."},{"Start":"00:04.880 ","End":"00:07.200","Text":"In each of them, not only do we have to solve them,"},{"Start":"00:07.200 ","End":"00:11.640","Text":"but also to check or verify the solution at the end."},{"Start":"00:11.640 ","End":"00:14.655","Text":"Let\u0027s start with Number 1."},{"Start":"00:14.655 ","End":"00:18.240","Text":"I\u0027ve written them again with more room here."},{"Start":"00:18.240 ","End":"00:21.975","Text":"Let\u0027s start by expanding the brackets."},{"Start":"00:21.975 ","End":"00:27.180","Text":"We have 4x plus 1 equals 7 times 2 is"},{"Start":"00:27.180 ","End":"00:32.850","Text":"14 minus 7x plus 3x plus 3."},{"Start":"00:32.850 ","End":"00:36.510","Text":"Then we separate the variables and the numbers,"},{"Start":"00:36.510 ","End":"00:38.245","Text":"variables to the left."},{"Start":"00:38.245 ","End":"00:40.620","Text":"What we have on the left is 4x,"},{"Start":"00:40.620 ","End":"00:46.640","Text":"and then the 7x comes over as plus and the 3x as minus."},{"Start":"00:46.640 ","End":"00:50.030","Text":"On the right-hand side we have the 14 from here,"},{"Start":"00:50.030 ","End":"00:54.665","Text":"we have the 3 from here and the 1 comes over as minus,"},{"Start":"00:54.665 ","End":"01:00.420","Text":"then we"},{"Start":"01:00.420 ","End":"01:05.385","Text":"get 4 plus 7 minus 3 is,"},{"Start":"01:05.385 ","End":"01:08.310","Text":"see 11 minus 3 is 8x."},{"Start":"01:08.310 ","End":"01:16.335","Text":"8x is equal to 14 plus 3 minus 1,"},{"Start":"01:16.335 ","End":"01:19.500","Text":"17 minus 1 is 16,"},{"Start":"01:19.500 ","End":"01:29.260","Text":"and then divide by 8."},{"Start":"01:29.260 ","End":"01:35.870","Text":"Then clearly, we should divide by 8 and get that x=2."},{"Start":"01:35.870 ","End":"01:38.135","Text":"That\u0027s the solving part."},{"Start":"01:38.135 ","End":"01:41.230","Text":"Now let\u0027s go to the verification part."},{"Start":"01:41.230 ","End":"01:49.790","Text":"Now I have this equation and I want to substitute x=2 and see if it comes out true."},{"Start":"01:49.790 ","End":"01:55.015","Text":"I get 4 times 2 plus 1 equals,"},{"Start":"01:55.015 ","End":"01:58.115","Text":"but equals with a question mark because now I\u0027m checking it."},{"Start":"01:58.115 ","End":"02:00.870","Text":"I could\u0027ve made a mistake here,"},{"Start":"02:01.150 ","End":"02:06.500","Text":"equals 7 times 2 minus 2."},{"Start":"02:06.500 ","End":"02:14.190","Text":"Everywhere I see x, I put 2 plus 3 times 2 plus 3."},{"Start":"02:14.190 ","End":"02:20.835","Text":"Let\u0027s see, the left-hand side is 4 times 2 is 8 plus 1 is 9."},{"Start":"02:20.835 ","End":"02:24.290","Text":"Let\u0027s see if this equals the right-hand side."},{"Start":"02:24.290 ","End":"02:26.750","Text":"The right-hand side, well, 2 minus 2 is 0,"},{"Start":"02:26.750 ","End":"02:28.535","Text":"so this whole bit is 0."},{"Start":"02:28.535 ","End":"02:30.560","Text":"I\u0027ve got 3 times 2 plus 3,"},{"Start":"02:30.560 ","End":"02:31.940","Text":"which is 6 plus 3,"},{"Start":"02:31.940 ","End":"02:33.155","Text":"which is 9,"},{"Start":"02:33.155 ","End":"02:35.240","Text":"and the answer is yes,"},{"Start":"02:35.240 ","End":"02:42.630","Text":"it is equal 2 and so x=2 is a solution verified."},{"Start":"02:42.630 ","End":"02:45.345","Text":"Onto the next one."},{"Start":"02:45.345 ","End":"02:48.090","Text":"This equation is not an x,"},{"Start":"02:48.090 ","End":"02:50.510","Text":"it\u0027s an m. No problem."},{"Start":"02:50.510 ","End":"02:52.370","Text":"Open brackets."},{"Start":"02:52.370 ","End":"02:57.775","Text":"Twice m plus 4 is 2m plus 8."},{"Start":"02:57.775 ","End":"03:05.685","Text":"Then plus 6 equals 4 times 10 is 40 minus 4 times 2 is 8m."},{"Start":"03:05.685 ","End":"03:08.805","Text":"M\u0027s to the left, numbers to the right."},{"Start":"03:08.805 ","End":"03:15.005","Text":"2m from here, plus 8m from here."},{"Start":"03:15.005 ","End":"03:17.585","Text":"Then here we already had the 40,"},{"Start":"03:17.585 ","End":"03:21.350","Text":"8 comes over as minus 6 comes over as minus."},{"Start":"03:21.350 ","End":"03:30.350","Text":"What we have here is 10m equals, let\u0027s see now."},{"Start":"03:30.350 ","End":"03:39.020","Text":"40 minus 14 is 26."},{"Start":"03:39.020 ","End":"03:44.060","Text":"Then divide both sides by m. You know what?"},{"Start":"03:44.060 ","End":"03:48.680","Text":"I\u0027ll do it as a decimal because 10 is so easy to divide by the decimal."},{"Start":"03:48.680 ","End":"03:53.900","Text":"It\u0027s 2.6. Now, I want to check if this is"},{"Start":"03:53.900 ","End":"04:00.530","Text":"a solution to this equation."},{"Start":"04:00.530 ","End":"04:06.005","Text":"Let\u0027s substitute m=2.6 here."},{"Start":"04:06.005 ","End":"04:10.430","Text":"I\u0027ve got twice. Now m is 2.6."},{"Start":"04:10.430 ","End":"04:13.380","Text":"This is a decimal not a multiplication,"},{"Start":"04:13.380 ","End":"04:16.030","Text":"lets just be careful here."},{"Start":"04:16.400 ","End":"04:25.310","Text":"Plus 4 plus 6 equals question mark because that\u0027s what I\u0027m checking."},{"Start":"04:25.310 ","End":"04:31.915","Text":"Is it equal to 4 times 10 plus 2 times,"},{"Start":"04:31.915 ","End":"04:37.310","Text":"here I\u0027ll emphasize is a multiplication and this is 2.6."},{"Start":"04:37.310 ","End":"04:40.085","Text":"Perhaps I should have done it in fractions"},{"Start":"04:40.085 ","End":"04:45.320","Text":"because we can confuse the multiplication point with a decimal point."},{"Start":"04:45.320 ","End":"04:48.185","Text":"Let\u0027s be careful and not confuse them."},{"Start":"04:48.185 ","End":"04:53.420","Text":"2.6 plus 4 is 6.6,"},{"Start":"04:53.420 ","End":"04:57.230","Text":"6.6 times 2 is 13.2."},{"Start":"04:57.230 ","End":"04:59.300","Text":"If you\u0027re not good at doing this sort of thing in your head,"},{"Start":"04:59.300 ","End":"05:01.820","Text":"you can use the calculator."},{"Start":"05:01.820 ","End":"05:04.895","Text":"Anyway, 13.2, we said,"},{"Start":"05:04.895 ","End":"05:09.020","Text":"plus 6, that makes this 19.2."},{"Start":"05:09.020 ","End":"05:12.780","Text":"Now let\u0027s see if this equals the right-hand side."},{"Start":"05:14.030 ","End":"05:17.190","Text":"This is not a plus,"},{"Start":"05:17.190 ","End":"05:19.245","Text":"this is a minus I\u0027m sorry."},{"Start":"05:19.245 ","End":"05:24.690","Text":"Well, that\u0027s better. Twice 2.6 is 5.2."},{"Start":"05:24.690 ","End":"05:28.105","Text":"5.2 from 10 is 4.8."},{"Start":"05:28.105 ","End":"05:29.710","Text":"I\u0027ll just make a note of that."},{"Start":"05:29.710 ","End":"05:31.165","Text":"This part is 4.8."},{"Start":"05:31.165 ","End":"05:37.815","Text":"4 times 4.8 is indeed 19.2,"},{"Start":"05:37.815 ","End":"05:45.440","Text":"so yes, I would say these are equal and so this is a solution."},{"Start":"05:45.510 ","End":"05:50.210","Text":"2 down 4 more to go."},{"Start":"05:50.660 ","End":"05:53.740","Text":"Number 3 has fractions in it."},{"Start":"05:53.740 ","End":"05:56.770","Text":"The first thing we want to do is get rid of the fractions"},{"Start":"05:56.770 ","End":"06:00.455","Text":"by multiplying by a common denominator."},{"Start":"06:00.455 ","End":"06:02.805","Text":"If you look at 6, 3,"},{"Start":"06:02.805 ","End":"06:04.760","Text":"and 4 and think about it for a moment,"},{"Start":"06:04.760 ","End":"06:10.465","Text":"we\u0027ll see that 12 is the best we can do the smallest common denominator."},{"Start":"06:10.465 ","End":"06:16.120","Text":"Let\u0027s multiply everything by 12."},{"Start":"06:16.250 ","End":"06:23.430","Text":"What we do is we just put some separators here, and what you do is,"},{"Start":"06:23.430 ","End":"06:26.640","Text":"you say 6 into 12 goes twice,"},{"Start":"06:26.640 ","End":"06:29.865","Text":"3 into 12 goes 4 times,"},{"Start":"06:29.865 ","End":"06:32.105","Text":"4 into 12 goes 3 times."},{"Start":"06:32.105 ","End":"06:37.685","Text":"In each case, we saw how many times the denominator goes into this common denominator?"},{"Start":"06:37.685 ","End":"06:39.260","Text":"Because this goes into this twice,"},{"Start":"06:39.260 ","End":"06:44.835","Text":"we multiply this numerator by 2 here by 4 here, by 3, and so on."},{"Start":"06:44.835 ","End":"06:47.400","Text":"Then we can get rid of the denominator."},{"Start":"06:47.510 ","End":"06:51.250","Text":"What we get is twice 1 minus 2z,"},{"Start":"06:51.250 ","End":"06:55.865","Text":"but we need brackets because it\u0027s the whole numerator."},{"Start":"06:55.865 ","End":"07:04.060","Text":"Then plus 4 times 2 equals 3 times 3z."},{"Start":"07:04.060 ","End":"07:06.440","Text":"I\u0027ll just leave it as that for the moment."},{"Start":"07:06.440 ","End":"07:08.480","Text":"Now let\u0027s multiply out."},{"Start":"07:08.480 ","End":"07:16.200","Text":"We get 2 minus 4z plus 4 times 2 is 8,"},{"Start":"07:16.200 ","End":"07:19.870","Text":"and then 3 times 3z is 9z."},{"Start":"07:19.870 ","End":"07:25.295","Text":"Let\u0027s put the variables on the left and the numbers on the right."},{"Start":"07:25.295 ","End":"07:28.925","Text":"We\u0027ve got minus 4z,"},{"Start":"07:28.925 ","End":"07:35.810","Text":"minus 9z equals on"},{"Start":"07:35.810 ","End":"07:41.470","Text":"the right we have minus 2, minus 8."},{"Start":"07:41.470 ","End":"07:47.655","Text":"Collecting, we have"},{"Start":"07:47.655 ","End":"07:56.700","Text":"minus 4 minus 9 minus 13z is equal to minus 10,"},{"Start":"07:56.700 ","End":"08:04.700","Text":"and finally, the coefficient of z we divide by minus 13."},{"Start":"08:04.700 ","End":"08:10.550","Text":"We\u0027ve got that z is equal to minus 10 over minus 13."},{"Start":"08:10.550 ","End":"08:15.420","Text":"The minus cancels and it\u0027s 10 over 13, a fraction."},{"Start":"08:15.880 ","End":"08:23.720","Text":"What we need to do now is verify the solution by substituting in the original equation."},{"Start":"08:23.720 ","End":"08:25.460","Text":"Let\u0027s substitute over here,"},{"Start":"08:25.460 ","End":"08:28.370","Text":"ignore the orange writing."},{"Start":"08:28.370 ","End":"08:35.990","Text":"Let\u0027s just go to the original equation and put z equals 10 over 13 in it."},{"Start":"08:35.990 ","End":"08:45.525","Text":"What we get is 1 minus twice 10 over 13 over 6,"},{"Start":"08:45.525 ","End":"08:49.620","Text":"plus 2 over 3."},{"Start":"08:49.620 ","End":"09:01.270","Text":"Is this equal to 3 times 10 over 13 over 4?"},{"Start":"09:01.270 ","End":"09:05.120","Text":"We have a bit of work with fractions to do."},{"Start":"09:05.120 ","End":"09:08.270","Text":"What we want to do is work on each one"},{"Start":"09:08.270 ","End":"09:11.360","Text":"separately and try and get it into a simple fraction."},{"Start":"09:11.360 ","End":"09:16.385","Text":"This one, if I just do the numerator,"},{"Start":"09:16.385 ","End":"09:20.305","Text":"the numerator here is,"},{"Start":"09:20.305 ","End":"09:23.300","Text":"one is 13 over 13,"},{"Start":"09:23.300 ","End":"09:26.165","Text":"and this is minus 20 over 13."},{"Start":"09:26.165 ","End":"09:28.775","Text":"Let me do the numerator in 1 step."},{"Start":"09:28.775 ","End":"09:37.910","Text":"Again. 13 over 13 minus twice 10 over 13 is 13 minus 20 is minus 7 over 13."},{"Start":"09:37.910 ","End":"09:44.970","Text":"That\u0027s what I get when I do the numerator over 6 plus 2 over 3."},{"Start":"09:45.010 ","End":"09:47.300","Text":"Is this equal to?"},{"Start":"09:47.300 ","End":"09:49.325","Text":"Again, we\u0027ll just see what the numerator is."},{"Start":"09:49.325 ","End":"09:56.850","Text":"The numerator is 30 over 13 over 4."},{"Start":"09:57.610 ","End":"10:00.470","Text":"I want to compute this."},{"Start":"10:00.470 ","End":"10:01.700","Text":"What we do is we take"},{"Start":"10:01.700 ","End":"10:06.410","Text":"the denominator of the numerator and put it down to the denominator."},{"Start":"10:06.410 ","End":"10:08.360","Text":"What I\u0027m asking is,"},{"Start":"10:08.360 ","End":"10:09.979","Text":"is minus"},{"Start":"10:09.979 ","End":"10:14.450","Text":"7 over 13 times 6"},{"Start":"10:14.450 ","End":"10:20.752","Text":"plus 2 thirds,"},{"Start":"10:20.752 ","End":"10:22.565","Text":"is this equal to?"},{"Start":"10:22.565 ","End":"10:24.830","Text":"Again, I put the 13 down at the bottom."},{"Start":"10:24.830 ","End":"10:30.600","Text":"Is this equal to 30 over 13 times 4."},{"Start":"10:31.540 ","End":"10:34.130","Text":"Let\u0027s see what we get."},{"Start":"10:34.130 ","End":"10:36.870","Text":"Let me just continue over here."},{"Start":"10:37.090 ","End":"10:46.875","Text":"We\u0027ve got minus 7 over 13 times 6 is 78,"},{"Start":"10:46.875 ","End":"10:53.860","Text":"so minus 7 over 78 plus,"},{"Start":"10:54.620 ","End":"11:02.560","Text":"let\u0027s see if we can take 2 on the last bit."},{"Start":"11:03.140 ","End":"11:07.950","Text":"Minus 7 over 78 plus 2 over 3."},{"Start":"11:07.950 ","End":"11:15.010","Text":"Is this equal to 30 over 52?"},{"Start":"11:16.330 ","End":"11:19.250","Text":"Of course you could do it on the calculator,"},{"Start":"11:19.250 ","End":"11:23.030","Text":"but you\u0027re supposed to not do it on the calculator"},{"Start":"11:23.030 ","End":"11:30.810","Text":"because we want to learn fractions basically, get practice in."},{"Start":"11:30.910 ","End":"11:34.745","Text":"Let\u0027s see if we can do a common denominator."},{"Start":"11:34.745 ","End":"11:37.165","Text":"A common denominator,"},{"Start":"11:37.165 ","End":"11:39.020","Text":"if I look back here,"},{"Start":"11:39.020 ","End":"11:44.495","Text":"really would be easier to say that 13 needs to be in there,"},{"Start":"11:44.495 ","End":"11:48.530","Text":"and I need something with 6 and 4 in it."},{"Start":"11:48.530 ","End":"11:52.410","Text":"I need 13 times 12."},{"Start":"11:52.410 ","End":"11:53.955","Text":"12 will do for 6 and 4."},{"Start":"11:53.955 ","End":"12:02.655","Text":"13 times 12 is 156."},{"Start":"12:02.655 ","End":"12:08.765","Text":"The least common denominator is a 156."},{"Start":"12:08.765 ","End":"12:13.200","Text":"Let\u0027s put each one as a fraction over a 156."},{"Start":"12:13.770 ","End":"12:16.060","Text":"In the case of the first one,"},{"Start":"12:16.060 ","End":"12:21.730","Text":"I have to multiply top and bottom by 2 because 78 goes into a 156 twice."},{"Start":"12:21.730 ","End":"12:27.160","Text":"This one becomes minus 14 over a 156."},{"Start":"12:27.160 ","End":"12:32.490","Text":"This one, 3 goes into 156, 52 times."},{"Start":"12:32.490 ","End":"12:37.630","Text":"52 times 2 is 104."},{"Start":"12:37.630 ","End":"12:42.235","Text":"This is 104 over 156."},{"Start":"12:42.235 ","End":"12:47.780","Text":"These are equal, I\u0027m just rewriting the fraction, expanding the fraction."},{"Start":"12:49.400 ","End":"12:54.015","Text":"Now the right-hand side equals question mark,"},{"Start":"12:54.015 ","End":"12:56.715","Text":"this into this goes 3 times,"},{"Start":"12:56.715 ","End":"13:01.240","Text":"3 times 30 is 90."},{"Start":"13:01.250 ","End":"13:04.610","Text":"What I want to know is, is this equal to this?"},{"Start":"13:04.610 ","End":"13:06.334","Text":"Now on the left-hand side,"},{"Start":"13:06.334 ","End":"13:09.410","Text":"I have a fraction with the same denominator."},{"Start":"13:09.410 ","End":"13:11.120","Text":"When it\u0027s the same denominator,"},{"Start":"13:11.120 ","End":"13:13.715","Text":"we just add or subtract the numerators."},{"Start":"13:13.715 ","End":"13:17.945","Text":"Minus 14 plus 104 is 90."},{"Start":"13:17.945 ","End":"13:21.830","Text":"This side is 90 over 156."},{"Start":"13:21.830 ","End":"13:25.025","Text":"This side is 90 over 156."},{"Start":"13:25.025 ","End":"13:27.380","Text":"They look equal to me,"},{"Start":"13:27.380 ","End":"13:32.130","Text":"and so 10 over 13 has been verified."}],"ID":4955},{"Watched":false,"Name":"Exercise 14 parts 4-6","Duration":"15m 16s","ChapterTopicVideoID":4956,"CourseChapterTopicPlaylistID":3864,"HasSubtitles":false,"VideoComments":[],"Subtitles":[],"ID":4956}],"Thumbnail":null,"ID":3864},{"Name":"Linear Two Unknowns","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Substitution - Part a","Duration":"10m 28s","ChapterTopicVideoID":4971,"CourseChapterTopicPlaylistID":3865,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.060","Text":"In this clip, we\u0027re going to learn about equations in 2 unknowns."},{"Start":"00:05.060 ","End":"00:09.845","Text":"Up to now, we\u0027ve had just x so there any other letter could\u0027ve done."},{"Start":"00:09.845 ","End":"00:12.839","Text":"This time we\u0027re going to have 2 unknowns, 2 variables,"},{"Start":"00:12.839 ","End":"00:14.760","Text":"and they will typically be x and y,"},{"Start":"00:14.760 ","End":"00:17.770","Text":"but any 2 letters will do."},{"Start":"00:18.350 ","End":"00:22.680","Text":"Just to show you what we\u0027re up against,"},{"Start":"00:22.680 ","End":"00:27.390","Text":"I could easily give you an equation and 2 unknowns and this 1 is just a, for instance,"},{"Start":"00:27.390 ","End":"00:31.755","Text":"has even take it from real life,"},{"Start":"00:31.755 ","End":"00:37.290","Text":"2 brothers, I know that the sum of their ages is 10."},{"Start":"00:37.290 ","End":"00:39.690","Text":"How old is each?"},{"Start":"00:39.690 ","End":"00:43.400","Text":"You might say, well let x be the age 1 of them,"},{"Start":"00:43.400 ","End":"00:48.365","Text":"and y be the age of the other and if the sum of their ages is 10,"},{"Start":"00:48.365 ","End":"00:50.885","Text":"I would write x plus y is 10."},{"Start":"00:50.885 ","End":"00:53.420","Text":"But I think you get the feeling that there\u0027s some information"},{"Start":"00:53.420 ","End":"00:56.360","Text":"missing here because there could be many possibilities."},{"Start":"00:56.360 ","End":"00:59.780","Text":"One could be 6-year-old and the other 4-year-old,"},{"Start":"00:59.780 ","End":"01:02.030","Text":"one could be 7 and the other one 3,"},{"Start":"01:02.030 ","End":"01:04.290","Text":"or it could be 8 and 2,"},{"Start":"01:04.290 ","End":"01:05.460","Text":"could be 2 and 8."},{"Start":"01:05.460 ","End":"01:09.575","Text":"A lot of possibilities to get 10."},{"Start":"01:09.575 ","End":"01:11.955","Text":"You see we need an extra clue."},{"Start":"01:11.955 ","End":"01:13.550","Text":"If it\u0027s 2 unknowns,"},{"Start":"01:13.550 ","End":"01:15.364","Text":"we need 2 pieces of information."},{"Start":"01:15.364 ","End":"01:19.025","Text":"Suppose now that I tell you that the difference of their ages,"},{"Start":"01:19.025 ","End":"01:22.040","Text":"the older less the younger is 2 years,"},{"Start":"01:22.040 ","End":"01:23.540","Text":"they are 2 years apart."},{"Start":"01:23.540 ","End":"01:25.940","Text":"Now we\u0027ll say let x be the older one,"},{"Start":"01:25.940 ","End":"01:27.680","Text":"and y is the younger one."},{"Start":"01:27.680 ","End":"01:29.060","Text":"We can always assume that,"},{"Start":"01:29.060 ","End":"01:31.810","Text":"that x minus y=2."},{"Start":"01:31.810 ","End":"01:34.010","Text":"I find now this then in fact,"},{"Start":"01:34.010 ","End":"01:36.980","Text":"one of the possibilities we tried which was 6 and 4,"},{"Start":"01:36.980 ","End":"01:43.775","Text":"actually works because sum of the ages is 10 and the difference of 6 and 4 is 2."},{"Start":"01:43.775 ","End":"01:45.590","Text":"This is typically what we have."},{"Start":"01:45.590 ","End":"01:46.610","Text":"For 2 unknowns,"},{"Start":"01:46.610 ","End":"01:48.470","Text":"we need 2 pieces of information in other words,"},{"Start":"01:48.470 ","End":"01:55.010","Text":"2 equations and together we usually put the one above the other with curly braces around"},{"Start":"01:55.010 ","End":"01:57.680","Text":"and this is called a system of"},{"Start":"01:57.680 ","End":"02:02.915","Text":"linear equations or a system of 2 equations and 2 unknowns x and y."},{"Start":"02:02.915 ","End":"02:06.005","Text":"That\u0027s just the introduction."},{"Start":"02:06.005 ","End":"02:08.615","Text":"Maybe I\u0027ll actually solve this."},{"Start":"02:08.615 ","End":"02:10.835","Text":"I\u0027ll leave it up there meanwhile, while I decide."},{"Start":"02:10.835 ","End":"02:16.135","Text":"But I\u0027ll start first with an easier example."},{"Start":"02:16.135 ","End":"02:20.585","Text":"We\u0027re going to use something called the method of substitution."},{"Start":"02:20.585 ","End":"02:22.895","Text":"There\u0027s going to be 2 main methods."},{"Start":"02:22.895 ","End":"02:25.880","Text":"One is going to be called substitution and"},{"Start":"02:25.880 ","End":"02:30.540","Text":"the other one is going to be called elimination and we have to know both."},{"Start":"02:30.540 ","End":"02:33.170","Text":"We\u0027re going to start off with the method of substitution."},{"Start":"02:33.170 ","End":"02:35.600","Text":"I\u0027ll give the easiest possible example."},{"Start":"02:35.600 ","End":"02:38.890","Text":"We start off slowly and gradually add features."},{"Start":"02:38.890 ","End":"02:43.625","Text":"Let me start with curly braces and I\u0027ll take the first one."},{"Start":"02:43.625 ","End":"02:52.470","Text":"Let\u0027s say 4x plus y=11"},{"Start":"02:52.470 ","End":"02:54.590","Text":"and for the second equation,"},{"Start":"02:54.590 ","End":"03:03.835","Text":"I will take y=3."},{"Start":"03:03.835 ","End":"03:06.655","Text":"Happen to place the equals one above the other."},{"Start":"03:06.655 ","End":"03:08.980","Text":"Notice that in the second equation,"},{"Start":"03:08.980 ","End":"03:10.600","Text":"x doesn\u0027t appear not so right,"},{"Start":"03:10.600 ","End":"03:14.215","Text":"I don\u0027t have to have both of them appearing in both equations."},{"Start":"03:14.215 ","End":"03:20.140","Text":"In fact, this is what makes the second equation easy to solve because in this case,"},{"Start":"03:20.140 ","End":"03:22.180","Text":"though it\u0027s a bit artificial,"},{"Start":"03:22.180 ","End":"03:24.750","Text":"I\u0027ve already given you what y is,"},{"Start":"03:24.750 ","End":"03:26.475","Text":"that y is 3."},{"Start":"03:26.475 ","End":"03:28.690","Text":"Hence the method of substitution."},{"Start":"03:28.690 ","End":"03:34.200","Text":"If I actually find one of the variables in this case y I didn\u0027t find it,"},{"Start":"03:34.200 ","End":"03:36.015","Text":"it was given but anyway,"},{"Start":"03:36.015 ","End":"03:39.640","Text":"so all I have to do is put this into this equation."},{"Start":"03:39.640 ","End":"03:43.275","Text":"We substitute y=3 in the first one."},{"Start":"03:43.275 ","End":"03:50.435","Text":"Then I copy the first equation and say 4x and instead of y, I put 3."},{"Start":"03:50.435 ","End":"03:58.100","Text":"This is what I substituted and then equals 11 and now I have one equation in x so"},{"Start":"03:58.100 ","End":"04:01.970","Text":"the usual variables on the left numbers on the right so"},{"Start":"04:01.970 ","End":"04:08.465","Text":"4x=11,-3 is 8 dividing by 4, x=2."},{"Start":"04:08.465 ","End":"04:10.850","Text":"Now, what do I do with this?"},{"Start":"04:10.850 ","End":"04:14.330","Text":"It\u0027s customary that when you give the answer,"},{"Start":"04:14.330 ","End":"04:16.580","Text":"you give the answer as a pair of x and"},{"Start":"04:16.580 ","End":"04:20.315","Text":"y. I like to also use the curly braces with the answer."},{"Start":"04:20.315 ","End":"04:24.470","Text":"I say the answer is x=2,"},{"Start":"04:24.470 ","End":"04:31.205","Text":"y=3 so an answer or solution is a pair of equalities what x is and what y is,"},{"Start":"04:31.205 ","End":"04:33.305","Text":"if x and y are the variables."},{"Start":"04:33.305 ","End":"04:35.780","Text":"That\u0027s the very first one,"},{"Start":"04:35.780 ","End":"04:39.155","Text":"very easy because we were given one of the variables."},{"Start":"04:39.155 ","End":"04:43.310","Text":"But this actually is a building block to be used later because often it\u0027s not so simple,"},{"Start":"04:43.310 ","End":"04:45.050","Text":"but somehow after hard work,"},{"Start":"04:45.050 ","End":"04:48.230","Text":"we find one of the variables then we can do"},{"Start":"04:48.230 ","End":"04:53.210","Text":"the substitution and find the other and this is the method of substitution."},{"Start":"04:53.210 ","End":"04:56.515","Text":"Let\u0027s take another one with this method."},{"Start":"04:56.515 ","End":"04:59.969","Text":"Next example will be"},{"Start":"04:59.969 ","End":"05:04.335","Text":"-2x plus"},{"Start":"05:04.335 ","End":"05:11.865","Text":"3y=13 curly braces,"},{"Start":"05:11.865 ","End":"05:13.710","Text":"and the second equation will"},{"Start":"05:13.710 ","End":"05:23.200","Text":"be x=y, -4."},{"Start":"05:24.880 ","End":"05:28.460","Text":"This time, I don\u0027t have exactly what one of"},{"Start":"05:28.460 ","End":"05:31.415","Text":"the variables is equal to that\u0027s too much to hope for."},{"Start":"05:31.415 ","End":"05:34.925","Text":"But I do have one variable isolated,"},{"Start":"05:34.925 ","End":"05:40.540","Text":"meaning on its own equaling something without x."},{"Start":"05:45.080 ","End":"05:51.980","Text":"What I can do is to put x into here again by substitution,"},{"Start":"05:51.980 ","End":"05:54.440","Text":"even though it\u0027s not a number, x is still some expression."},{"Start":"05:54.440 ","End":"05:57.275","Text":"I can plug it into the other equation."},{"Start":"05:57.275 ","End":"06:02.135","Text":"Now, this time what I get is -2."},{"Start":"06:02.135 ","End":"06:04.055","Text":"This time I\u0027m replacing x."},{"Start":"06:04.055 ","End":"06:05.420","Text":"Previously I replaced y."},{"Start":"06:05.420 ","End":"06:07.085","Text":"Here, I\u0027m substituting x,"},{"Start":"06:07.085 ","End":"06:10.435","Text":"instead of x, I\u0027ll put y, -4."},{"Start":"06:10.435 ","End":"06:12.205","Text":"That\u0027s this x here,"},{"Start":"06:12.205 ","End":"06:18.000","Text":"I\u0027m putting something equal to it and then the rest of it I copy, +3y=13."},{"Start":"06:18.340 ","End":"06:22.665","Text":"Now, I have an equation in y just one variable."},{"Start":"06:22.665 ","End":"06:27.270","Text":"Open brackets -2y - times - is +,"},{"Start":"06:27.270 ","End":"06:33.875","Text":"8+3y=13 variables on the left, numbers on the right,"},{"Start":"06:33.875 ","End":"06:36.890","Text":"the variable is y this time."},{"Start":"06:36.890 ","End":"06:46.180","Text":"What do we have? We have minus 2y+3y=13 minus 8."},{"Start":"06:46.180 ","End":"06:51.060","Text":"Minus 2y plus 3y is just y,"},{"Start":"06:51.060 ","End":"06:55.680","Text":"1y, which is y and 13-8 is 5."},{"Start":"06:55.680 ","End":"07:03.795","Text":"Now I have y=5 and now that I have y=5,"},{"Start":"07:03.795 ","End":"07:06.489","Text":"like in the previous case just substitute it,"},{"Start":"07:06.489 ","End":"07:08.635","Text":"but where should I substitute it?"},{"Start":"07:08.635 ","End":"07:11.545","Text":"I could substitute it in either one of these 2."},{"Start":"07:11.545 ","End":"07:14.290","Text":"But this somehow looks simpler,"},{"Start":"07:14.290 ","End":"07:17.860","Text":"smaller numbers and also it gives me immediately what"},{"Start":"07:17.860 ","End":"07:21.550","Text":"x is in terms of y so I\u0027ll substitute this one into"},{"Start":"07:21.550 ","End":"07:30.375","Text":"this equation and rewrite this one over here as x = instead of y -4,"},{"Start":"07:30.375 ","End":"07:32.220","Text":"I have y is 5,"},{"Start":"07:32.220 ","End":"07:35.040","Text":"so it\u0027s 5 -4."},{"Start":"07:35.040 ","End":"07:41.280","Text":"That means that x=1."},{"Start":"07:41.280 ","End":"07:45.740","Text":"As usual, like to write the answer with curly brackets."},{"Start":"07:45.740 ","End":"07:47.185","Text":"Let\u0027s say right there over here,"},{"Start":"07:47.185 ","End":"07:50.180","Text":"so the solution is a pair of numbers,"},{"Start":"07:50.180 ","End":"07:53.575","Text":"x and y. X is 1,"},{"Start":"07:53.575 ","End":"07:56.450","Text":"y is 5,"},{"Start":"07:56.450 ","End":"07:58.325","Text":"and that\u0027s the answer to this one."},{"Start":"07:58.325 ","End":"08:00.610","Text":"For the last one,"},{"Start":"08:00.610 ","End":"08:03.980","Text":"why don\u0027t I just take this because I already in"},{"Start":"08:03.980 ","End":"08:07.010","Text":"my head know the answer maybe you do too and maybe you\u0027ve forgotten it."},{"Start":"08:07.010 ","End":"08:09.300","Text":"I won\u0027t tell you again."},{"Start":"08:10.250 ","End":"08:13.320","Text":"Remember this was a story problem, 2 brothers,"},{"Start":"08:13.320 ","End":"08:18.885","Text":"the sum of their ages is 10 and the difference is 2 we did it abstractly here."},{"Start":"08:18.885 ","End":"08:24.520","Text":"I can use this trick here each time I\u0027m building a level."},{"Start":"08:24.520 ","End":"08:28.265","Text":"This time I don\u0027t even have one in terms of the other,"},{"Start":"08:28.265 ","End":"08:29.690","Text":"but with a little effort,"},{"Start":"08:29.690 ","End":"08:32.180","Text":"very small effort I can make it so."},{"Start":"08:32.180 ","End":"08:36.545","Text":"For example, if I took the second equation and brought y to the other side,"},{"Start":"08:36.545 ","End":"08:42.050","Text":"the second equation could become x = bringing"},{"Start":"08:42.050 ","End":"08:48.410","Text":"this over y plus 2 just by taking the y over to the other side."},{"Start":"08:48.410 ","End":"08:51.410","Text":"Now I can use the trick like here,"},{"Start":"08:51.410 ","End":"08:54.020","Text":"substituting this in here again,"},{"Start":"08:54.020 ","End":"08:56.915","Text":"I\u0027m substituting x, we\u0027ll get an equation in y."},{"Start":"08:56.915 ","End":"08:59.690","Text":"Instead of x, I write y+2."},{"Start":"08:59.690 ","End":"09:05.030","Text":"Y+2, I\u0027m just putting it in parentheses for emphasis that this was x+ y=10."},{"Start":"09:05.030 ","End":"09:07.420","Text":"Let\u0027s see, 2y+2,"},{"Start":"09:07.420 ","End":"09:12.420","Text":"y+y=10, 2y=10 minus 2 is 8 so y=4."},{"Start":"09:12.420 ","End":"09:16.056","Text":"I remember that\u0027s what we had,"},{"Start":"09:16.056 ","End":"09:19.292","Text":"the younger brother was 4."},{"Start":"09:19.292 ","End":"09:24.371","Text":"Now again, we have to substitute and it\u0027s"},{"Start":"09:24.371 ","End":"09:30.328","Text":"going to be easier to substitute in this one here,"},{"Start":"09:30.328 ","End":"09:35.486","Text":"because then it\u0027ll give me x right away."},{"Start":"09:35.486 ","End":"09:38.848","Text":"I\u0027ll put this one in here,"},{"Start":"09:38.848 ","End":"09:41.616","Text":"and I\u0027ve got now,"},{"Start":"09:41.616 ","End":"09:46.939","Text":"this one becomes x equals instead of y,"},{"Start":"09:46.939 ","End":"09:48.824","Text":"I put 4 and +2."},{"Start":"09:48.824 ","End":"09:53.050","Text":"Now I just compute this, x=6."},{"Start":"09:53.050 ","End":"09:54.679","Text":"Y is 4, x=6."},{"Start":"09:54.679 ","End":"09:58.216","Text":"If I write it properly,"},{"Start":"09:58.216 ","End":"10:03.778","Text":"2 equalities, but"},{"Start":"10:03.778 ","End":"10:10.963","Text":"it\u0027s all part of one solution, x=6 and y=4."},{"Start":"10:10.963 ","End":"10:14.309","Text":"That\u0027s it for now."},{"Start":"10:14.309 ","End":"10:19.921","Text":"More difficult ones coming"},{"Start":"10:19.921 ","End":"10:26.790","Text":"up don\u0027t worry though."}],"ID":4959},{"Watched":false,"Name":"Substitution - Part b","Duration":"5m 44s","ChapterTopicVideoID":4969,"CourseChapterTopicPlaylistID":3865,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"This is a continuation of the previous clip on"},{"Start":"00:03.570 ","End":"00:06.975","Text":"the method of substitution for linear equations in 2 unknowns."},{"Start":"00:06.975 ","End":"00:08.730","Text":"It\u0027s a continuation."},{"Start":"00:08.730 ","End":"00:12.075","Text":"I\u0027ll indicate that with Part 2."},{"Start":"00:12.075 ","End":"00:14.880","Text":"Let\u0027s do some more example."},{"Start":"00:14.880 ","End":"00:18.510","Text":"Let\u0027s see 2x plus"},{"Start":"00:18.510 ","End":"00:28.600","Text":"3y=5 and 5x plus 7y=11."},{"Start":"00:28.810 ","End":"00:36.110","Text":"As you can see, it\u0027s not very easy to get 1 variable in terms of the other."},{"Start":"00:36.110 ","End":"00:38.660","Text":"All the coefficients are not 1."},{"Start":"00:38.660 ","End":"00:41.990","Text":"There\u0027s no easy way to say x equals or y equals."},{"Start":"00:41.990 ","End":"00:43.565","Text":"We\u0027ll have to work a bit."},{"Start":"00:43.565 ","End":"00:47.060","Text":"In actual fact, this particular system"},{"Start":"00:47.060 ","End":"00:50.150","Text":"of equations is better done with the other method that we\u0027ll learn."},{"Start":"00:50.150 ","End":"00:51.800","Text":"It\u0027s called method of elimination,"},{"Start":"00:51.800 ","End":"00:55.820","Text":"but I want to show you that you can always just use the method of substitution."},{"Start":"00:55.820 ","End":"00:58.160","Text":"It\u0027s just going to be a bit more cumbersome here."},{"Start":"00:58.160 ","End":"01:01.200","Text":"But here it goes."},{"Start":"01:01.200 ","End":"01:03.195","Text":"I don\u0027t have an easy way."},{"Start":"01:03.195 ","End":"01:06.435","Text":"I look at what seems to me the easiest."},{"Start":"01:06.435 ","End":"01:08.950","Text":"I think I\u0027ll take x from the first equation."},{"Start":"01:08.950 ","End":"01:10.795","Text":"I only have to divide it by 2."},{"Start":"01:10.795 ","End":"01:13.540","Text":"Whatever way I do it, I\u0027m going to get fractions."},{"Start":"01:13.540 ","End":"01:14.860","Text":"Here\u0027s how we go about it."},{"Start":"01:14.860 ","End":"01:16.795","Text":"I\u0027ll take the first equation."},{"Start":"01:16.795 ","End":"01:19.570","Text":"In fact, sometimes I even label the equations."},{"Start":"01:19.570 ","End":"01:22.685","Text":"This is Equation 1 and this is Equation 2."},{"Start":"01:22.685 ","End":"01:26.050","Text":"From Equation 1,"},{"Start":"01:26.280 ","End":"01:30.960","Text":"I say 2x equals,"},{"Start":"01:30.960 ","End":"01:33.940","Text":"I bring the 3y over to the other side,"},{"Start":"01:33.940 ","End":"01:37.060","Text":"minus 3y plus 5,"},{"Start":"01:37.060 ","End":"01:39.160","Text":"and then divide by 2,"},{"Start":"01:39.160 ","End":"01:40.420","Text":"bring the 2 over."},{"Start":"01:40.420 ","End":"01:47.510","Text":"I get x equals minus 3y plus 5/2."},{"Start":"01:47.680 ","End":"01:52.429","Text":"Now that I\u0027ve extracted x from the first equation,"},{"Start":"01:52.429 ","End":"01:55.415","Text":"I can now substitute it in the second equation,"},{"Start":"01:55.415 ","End":"01:57.145","Text":"and let\u0027s see what we get."},{"Start":"01:57.145 ","End":"02:01.685","Text":"Here is 5 and then instead of x, I put this,"},{"Start":"02:01.685 ","End":"02:05.910","Text":"which is minus 3y plus"},{"Start":"02:06.250 ","End":"02:13.180","Text":"5/2 and the rest of it as is, plus 7y=11."},{"Start":"02:15.290 ","End":"02:20.240","Text":"What I can see is an integer times a fraction."},{"Start":"02:20.240 ","End":"02:23.885","Text":"I multiply the integer by the numerator."},{"Start":"02:23.885 ","End":"02:26.420","Text":"This is just a slight rewrite of this."},{"Start":"02:26.420 ","End":"02:30.080","Text":"5 times minus 3y plus 5,"},{"Start":"02:30.080 ","End":"02:37.180","Text":"all over 2 plus 7y=11."},{"Start":"02:37.180 ","End":"02:42.755","Text":"Now what I want to do is I see a denominator."},{"Start":"02:42.755 ","End":"02:46.565","Text":"I want to multiply by this denominator,"},{"Start":"02:46.565 ","End":"02:48.770","Text":"just like we used to do in 1 variable."},{"Start":"02:48.770 ","End":"02:53.280","Text":"I write that my common denominator is 2,"},{"Start":"02:53.280 ","End":"02:56.625","Text":"and I\u0027m going to multiply by 2 on the left and on the right."},{"Start":"02:56.625 ","End":"02:59.865","Text":"Multiplying on the left means multiplying each one."},{"Start":"02:59.865 ","End":"03:02.930","Text":"What we do is like we used to do, 2 into 2,"},{"Start":"03:02.930 ","End":"03:06.820","Text":"goes 1 time and write the 1 here."},{"Start":"03:06.820 ","End":"03:09.470","Text":"This is not a fraction,"},{"Start":"03:09.470 ","End":"03:14.105","Text":"so I need to multiply it by the full 2 and here by 2."},{"Start":"03:14.105 ","End":"03:19.070","Text":"Now we get 1 times this is just whatever it is,"},{"Start":"03:19.070 ","End":"03:23.215","Text":"5 times minus 3y plus 5."},{"Start":"03:23.215 ","End":"03:30.035","Text":"Twice 7y is 14y and twice 11 is 22."},{"Start":"03:30.035 ","End":"03:32.030","Text":"Now we\u0027re in familiar territory."},{"Start":"03:32.030 ","End":"03:35.180","Text":"Expand the brackets, 5 by each of these,"},{"Start":"03:35.180 ","End":"03:39.305","Text":"so we get minus 15y plus 25,"},{"Start":"03:39.305 ","End":"03:43.640","Text":"then plus 14y equals 22, as is."},{"Start":"03:43.640 ","End":"03:47.310","Text":"Now variables on the left numbers on the right."},{"Start":"03:47.380 ","End":"03:54.995","Text":"Minus 15y plus 14y equals 22 minus 25."},{"Start":"03:54.995 ","End":"03:57.635","Text":"Let me continue over here."},{"Start":"03:57.635 ","End":"04:04.790","Text":"What we have is minus y here because minus 15 plus 14 is minus 1."},{"Start":"04:04.790 ","End":"04:06.395","Text":"I don\u0027t need to write the 1,"},{"Start":"04:06.395 ","End":"04:12.250","Text":"so it\u0027s just minus y equals minus 3."},{"Start":"04:13.210 ","End":"04:17.150","Text":"Formally, you can just cancel the minuses."},{"Start":"04:17.150 ","End":"04:21.245","Text":"But if you want, it\u0027s dividing both sides by minus 1."},{"Start":"04:21.245 ","End":"04:23.840","Text":"Y=3, and that\u0027s good,"},{"Start":"04:23.840 ","End":"04:26.045","Text":"but we still need to find x."},{"Start":"04:26.045 ","End":"04:31.940","Text":"The easiest thing to do is to substitute it in the place where we extracted x."},{"Start":"04:31.940 ","End":"04:40.156","Text":"In other words a substituted in here and then we just get x straight away, almost."},{"Start":"04:40.156 ","End":"04:44.195","Text":"Because we get x equals just copying but substituting y,"},{"Start":"04:44.195 ","End":"04:47.090","Text":"minus 3 times y,"},{"Start":"04:47.090 ","End":"04:52.910","Text":"which is 3 plus 5/2."},{"Start":"04:52.910 ","End":"05:01.265","Text":"This is equal to minus 9 plus 5/2."},{"Start":"05:01.265 ","End":"05:07.200","Text":"Minus 9 plus 5 is minus 4."},{"Start":"05:07.420 ","End":"05:10.875","Text":"Minus 4/2, the minus stays."},{"Start":"05:10.875 ","End":"05:13.290","Text":"Negative over positive and 4/2 is 2,"},{"Start":"05:13.290 ","End":"05:16.125","Text":"so it\u0027s just minus 2."},{"Start":"05:16.125 ","End":"05:21.150","Text":"Now I have that x equals minus 2."},{"Start":"05:21.150 ","End":"05:25.530","Text":"I have underlined them both x and y."},{"Start":"05:25.530 ","End":"05:29.555","Text":"Finally, somewhere at the side or anywhere I can find space,"},{"Start":"05:29.555 ","End":"05:34.505","Text":"I write the answer is x equals minus 2."},{"Start":"05:34.505 ","End":"05:40.970","Text":"Usually put the x first and the y second and the curly braces and y=3."},{"Start":"05:40.970 ","End":"05:45.359","Text":"This is the solution."}],"ID":4960},{"Watched":false,"Name":"Substitution - Part c","Duration":"6m 20s","ChapterTopicVideoID":4970,"CourseChapterTopicPlaylistID":3865,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"This is a continuation of the previous clip,"},{"Start":"00:03.120 ","End":"00:04.740","Text":"linear equation in 2 unknowns,"},{"Start":"00:04.740 ","End":"00:06.360","Text":"a method of substitution."},{"Start":"00:06.360 ","End":"00:10.215","Text":"We did part 2, now part 3 and the last part."},{"Start":"00:10.215 ","End":"00:15.630","Text":"Let\u0027s take another example,"},{"Start":"00:15.630 ","End":"00:23.370","Text":"y=2x minus 3, y=x plus 2."},{"Start":"00:23.370 ","End":"00:27.600","Text":"This kind of situation occurs often in mathematics."},{"Start":"00:27.600 ","End":"00:33.990","Text":"You\u0027ll see it later on maybe in analytic geometry but never mind that."},{"Start":"00:33.990 ","End":"00:37.500","Text":"Let\u0027s just take it as given this situation occurs,"},{"Start":"00:37.500 ","End":"00:41.875","Text":"which is why I brought an example of y equals something and y equals something."},{"Start":"00:41.875 ","End":"00:44.915","Text":"Now, we can do it by substitution and say,"},{"Start":"00:44.915 ","End":"00:46.505","Text":"\"Okay, the first one,"},{"Start":"00:46.505 ","End":"00:48.755","Text":"y is already in terms of x,"},{"Start":"00:48.755 ","End":"00:55.270","Text":"let\u0027s substitute it in the second and we get 2x minus 3=x plus 2.\""},{"Start":"00:55.270 ","End":"00:58.360","Text":"Of course, you could also look at it in a commonsense way and say,"},{"Start":"00:58.360 ","End":"01:01.670","Text":"\"If these 2 things are each equal to a common thing,"},{"Start":"01:01.670 ","End":"01:04.094","Text":"y, then they\u0027re equal among themselves.\""},{"Start":"01:04.094 ","End":"01:06.980","Text":"You could look at it that way or you could do it by substitution."},{"Start":"01:06.980 ","End":"01:11.075","Text":"In any event, this is now 1 equation with 1 unknown."},{"Start":"01:11.075 ","End":"01:13.340","Text":"Let\u0027s bring the x over to the other side."},{"Start":"01:13.340 ","End":"01:15.380","Text":"You know what, we already can do it in our heads,"},{"Start":"01:15.380 ","End":"01:18.140","Text":"2x minus x is x."},{"Start":"01:18.140 ","End":"01:22.510","Text":"Bring the 3 over, it\u0027s 2 plus 3=5,"},{"Start":"01:22.510 ","End":"01:26.240","Text":"and that\u0027s it for x."},{"Start":"01:26.240 ","End":"01:29.060","Text":"Don\u0027t forget that we also have to find y."},{"Start":"01:29.060 ","End":"01:31.130","Text":"Now, once you\u0027ve got x,"},{"Start":"01:31.130 ","End":"01:34.715","Text":"you can put y either here or here, whichever suits you."},{"Start":"01:34.715 ","End":"01:37.490","Text":"It\u0027s going to be slightly easier with the second I think."},{"Start":"01:37.490 ","End":"01:40.040","Text":"Let\u0027s say that now y=x,"},{"Start":"01:40.040 ","End":"01:46.420","Text":"which is 5 plus 2, so y=7."},{"Start":"01:46.420 ","End":"01:49.860","Text":"I\u0027m not going to bother with the curly braces x=5,"},{"Start":"01:49.860 ","End":"01:52.185","Text":"y=7, that\u0027s the solution."},{"Start":"01:52.185 ","End":"01:56.305","Text":"Now, one more and last example."},{"Start":"01:56.305 ","End":"02:01.655","Text":"I just went ahead and wrote it out to save time."},{"Start":"02:01.655 ","End":"02:06.200","Text":"The thing here is just they make you work a little bit."},{"Start":"02:06.200 ","End":"02:07.670","Text":"If you get an exercise like this,"},{"Start":"02:07.670 ","End":"02:11.120","Text":"it\u0027s just not neatly arranged."},{"Start":"02:11.120 ","End":"02:13.820","Text":"That\u0027s all the difference there is."},{"Start":"02:13.820 ","End":"02:18.980","Text":"Let\u0027s see what happens if we bring everything to the left,"},{"Start":"02:18.980 ","End":"02:23.390","Text":"variables to the left, numbers to the right, or whatever."},{"Start":"02:23.390 ","End":"02:25.845","Text":"Take the first one."},{"Start":"02:25.845 ","End":"02:28.050","Text":"Let\u0027s put the x\u0027s first."},{"Start":"02:28.050 ","End":"02:35.670","Text":"We can do it mentally 4x plus 2x on the left-hand side is 6x."},{"Start":"02:35.670 ","End":"02:39.660","Text":"Now, the y terms,"},{"Start":"02:39.660 ","End":"02:41.310","Text":"I\u0027m bringing them also to the left,"},{"Start":"02:41.310 ","End":"02:44.820","Text":"I\u0027ve got a minus y and I\u0027ve got a minus 2y."},{"Start":"02:44.820 ","End":"02:47.235","Text":"That\u0027s minus 3y."},{"Start":"02:47.235 ","End":"02:48.915","Text":"Numbers to the right,"},{"Start":"02:48.915 ","End":"02:54.195","Text":"plus 15 minus 3 equals 12."},{"Start":"02:54.195 ","End":"02:57.780","Text":"Now, the second 1. To the x\u0027s first,"},{"Start":"02:57.780 ","End":"03:04.240","Text":"minus 4x plus x is minus 3x."},{"Start":"03:04.880 ","End":"03:08.600","Text":"Let\u0027s see the y, we\u0027ve got here plus 2y,"},{"Start":"03:08.600 ","End":"03:15.330","Text":"then we\u0027ll get a minus y is plus y."},{"Start":"03:16.340 ","End":"03:19.920","Text":"Then numbers on the right,"},{"Start":"03:19.920 ","End":"03:25.184","Text":"we have minus 6 and then minus 1 is minus"},{"Start":"03:25.184 ","End":"03:30.860","Text":"7 This is what we get after tidying up."},{"Start":"03:30.860 ","End":"03:34.460","Text":"Now, there\u0027s a recommendation I\u0027d like to make."},{"Start":"03:34.460 ","End":"03:36.920","Text":"That is if you happen to notice,"},{"Start":"03:36.920 ","End":"03:41.060","Text":"I just noticed that everything here is divisible by 3."},{"Start":"03:41.060 ","End":"03:45.290","Text":"Then I recommend doing that and work will go quicker,"},{"Start":"03:45.290 ","End":"03:50.045","Text":"but I\u0027m going to assume that you didn\u0027t notice it and just proceed as usual."},{"Start":"03:50.045 ","End":"03:52.915","Text":"But I\u0027ll just write that down"},{"Start":"03:52.915 ","End":"03:55.910","Text":"and you might want to try this on your own later to continue,"},{"Start":"03:55.910 ","End":"04:00.860","Text":"but I could have written this now is 2x minus y=4,"},{"Start":"04:00.860 ","End":"04:02.750","Text":"if I divide it by 3,"},{"Start":"04:02.750 ","End":"04:05.945","Text":"but I\u0027ll assume that you didn\u0027t notice it."},{"Start":"04:05.945 ","End":"04:07.910","Text":"Now, if I look at this,"},{"Start":"04:07.910 ","End":"04:12.260","Text":"I see that I have a single y here, coefficient 1."},{"Start":"04:12.260 ","End":"04:15.035","Text":"What I\u0027d like to do is isolate y,"},{"Start":"04:15.035 ","End":"04:17.255","Text":"or extract y, and then substitute."},{"Start":"04:17.255 ","End":"04:19.280","Text":"From the second equation,"},{"Start":"04:19.280 ","End":"04:25.830","Text":"I get y equals minus 7 plus 3x."},{"Start":"04:25.830 ","End":"04:27.450","Text":"We\u0027ll write the x\u0027s first,"},{"Start":"04:27.450 ","End":"04:29.940","Text":"plus 3x minus 7,"},{"Start":"04:29.940 ","End":"04:32.595","Text":"that\u0027s by bringing the 3x to this side."},{"Start":"04:32.595 ","End":"04:38.755","Text":"Now, I substitute this after substituting the other 1 in the first equation in here."},{"Start":"04:38.755 ","End":"04:47.310","Text":"Then I will get 6x minus 3 times y is this."},{"Start":"04:47.310 ","End":"04:51.670","Text":"3x minus 7=12."},{"Start":"04:51.670 ","End":"04:55.170","Text":"Expand the brackets."},{"Start":"04:55.170 ","End":"05:00.280","Text":"We\u0027ve got 6x minus 3 times 3 is 9x,"},{"Start":"05:00.280 ","End":"05:05.420","Text":"minus minus is plus 21=12."},{"Start":"05:05.420 ","End":"05:08.000","Text":"x\u0027s to the left, numbers on the right,"},{"Start":"05:08.000 ","End":"05:12.020","Text":"6x minus 9x is minus 3x."},{"Start":"05:12.020 ","End":"05:19.350","Text":"On the right, 12 minus 21 is minus 9."},{"Start":"05:19.660 ","End":"05:21.800","Text":"That gives us,"},{"Start":"05:21.800 ","End":"05:23.540","Text":"I\u0027m running out of space here,"},{"Start":"05:23.540 ","End":"05:29.615","Text":"so gives us that x equals minus 9 over minus 3,"},{"Start":"05:29.615 ","End":"05:31.840","Text":"which is plus 3."},{"Start":"05:31.840 ","End":"05:33.615","Text":"After I have this,"},{"Start":"05:33.615 ","End":"05:37.720","Text":"I now substitute it in this one here."},{"Start":"05:37.720 ","End":"05:40.580","Text":"Because that will give me y straightaway."},{"Start":"05:40.580 ","End":"05:47.135","Text":"I get that y=3 times and x is,"},{"Start":"05:47.135 ","End":"05:50.345","Text":"yeah, I meant to say 3."},{"Start":"05:50.345 ","End":"05:56.220","Text":"There we go. Then minus 7,"},{"Start":"05:56.220 ","End":"05:58.650","Text":"so 3 times 3 is 9,"},{"Start":"05:58.650 ","End":"06:00.705","Text":"9 minus 7 is 2."},{"Start":"06:00.705 ","End":"06:02.895","Text":"So y=2."},{"Start":"06:02.895 ","End":"06:06.040","Text":"So x=3, y=2."},{"Start":"06:06.040 ","End":"06:13.325","Text":"I\u0027m going to skip the curly braces part and we are done with this topic."},{"Start":"06:13.325 ","End":"06:15.305","Text":"Next coming up."},{"Start":"06:15.305 ","End":"06:19.470","Text":"Method of elimination. That\u0027s all."}],"ID":4961},{"Watched":false,"Name":"Elimination - Part a","Duration":"6m 26s","ChapterTopicVideoID":4968,"CourseChapterTopicPlaylistID":3865,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"Here we\u0027re going to continue with linear equation in 2 unknowns, but don\u0027t worry,"},{"Start":"00:03.960 ","End":"00:06.540","Text":"we\u0027re done with the method of substitution,"},{"Start":"00:06.540 ","End":"00:10.120","Text":"and instead of that, we\u0027re going to learn elimination."},{"Start":"00:10.120 ","End":"00:12.825","Text":"We\u0027ll begin with an easy example,"},{"Start":"00:12.825 ","End":"00:15.910","Text":"a basic example for elimination."},{"Start":"00:15.910 ","End":"00:24.450","Text":"Let\u0027s say 2x+3= 5"},{"Start":"00:24.450 ","End":"00:32.170","Text":", and that 2x-3y=3."},{"Start":"00:32.170 ","End":"00:37.430","Text":"What I want you to note here is that,"},{"Start":"00:37.430 ","End":"00:44.555","Text":"we have the same term in opposite signs in these 2 equations."},{"Start":"00:44.555 ","End":"00:47.930","Text":"This is the best case and in this case,"},{"Start":"00:47.930 ","End":"00:51.770","Text":"you almost have to do it by elimination. Well, you don\u0027t have to."},{"Start":"00:51.770 ","End":"00:58.010","Text":"You can always just use substitution but it becomes so easy to do it by elimination."},{"Start":"00:58.010 ","End":"01:01.530","Text":"The idea is this."},{"Start":"01:01.530 ","End":"01:04.900","Text":"Something else you can do with equations that we haven\u0027t talked about before."},{"Start":"01:04.900 ","End":"01:07.150","Text":"If you have 2 equations, you can add them,"},{"Start":"01:07.150 ","End":"01:08.920","Text":"meaning you can add the left-hand side,"},{"Start":"01:08.920 ","End":"01:10.600","Text":"and add the right-hand sides."},{"Start":"01:10.600 ","End":"01:12.640","Text":"If 2 things are equal,"},{"Start":"01:12.640 ","End":"01:15.385","Text":"and these 2 other things are equal,"},{"Start":"01:15.385 ","End":"01:17.200","Text":"then if you add them, they\u0027re equal."},{"Start":"01:17.200 ","End":"01:20.905","Text":"It\u0027s just like saying if 3 equals 3,"},{"Start":"01:20.905 ","End":"01:22.540","Text":"and 6 equals 6,"},{"Start":"01:22.540 ","End":"01:25.925","Text":"and we add them, then 9 equals 9."},{"Start":"01:25.925 ","End":"01:29.830","Text":"It doesn\u0027t make sense in algebra so much, but that\u0027s the idea."},{"Start":"01:29.830 ","End":"01:32.469","Text":"We\u0027re going to add these 2 equations."},{"Start":"01:32.469 ","End":"01:35.290","Text":"The first time I\u0027ll emphasize this,"},{"Start":"01:35.290 ","End":"01:36.880","Text":"I\u0027ll just say, \"Okay,"},{"Start":"01:36.880 ","End":"01:37.960","Text":"I\u0027m putting a plus,"},{"Start":"01:37.960 ","End":"01:41.233","Text":"I\u0027m adding these 2 equations just like we do with arithmetic,"},{"Start":"01:41.233 ","End":"01:43.305","Text":"and put an equals sign.\""},{"Start":"01:43.305 ","End":"01:50.550","Text":"Then what we get if we do the addition is 2x+2x=4x."},{"Start":"01:51.320 ","End":"01:54.830","Text":"The plus 3y and the minus 3y, if you add them,"},{"Start":"01:54.830 ","End":"01:56.750","Text":"cancel each other out or it\u0027s 0,"},{"Start":"01:56.750 ","End":"02:00.140","Text":"but I\u0027m not going to put plus 0, or whatever."},{"Start":"02:00.140 ","End":"02:01.310","Text":"This is the elimination,"},{"Start":"02:01.310 ","End":"02:02.795","Text":"y is eliminated,"},{"Start":"02:02.795 ","End":"02:05.090","Text":"5 plus 3 is 8."},{"Start":"02:05.090 ","End":"02:06.875","Text":"When you have this case,"},{"Start":"02:06.875 ","End":"02:10.160","Text":"add them, eliminated y."},{"Start":"02:10.160 ","End":"02:13.670","Text":"Now we\u0027ve got an equation in 1 variable 4x=8,"},{"Start":"02:13.670 ","End":"02:16.175","Text":"so clearly, x=2."},{"Start":"02:16.175 ","End":"02:18.140","Text":"That\u0027s x found."},{"Start":"02:18.140 ","End":"02:20.130","Text":"Now, we eliminated y,"},{"Start":"02:20.130 ","End":"02:21.840","Text":"but we still want to find it."},{"Start":"02:21.840 ","End":"02:25.325","Text":"I take this one and put it in one of the two above."},{"Start":"02:25.325 ","End":"02:26.690","Text":"I don\u0027t know which,"},{"Start":"02:26.690 ","End":"02:30.830","Text":"let\u0027s just say we put it in this one, it\u0027s closer."},{"Start":"02:30.830 ","End":"02:34.100","Text":"I don\u0027t know. Either one will do of course."},{"Start":"02:34.100 ","End":"02:36.170","Text":"We get in the second one,"},{"Start":"02:36.170 ","End":"02:39.665","Text":"twice x twice 2,"},{"Start":"02:39.665 ","End":"02:45.970","Text":"put it in brackets for emphasis, minus 3y=3."},{"Start":"02:45.970 ","End":"02:48.610","Text":"Just putting the x=2 here,"},{"Start":"02:49.070 ","End":"02:52.830","Text":"2 times 2 is 4."},{"Start":"02:52.830 ","End":"02:56.385","Text":"Let\u0027s bring it to the other side also,"},{"Start":"02:56.385 ","End":"03:03.670","Text":"I have -3y=3-4,"},{"Start":"03:03.920 ","End":"03:07.905","Text":"which is minus 1."},{"Start":"03:07.905 ","End":"03:13.823","Text":"Then divide both sides by 3,"},{"Start":"03:13.823 ","End":"03:22.520","Text":"and I\u0027ve got the y equals minus 1 over minus 3."},{"Start":"03:22.520 ","End":"03:25.355","Text":"Minus over minus is plus, so it\u0027s a 1/3."},{"Start":"03:25.355 ","End":"03:31.280","Text":"I could write my answer as curly braces."},{"Start":"03:31.280 ","End":"03:34.730","Text":"We\u0027ll put it over here, x=2,"},{"Start":"03:34.730 ","End":"03:39.905","Text":"y=1/3, that\u0027s the answer."},{"Start":"03:39.905 ","End":"03:43.010","Text":"This is the basic case."},{"Start":"03:43.010 ","End":"03:48.939","Text":"Let\u0027s do another."},{"Start":"03:48.939 ","End":"03:54.623","Text":"5x+y=5,"},{"Start":"03:54.623 ","End":"04:02.790","Text":"and the other equation, 5x+2y=14."},{"Start":"04:02.790 ","End":"04:08.225","Text":"We look at this and we\u0027re not quite in the case where we have an opposite pair,"},{"Start":"04:08.225 ","End":"04:11.270","Text":"like here and here, but we\u0027re close to it."},{"Start":"04:11.270 ","End":"04:13.790","Text":"With the y\u0027s there\u0027s no match with the x there\u0027s a match,"},{"Start":"04:13.790 ","End":"04:15.500","Text":"but not with the sign."},{"Start":"04:15.500 ","End":"04:17.360","Text":"That\u0027s easily fixed."},{"Start":"04:17.360 ","End":"04:20.810","Text":"All we do is to multiply one of them,"},{"Start":"04:20.810 ","End":"04:24.725","Text":"so the second one by minus 1,"},{"Start":"04:24.725 ","End":"04:33.710","Text":"then this one becomes, -5x-2y=-14."},{"Start":"04:33.710 ","End":"04:35.648","Text":"Just reverse the signs of everything."},{"Start":"04:35.648 ","End":"04:36.769","Text":"They all happen to be plus,"},{"Start":"04:36.769 ","End":"04:38.675","Text":"so we get all minuses."},{"Start":"04:38.675 ","End":"04:40.999","Text":"Again, we do an addition,"},{"Start":"04:40.999 ","End":"04:42.785","Text":"this time not with these two,"},{"Start":"04:42.785 ","End":"04:44.255","Text":"but with these two."},{"Start":"04:44.255 ","End":"04:46.818","Text":"We add these two together,"},{"Start":"04:46.818 ","End":"04:49.350","Text":"and what we will get is as follows;"},{"Start":"04:49.350 ","End":"04:52.510","Text":"5x and minus 5x is 0."},{"Start":"04:52.510 ","End":"04:53.885","Text":"I don\u0027t write anything,"},{"Start":"04:53.885 ","End":"04:55.100","Text":"they cancel each other out."},{"Start":"04:55.100 ","End":"05:05.153","Text":"We\u0027ve eliminated x. Y and minus 2y is minus y,"},{"Start":"05:05.153 ","End":"05:11.730","Text":"and 5 and minus 14 is minus 9."},{"Start":"05:11.730 ","End":"05:15.290","Text":"We said before if you have 2 minuses you can just throw out the minuses,"},{"Start":"05:15.290 ","End":"05:20.615","Text":"but what we\u0027re really doing is dividing or multiplying by minus 1, and so that\u0027s y."},{"Start":"05:20.615 ","End":"05:25.160","Text":"Now, I have to find x. I\u0027ll choose either one of these."},{"Start":"05:25.160 ","End":"05:26.750","Text":"I go for the ones with the smaller number,"},{"Start":"05:26.750 ","End":"05:31.530","Text":"so I\u0027ll put it into the first one to here."},{"Start":"05:31.530 ","End":"05:37.860","Text":"Then if y is 9 I\u0027ve got 5x plus, instead of y,"},{"Start":"05:37.860 ","End":"05:47.040","Text":"I\u0027ll put 9, the bracket is just for emphasis is equal to 5."},{"Start":"05:47.040 ","End":"05:52.430","Text":"If I bring the number over to the other side,"},{"Start":"05:52.430 ","End":"05:57.155","Text":"I\u0027ve got 5x=5-9 is minus 4."},{"Start":"05:57.155 ","End":"06:03.515","Text":"Dividing by 5, x equals minus 4/5."},{"Start":"06:03.515 ","End":"06:09.390","Text":"That\u0027s x, and I\u0027ll just put the solution."},{"Start":"06:09.390 ","End":"06:16.185","Text":"The solution is x equals minus 4/5,"},{"Start":"06:16.185 ","End":"06:20.395","Text":"and y equals 9."},{"Start":"06:20.395 ","End":"06:26.580","Text":"Let\u0027s take a break here and another couple of examples in the next clip."}],"ID":4957},{"Watched":false,"Name":"Elimination - Part b","Duration":"7m 28s","ChapterTopicVideoID":4967,"CourseChapterTopicPlaylistID":3865,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.010","Text":"Here we are back after the break."},{"Start":"00:02.010 ","End":"00:04.499","Text":"Let me erase the board."},{"Start":"00:04.499 ","End":"00:08.680","Text":"Let\u0027s just do it the electronic way."},{"Start":"00:10.580 ","End":"00:15.120","Text":"I\u0027ll indicate that this is Part 2 or continuation,"},{"Start":"00:15.120 ","End":"00:19.530","Text":"do another couple of examples of this method of elimination."},{"Start":"00:19.530 ","End":"00:22.560","Text":"To save time, I wrote it out for you,"},{"Start":"00:22.560 ","End":"00:28.725","Text":"2x plus 5 y is minus 8 and 3x plus y is 14,"},{"Start":"00:28.725 ","End":"00:31.840","Text":"and we want to use elimination."},{"Start":"00:32.150 ","End":"00:39.440","Text":"Now, we don\u0027t have the case where we have equal coefficients or opposite coefficients."},{"Start":"00:39.440 ","End":"00:42.965","Text":"If it was one minus the other or plus the other,"},{"Start":"00:42.965 ","End":"00:46.970","Text":"so we\u0027re going to have to work a little bit"},{"Start":"00:46.970 ","End":"00:52.790","Text":"more to make coefficients equal or equal with opposite sign."},{"Start":"00:52.790 ","End":"00:56.750","Text":"The thing to do is to look and see if you have one which"},{"Start":"00:56.750 ","End":"01:01.400","Text":"is a plus or minus 1 coefficient and I do the plus y."},{"Start":"01:01.400 ","End":"01:06.980","Text":"That makes it a little bit easier because then all I have to do is multiply this by 5."},{"Start":"01:06.980 ","End":"01:10.550","Text":"Well, actually better to multiply it by negative 5"},{"Start":"01:10.550 ","End":"01:14.435","Text":"and then I\u0027ll get opposite coefficients."},{"Start":"01:14.435 ","End":"01:16.550","Text":"So let\u0027s do that."},{"Start":"01:16.550 ","End":"01:22.370","Text":"I\u0027ll indicate at the side that I\u0027m multiplying the second one by minus 5."},{"Start":"01:22.370 ","End":"01:27.110","Text":"The second one is this one number 2,"},{"Start":"01:27.110 ","End":"01:33.660","Text":"number 1, and multiplying by minus 5."},{"Start":"01:33.660 ","End":"01:42.410","Text":"So I\u0027ll get minus 15x"},{"Start":"01:42.410 ","End":"01:50.405","Text":"minus 5y equals minus 70."},{"Start":"01:50.405 ","End":"01:53.455","Text":"Wrong color. Never mind."},{"Start":"01:53.455 ","End":"01:57.690","Text":"Now what I\u0027m going to do is add this one to this one."},{"Start":"01:57.690 ","End":"01:59.655","Text":"You see the 5y and the minus 5y."},{"Start":"01:59.655 ","End":"02:03.495","Text":"These 2, I\u0027m going to add them together."},{"Start":"02:03.495 ","End":"02:06.990","Text":"What I\u0027ll get is as follows,"},{"Start":"02:06.990 ","End":"02:13.620","Text":"2x and minus 15x is minus 13x;"},{"Start":"02:13.620 ","End":"02:15.950","Text":"5y and minus 5y is nothing."},{"Start":"02:15.950 ","End":"02:18.260","Text":"We don\u0027t write anything. That is the elimination."},{"Start":"02:18.260 ","End":"02:20.030","Text":"We eliminated y,"},{"Start":"02:20.030 ","End":"02:27.905","Text":"and this is equal to: minus 8 minus 70 is minus 78."},{"Start":"02:27.905 ","End":"02:34.080","Text":"So x is equal to minus 78 over minus 13."},{"Start":"02:34.080 ","End":"02:36.195","Text":"Minus over minus is plus,"},{"Start":"02:36.195 ","End":"02:39.404","Text":"and 13 goes into 78, 6 times."},{"Start":"02:39.404 ","End":"02:41.770","Text":"So x equals 6."},{"Start":"02:41.770 ","End":"02:45.220","Text":"Now we just substitute into either one."},{"Start":"02:45.220 ","End":"02:49.350","Text":"I\u0027ll substitute it into the second one,"},{"Start":"02:49.350 ","End":"02:51.700","Text":"into this one here."},{"Start":"02:51.860 ","End":"02:55.200","Text":"We\u0027ll get 3x,"},{"Start":"02:55.200 ","End":"02:58.545","Text":"which is 3 times 6,"},{"Start":"02:58.545 ","End":"03:04.655","Text":"because we found x equals 6 plus y equals 14."},{"Start":"03:04.655 ","End":"03:11.060","Text":"So y equals 14 minus this is 18."},{"Start":"03:11.060 ","End":"03:12.695","Text":"If I multiply it out,"},{"Start":"03:12.695 ","End":"03:16.660","Text":"which is minus 4,"},{"Start":"03:16.660 ","End":"03:18.785","Text":"that is our answer."},{"Start":"03:18.785 ","End":"03:25.710","Text":"We have that x equals 6 and y equals minus 4."},{"Start":"03:25.710 ","End":"03:27.260","Text":"I want to remind you."},{"Start":"03:27.260 ","End":"03:28.460","Text":"I remind you from time to time,"},{"Start":"03:28.460 ","End":"03:31.399","Text":"that you can always check your answer by substituting"},{"Start":"03:31.399 ","End":"03:35.915","Text":"these values in the original equations and seeing if it works."},{"Start":"03:35.915 ","End":"03:38.570","Text":"Just for instance, I\u0027ll just check the first one,"},{"Start":"03:38.570 ","End":"03:43.100","Text":"2x is 12, 5y is minus 20,"},{"Start":"03:43.100 ","End":"03:45.260","Text":"12 minus 20 equals minus 8."},{"Start":"03:45.260 ","End":"03:47.310","Text":"Yes, and so on."},{"Start":"03:47.310 ","End":"03:50.645","Text":"Remember you always have the option on an exam or something,"},{"Start":"03:50.645 ","End":"03:54.785","Text":"because you don\u0027t have the answers that you can check yourself by substitution."},{"Start":"03:54.785 ","End":"03:59.120","Text":"Let\u0027s do 1 more. Here it is."},{"Start":"03:59.120 ","End":"04:00.575","Text":"Once again, I wrote it out for you."},{"Start":"04:00.575 ","End":"04:02.630","Text":"5x plus 4y is 14,"},{"Start":"04:02.630 ","End":"04:05.585","Text":"8x plus 5y is 0."},{"Start":"04:05.585 ","End":"04:10.820","Text":"In this case, this is the hardest,"},{"Start":"04:10.820 ","End":"04:12.170","Text":"but it\u0027s still not hard,"},{"Start":"04:12.170 ","End":"04:16.160","Text":"is that we have to multiply both of them by something because they don\u0027t"},{"Start":"04:16.160 ","End":"04:20.180","Text":"have anything with coefficient 1 or minus 1,"},{"Start":"04:20.180 ","End":"04:26.000","Text":"and we don\u0027t have equality or minus equality,"},{"Start":"04:26.000 ","End":"04:28.160","Text":"but any other coefficients."},{"Start":"04:28.160 ","End":"04:33.575","Text":"So we\u0027re going to have to make the coefficients equal by multiplying both."},{"Start":"04:33.575 ","End":"04:36.540","Text":"In this case, for example,"},{"Start":"04:36.670 ","End":"04:40.310","Text":"we have to decide if we want to eliminate x or y."},{"Start":"04:40.310 ","End":"04:43.445","Text":"I don\u0027t see that one is more difficult than the other."},{"Start":"04:43.445 ","End":"04:46.285","Text":"Let\u0027s say I wanted to eliminate y."},{"Start":"04:46.285 ","End":"04:50.270","Text":"To eliminate y and e to make these 2 coefficients equal."},{"Start":"04:50.270 ","End":"04:54.290","Text":"So if I multiply this one by 5,"},{"Start":"04:54.290 ","End":"04:55.865","Text":"there\u0027ll be a 20 here."},{"Start":"04:55.865 ","End":"04:58.760","Text":"If I multiply this one by minus 4,"},{"Start":"04:58.760 ","End":"05:00.710","Text":"I will get a minus 20 here,"},{"Start":"05:00.710 ","End":"05:02.930","Text":"and that\u0027s very good for elimination."},{"Start":"05:02.930 ","End":"05:05.300","Text":"Basically I\u0027m taking this coefficient here,"},{"Start":"05:05.300 ","End":"05:08.345","Text":"this coefficient here, and reversing one of them."},{"Start":"05:08.345 ","End":"05:14.810","Text":"What I get are 2 equivalent equations to these, just multiplied out."},{"Start":"05:14.810 ","End":"05:17.285","Text":"Multiply everything here by 5,"},{"Start":"05:17.285 ","End":"05:23.720","Text":"25x plus 20y equals 14 times 5 is 70."},{"Start":"05:23.720 ","End":"05:26.560","Text":"Multiply everything here by minus 4."},{"Start":"05:26.560 ","End":"05:32.005","Text":"Minus 4 times 8 is minus 32x minus 20y."},{"Start":"05:32.005 ","End":"05:34.745","Text":"Anything times 0 is 0."},{"Start":"05:34.745 ","End":"05:38.350","Text":"Now, I\u0027ll add these 2."},{"Start":"05:38.350 ","End":"05:44.670","Text":"What I\u0027ll get is 25x minus 32x is minus 7x."},{"Start":"05:44.670 ","End":"05:49.620","Text":"The y is eliminated as we intended."},{"Start":"05:49.620 ","End":"05:55.905","Text":"70 plus 0 is 70 divide by minus 7,"},{"Start":"05:55.905 ","End":"06:02.225","Text":"x is 70 over minus 7 plus over minus is minus."},{"Start":"06:02.225 ","End":"06:04.595","Text":"So it\u0027s minus 10."},{"Start":"06:04.595 ","End":"06:13.610","Text":"Now I have x, I have to find y. I have to substitute in any one of these 2 equations,"},{"Start":"06:13.610 ","End":"06:16.500","Text":"I\u0027ll go with the first."},{"Start":"06:16.640 ","End":"06:18.945","Text":"The first one,"},{"Start":"06:18.945 ","End":"06:21.360","Text":"I\u0027ll get x is minus 10,"},{"Start":"06:21.360 ","End":"06:24.075","Text":"putting it into here."},{"Start":"06:24.075 ","End":"06:30.060","Text":"So 5 times minus"},{"Start":"06:30.060 ","End":"06:37.825","Text":"10 plus 4y equals 14."},{"Start":"06:37.825 ","End":"06:41.880","Text":"That means it will continue over here."},{"Start":"06:43.150 ","End":"06:53.555","Text":"So 4y is going to equal 14 plus 50 because here I\u0027ve got minus 50,"},{"Start":"06:53.555 ","End":"06:58.585","Text":"14 plus 50 is 64."},{"Start":"06:58.585 ","End":"07:02.190","Text":"Just write down, this was 14 plus 50."},{"Start":"07:02.190 ","End":"07:08.615","Text":"Then I divide by 4 and I\u0027ve got y equals 16."},{"Start":"07:08.615 ","End":"07:12.570","Text":"Now I have both x and y,"},{"Start":"07:12.570 ","End":"07:14.780","Text":"and I\u0027ll just write the solution."},{"Start":"07:14.780 ","End":"07:19.180","Text":"Solution x is equal to minus 10,"},{"Start":"07:19.180 ","End":"07:22.455","Text":"y equals 16,"},{"Start":"07:22.455 ","End":"07:28.140","Text":"and we are done with this topic of Method of Elimination."}],"ID":4958},{"Watched":false,"Name":"Exercise 1","Duration":"1m 24s","ChapterTopicVideoID":4957,"CourseChapterTopicPlaylistID":3865,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.950","Text":"In this exercise, we have to solve the system of equations."},{"Start":"00:04.950 ","End":"00:07.425","Text":"It\u0027s 2 equations in 2 unknowns."},{"Start":"00:07.425 ","End":"00:09.495","Text":"Let\u0027s do it over here."},{"Start":"00:09.495 ","End":"00:11.745","Text":"In this case, we have,"},{"Start":"00:11.745 ","End":"00:14.730","Text":"the second equation is just y=5."},{"Start":"00:14.730 ","End":"00:16.080","Text":"It tells us what y is."},{"Start":"00:16.080 ","End":"00:20.009","Text":"This is lucky for us. It\u0027s easy."},{"Start":"00:20.009 ","End":"00:25.785","Text":"All we have to do when we have 1 of the variables is substituted in the other equation."},{"Start":"00:25.785 ","End":"00:28.545","Text":"I take this equation and put y=5,"},{"Start":"00:28.545 ","End":"00:33.180","Text":"I get 3x plus instead of y I put 5."},{"Start":"00:33.180 ","End":"00:38.820","Text":"I\u0027m using parentheses it\u0027s just for emphasis to show that I substituted equals 11."},{"Start":"00:38.820 ","End":"00:43.350","Text":"Now we have an equation in 1 unknown x."},{"Start":"00:43.350 ","End":"00:46.380","Text":"So bring the 5 over to the other side."},{"Start":"00:46.380 ","End":"00:51.630","Text":"11-5, we do it in our heads is 6."},{"Start":"00:51.630 ","End":"00:54.930","Text":"Now we divide both sides by 3."},{"Start":"00:54.930 ","End":"00:57.490","Text":"We get that x=2."},{"Start":"00:58.000 ","End":"01:03.155","Text":"Once we have x, we now need to find y."},{"Start":"01:03.155 ","End":"01:07.235","Text":"Normally, yes, but in this case, we already have it."},{"Start":"01:07.235 ","End":"01:15.550","Text":"I like to write the solution with curly braces and say {x=2, y=5}."},{"Start":"01:15.550 ","End":"01:16.980","Text":"There\u0027s 2 things here,"},{"Start":"01:16.980 ","End":"01:18.405","Text":"but it\u0027s 1 solution,"},{"Start":"01:18.405 ","End":"01:20.085","Text":"and I\u0027ll highlight it."},{"Start":"01:20.085 ","End":"01:23.620","Text":"There we go, and we\u0027re done."}],"ID":4962},{"Watched":false,"Name":"Exercise 2","Duration":"3m 1s","ChapterTopicVideoID":4958,"CourseChapterTopicPlaylistID":3865,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.270","Text":"We have here a system of 2 equations and 2 unknowns and we want to solve it,"},{"Start":"00:05.270 ","End":"00:06.885","Text":"let\u0027s solve it over here."},{"Start":"00:06.885 ","End":"00:12.825","Text":"Here I noticed that the second equation is x in terms of y."},{"Start":"00:12.825 ","End":"00:14.820","Text":"What we do is substitution."},{"Start":"00:14.820 ","End":"00:18.570","Text":"We substitute this value of x in the other equation."},{"Start":"00:18.570 ","End":"00:21.810","Text":"Now I rewrite this one but with the substitution,"},{"Start":"00:21.810 ","End":"00:27.705","Text":"so instead of x, I put 5y plus 14."},{"Start":"00:27.705 ","End":"00:29.760","Text":"Then the rest as is,"},{"Start":"00:29.760 ","End":"00:33.790","Text":"plus 2y= minus 16."},{"Start":"00:33.790 ","End":"00:37.070","Text":"Now I have an equation in1 unknown y."},{"Start":"00:37.070 ","End":"00:38.900","Text":"Let\u0027s simplify it."},{"Start":"00:38.900 ","End":"00:40.330","Text":"Open the brackets,"},{"Start":"00:40.330 ","End":"00:44.110","Text":"minus 3 times 5y is minus 15y,"},{"Start":"00:44.110 ","End":"00:48.050","Text":"minus 3 times 14 minus 42,"},{"Start":"00:48.050 ","End":"00:49.310","Text":"that\u0027s the distributive law,"},{"Start":"00:49.310 ","End":"00:51.590","Text":"it\u0027s expansion of brackets,"},{"Start":"00:51.590 ","End":"00:57.365","Text":"then plus 2y equals minus 16."},{"Start":"00:57.365 ","End":"00:59.180","Text":"Now let\u0027s collect like terms."},{"Start":"00:59.180 ","End":"01:00.470","Text":"I want y on the left,"},{"Start":"01:00.470 ","End":"01:01.790","Text":"numbers on the right."},{"Start":"01:01.790 ","End":"01:03.680","Text":"The y\u0027s already are on the left,"},{"Start":"01:03.680 ","End":"01:07.690","Text":"minus 15y plus 2y."},{"Start":"01:07.690 ","End":"01:10.850","Text":"The numbers minus 16 as is,"},{"Start":"01:10.850 ","End":"01:12.680","Text":"but the 42 changes sides,"},{"Start":"01:12.680 ","End":"01:17.285","Text":"so it becomes plus 42. Let\u0027s see."},{"Start":"01:17.285 ","End":"01:19.250","Text":"Take y outside the brackets,"},{"Start":"01:19.250 ","End":"01:22.100","Text":"so factorize or collect however you want to call it,"},{"Start":"01:22.100 ","End":"01:26.715","Text":"we have minus 15 apples plus 2 apples,"},{"Start":"01:26.715 ","End":"01:29.164","Text":"so it\u0027s minus 13 apples."},{"Start":"01:29.164 ","End":"01:39.000","Text":"I mean, y of course and 42 minus 16 is 26."},{"Start":"01:39.170 ","End":"01:45.305","Text":"All I have to do now is divide both sides by the minus 13."},{"Start":"01:45.305 ","End":"01:50.449","Text":"I\u0027m left with y is 26 over minus 13."},{"Start":"01:50.449 ","End":"01:56.105","Text":"Some people say move the minus 13 to the other side and put it on the denominator."},{"Start":"01:56.105 ","End":"01:59.060","Text":"Either way, plus over minus is minus,"},{"Start":"01:59.060 ","End":"02:00.860","Text":"26 over 13 is 2,"},{"Start":"02:00.860 ","End":"02:03.185","Text":"so the answer is minus 2."},{"Start":"02:03.185 ","End":"02:06.620","Text":"Now I have the answer to why."},{"Start":"02:06.620 ","End":"02:12.425","Text":"What I do with this now is I substitute it over here,"},{"Start":"02:12.425 ","End":"02:19.100","Text":"where I have x in terms of y. I say x equals 5 times."},{"Start":"02:19.100 ","End":"02:24.825","Text":"Now y is minus 2 and then plus 14."},{"Start":"02:24.825 ","End":"02:26.775","Text":"Now it\u0027s just arithmetic,"},{"Start":"02:26.775 ","End":"02:29.835","Text":"5 times minus 2 is minus 10,"},{"Start":"02:29.835 ","End":"02:33.915","Text":"plus 14, and this is equal to 4."},{"Start":"02:33.915 ","End":"02:37.435","Text":"I now have the y is minus 2, x is 4."},{"Start":"02:37.435 ","End":"02:41.029","Text":"But I want to write that down as the solution."},{"Start":"02:41.029 ","End":"02:43.265","Text":"Note that solution is not just one thing,"},{"Start":"02:43.265 ","End":"02:45.110","Text":"the pair of things is a solution."},{"Start":"02:45.110 ","End":"02:47.285","Text":"What x equals and what y equals,"},{"Start":"02:47.285 ","End":"02:49.115","Text":"that is 1 solution."},{"Start":"02:49.115 ","End":"02:50.720","Text":"X equals 4,"},{"Start":"02:50.720 ","End":"02:52.235","Text":"I\u0027ll just copy from here,"},{"Start":"02:52.235 ","End":"02:56.070","Text":"and y equals minus 2 copied it from here and I like to"},{"Start":"02:56.070 ","End":"03:01.600","Text":"highlight the result and there we are."}],"ID":4963},{"Watched":false,"Name":"Exercise 3","Duration":"2m 30s","ChapterTopicVideoID":4959,"CourseChapterTopicPlaylistID":3865,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.445","Text":"Here we have to solve this system of 2 equations and 2 unknowns; x and y."},{"Start":"00:05.445 ","End":"00:07.725","Text":"Let\u0027s do it over here."},{"Start":"00:07.725 ","End":"00:11.895","Text":"I was hoping we might have 1 variable in terms of another."},{"Start":"00:11.895 ","End":"00:13.920","Text":"We don\u0027t, but we have close to it."},{"Start":"00:13.920 ","End":"00:16.170","Text":"Look, I have just x on its own."},{"Start":"00:16.170 ","End":"00:25.210","Text":"If I rewrite this slightly as x=-4y+4,"},{"Start":"00:25.340 ","End":"00:28.395","Text":"I brought the 4y to the other side,"},{"Start":"00:28.395 ","End":"00:30.465","Text":"now I have x in terms of y,"},{"Start":"00:30.465 ","End":"00:35.520","Text":"and then I can use the substitution of this into this."},{"Start":"00:35.520 ","End":"00:40.050","Text":"I\u0027m rewriting the first now as 5 times, and instead of x,"},{"Start":"00:40.050 ","End":"00:50.385","Text":"I put (-4y+4)-2y=-2, that\u0027s just from here."},{"Start":"00:50.385 ","End":"00:54.585","Text":"1 equation in 1 unknown y, let\u0027s simplify."},{"Start":"00:54.585 ","End":"01:00.730","Text":"First of all, expand -20y+20-2y=-2."},{"Start":"01:05.510 ","End":"01:10.690","Text":"Now, let\u0027s keep the y\u0027s on the left put the numbers on the right."},{"Start":"01:10.880 ","End":"01:19.200","Text":"-20y-2y= -2-20, the 20 changes sign when it goes over."},{"Start":"01:19.200 ","End":"01:23.205","Text":"Then, here altogether I have -22y,"},{"Start":"01:23.205 ","End":"01:25.875","Text":"and here I have -22."},{"Start":"01:25.875 ","End":"01:34.790","Text":"Now, if we just divide both sides by 22 or rather by -22,"},{"Start":"01:34.790 ","End":"01:39.230","Text":"I meant that\u0027s what we divide by then -22 over is just 1,"},{"Start":"01:39.230 ","End":"01:44.195","Text":"and -22 over -22 is just 1, so y=1."},{"Start":"01:44.195 ","End":"01:48.240","Text":"That gives us 1 of the variables,"},{"Start":"01:48.240 ","End":"01:49.890","Text":"now I need to find x."},{"Start":"01:49.890 ","End":"01:54.335","Text":"What I do is, I substituted here where I have x in terms of y,"},{"Start":"01:54.335 ","End":"01:58.895","Text":"and so this gives me x=-4(1)+4,"},{"Start":"01:58.895 ","End":"02:02.520","Text":"I put it in parentheses for emphasis."},{"Start":"02:04.430 ","End":"02:10.215","Text":"That\u0027s -4+4, and that\u0027s 0."},{"Start":"02:10.215 ","End":"02:13.280","Text":"So now, I have y over here and x over here,"},{"Start":"02:13.280 ","End":"02:20.200","Text":"I just have to collect them together in one place and say x=0,"},{"Start":"02:20.200 ","End":"02:25.355","Text":"y=1, and I like to highlight it."},{"Start":"02:25.355 ","End":"02:28.950","Text":"This is the solution, we\u0027re done."}],"ID":4964},{"Watched":false,"Name":"Exercise 4","Duration":"4m 40s","ChapterTopicVideoID":4960,"CourseChapterTopicPlaylistID":3865,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.270","Text":"Here I have a system of 2 equations and 2 unknowns."},{"Start":"00:05.270 ","End":"00:08.475","Text":"This time, we\u0027re going to use a different technique."},{"Start":"00:08.475 ","End":"00:12.240","Text":"Up to now, we use the technique called substitution,"},{"Start":"00:12.240 ","End":"00:15.510","Text":"but this time we\u0027re going to use a method called elimination."},{"Start":"00:15.510 ","End":"00:18.975","Text":"Let me eliminate the substitution and go for this."},{"Start":"00:18.975 ","End":"00:21.840","Text":"The elimination works like this."},{"Start":"00:21.840 ","End":"00:25.620","Text":"I want to multiply each of these equations,"},{"Start":"00:25.620 ","End":"00:28.950","Text":"perhaps one, perhaps both by some constant,"},{"Start":"00:28.950 ","End":"00:37.585","Text":"and I want the x\u0027s to have equal numbers but just opposite signs."},{"Start":"00:37.585 ","End":"00:39.480","Text":"I\u0027ll show you what I mean."},{"Start":"00:39.480 ","End":"00:42.230","Text":"If I multiply this one by,"},{"Start":"00:42.230 ","End":"00:46.855","Text":"let\u0027s say I take 5 times"},{"Start":"00:46.855 ","End":"00:53.745","Text":"this equation and I take minus 2 times this equation,"},{"Start":"00:53.745 ","End":"00:58.270","Text":"then here I\u0027ll get 10x and here I\u0027ll get minus 10x."},{"Start":"00:58.280 ","End":"01:03.020","Text":"I\u0027ll be able to eliminate x and that\u0027s why it\u0027s called elimination."},{"Start":"01:03.020 ","End":"01:07.340","Text":"Let\u0027s do this one. Then I\u0027ll mention now."},{"Start":"01:07.340 ","End":"01:08.540","Text":"It doesn\u0027t have to be for x."},{"Start":"01:08.540 ","End":"01:14.760","Text":"It could be elimination of y perhaps in another exercise we\u0027ll eliminate y and leave x."},{"Start":"01:14.760 ","End":"01:16.790","Text":"This time we\u0027re going to eliminate x."},{"Start":"01:16.790 ","End":"01:21.655","Text":"You\u0027re allowed to multiply both sides of an equation by the same number."},{"Start":"01:21.655 ","End":"01:29.770","Text":"If we multiply the left by 5 we get 5 times 2x is 10x."},{"Start":"01:29.770 ","End":"01:34.700","Text":"I\u0027ll just write this, 5 times 3y is 15y."},{"Start":"01:35.960 ","End":"01:41.045","Text":"We have to also multiply the other side of the equation by 5."},{"Start":"01:41.045 ","End":"01:44.240","Text":"So 5 times 5 is 25."},{"Start":"01:44.240 ","End":"01:50.130","Text":"The other equation, I\u0027m multiplying by minus 2 here,"},{"Start":"01:50.130 ","End":"01:55.385","Text":"here, and here, and here."},{"Start":"01:55.385 ","End":"02:00.520","Text":"We get minus 2 times 5x is minus 10x."},{"Start":"02:00.520 ","End":"02:02.555","Text":"This is the important thing."},{"Start":"02:02.555 ","End":"02:06.655","Text":"I wanted it so the numbers come out equal but opposite in sign,"},{"Start":"02:06.655 ","End":"02:08.480","Text":"and this I have already."},{"Start":"02:08.480 ","End":"02:11.010","Text":"Let\u0027s continue though."},{"Start":"02:13.790 ","End":"02:22.190","Text":"Minus 2 times 7y is minus 14y, minus 2 times 11 is minus 22."},{"Start":"02:22.190 ","End":"02:23.540","Text":"Now what you do with these,"},{"Start":"02:23.540 ","End":"02:26.210","Text":"it\u0027s just like an addition and arithmetic, I add the 2."},{"Start":"02:26.210 ","End":"02:28.490","Text":"I\u0027d like to put a line here and a plus."},{"Start":"02:28.490 ","End":"02:30.830","Text":"I can do this mentally."},{"Start":"02:30.830 ","End":"02:32.755","Text":"I did it this time."},{"Start":"02:32.755 ","End":"02:36.280","Text":"Let\u0027s see what we get when we add 10x minus 10x is 0."},{"Start":"02:36.280 ","End":"02:39.800","Text":"I don\u0027t even write anything, just zeros, nothing."},{"Start":"02:39.800 ","End":"02:42.300","Text":"That\u0027s what it\u0027s supposed to be."},{"Start":"02:43.950 ","End":"02:48.290","Text":"15y and minus 14y is 15 minus 14 is 1y,"},{"Start":"02:48.290 ","End":"02:51.650","Text":"just writing the 1 for emphasis though I don\u0027t need it,"},{"Start":"02:51.650 ","End":"02:56.910","Text":"and 25 and minus 22 is 3."},{"Start":"02:57.110 ","End":"03:02.770","Text":"So y=3, I\u0027ll just erase the 1."},{"Start":"03:02.770 ","End":"03:06.620","Text":"That\u0027s, found y. Now, what about x?"},{"Start":"03:06.620 ","End":"03:09.290","Text":"Well, I pick one of these 2 equations."},{"Start":"03:09.290 ","End":"03:11.135","Text":"It doesn\u0027t really matter which."},{"Start":"03:11.135 ","End":"03:15.470","Text":"I\u0027ll choose the first one because it has smaller numbers and put y in it."},{"Start":"03:15.470 ","End":"03:23.490","Text":"If I take this first equation and I put y=3 into it,"},{"Start":"03:23.490 ","End":"03:31.095","Text":"what I\u0027ll get is 2x plus 3y,"},{"Start":"03:31.095 ","End":"03:35.715","Text":"but y=3 equals 5."},{"Start":"03:35.715 ","End":"03:37.470","Text":"Now, it\u0027s like again,"},{"Start":"03:37.470 ","End":"03:39.000","Text":"1 equation and 1 unknown."},{"Start":"03:39.000 ","End":"03:41.380","Text":"So I get 2x plus 9 equals 5."},{"Start":"03:44.360 ","End":"03:49.295","Text":"Bring the 9 over to the other side,2x is 5 minus 9."},{"Start":"03:49.295 ","End":"03:50.890","Text":"Let\u0027s do it in our heads,"},{"Start":"03:50.890 ","End":"03:54.535","Text":"won\u0027t even write it, 5 minus 9 is minus 4."},{"Start":"03:54.535 ","End":"03:56.935","Text":"Now, divide by 2."},{"Start":"03:56.935 ","End":"04:02.725","Text":"We\u0027ve got x=minus 2."},{"Start":"04:02.725 ","End":"04:06.565","Text":"Now, I have y and I have x,"},{"Start":"04:06.565 ","End":"04:11.150","Text":"and I just collect them neatly in a solution and"},{"Start":"04:11.150 ","End":"04:19.650","Text":"say that x=minus 2 and y=3."},{"Start":"04:19.650 ","End":"04:23.435","Text":"Dimension at this point that there are people who like to write the solution,"},{"Start":"04:23.435 ","End":"04:28.380","Text":"something comma, something they would write, say minus 2,3."},{"Start":"04:28.380 ","End":"04:31.340","Text":"I don\u0027t particularly like using that."},{"Start":"04:31.340 ","End":"04:36.185","Text":"I prefer to say explicitly what each variable is and I\u0027ll even highlight it."},{"Start":"04:36.185 ","End":"04:39.660","Text":"There we go, and we\u0027re done."}],"ID":4965},{"Watched":false,"Name":"Exercise 5","Duration":"2m 13s","ChapterTopicVideoID":4961,"CourseChapterTopicPlaylistID":3865,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.350","Text":"Here we have to solve this system of 2 equations in 2 unknowns,"},{"Start":"00:04.350 ","End":"00:07.335","Text":"x and y. I will solve it over here."},{"Start":"00:07.335 ","End":"00:10.860","Text":"I noticed that the left-hand sides are both the same,"},{"Start":"00:10.860 ","End":"00:12.000","Text":"y equals something,"},{"Start":"00:12.000 ","End":"00:14.385","Text":"and y equals something."},{"Start":"00:14.385 ","End":"00:17.430","Text":"It seems that the obvious thing to do is to"},{"Start":"00:17.430 ","End":"00:21.120","Text":"equate this to this because they\u0027re both equal to the same thing,"},{"Start":"00:21.120 ","End":"00:28.440","Text":"and so what I get is x minus 3 equals 2x plus 4."},{"Start":"00:28.440 ","End":"00:32.840","Text":"This is obvious but if you want me to stick to formal methods,"},{"Start":"00:32.840 ","End":"00:35.450","Text":"I could say that this is a method of substitution."},{"Start":"00:35.450 ","End":"00:37.295","Text":"Also because I could say, okay,"},{"Start":"00:37.295 ","End":"00:38.870","Text":"y equals x minus 3,"},{"Start":"00:38.870 ","End":"00:41.510","Text":"and let\u0027s substitute y in the second equation."},{"Start":"00:41.510 ","End":"00:43.720","Text":"Then when I substitute this y,"},{"Start":"00:43.720 ","End":"00:44.970","Text":"as x minus 3,"},{"Start":"00:44.970 ","End":"00:46.605","Text":"I get exactly what\u0027s written here,"},{"Start":"00:46.605 ","End":"00:48.680","Text":"so in a way this is substitution,"},{"Start":"00:48.680 ","End":"00:50.650","Text":"but it\u0027s more basic than that."},{"Start":"00:50.650 ","End":"00:55.250","Text":"Two things that are each equal to the third thing are equal among themselves."},{"Start":"00:55.250 ","End":"01:00.260","Text":"Let\u0027s solve for x and just bring the x\u0027s on the left numbers on the right,"},{"Start":"01:00.260 ","End":"01:03.590","Text":"so I have x minus 2x,"},{"Start":"01:03.590 ","End":"01:09.780","Text":"and here equals 4 plus 3."},{"Start":"01:09.780 ","End":"01:15.780","Text":"What do I have? Minus x equals 7 divide by minus 1,"},{"Start":"01:15.780 ","End":"01:17.830","Text":"or you could multiply by minus 1,"},{"Start":"01:17.830 ","End":"01:22.290","Text":"it doesn\u0027t matter, and we get x equals minus 7."},{"Start":"01:22.290 ","End":"01:24.930","Text":"That\u0027s x. What about y?"},{"Start":"01:24.930 ","End":"01:31.420","Text":"Now we just substitute that over there in this equation,"},{"Start":"01:32.270 ","End":"01:38.550","Text":"and so we get that y equals twice, now,"},{"Start":"01:38.550 ","End":"01:42.615","Text":"x is minus 7 plus 4,"},{"Start":"01:42.615 ","End":"01:47.025","Text":"which is minus 14 plus 4,"},{"Start":"01:47.025 ","End":"01:50.440","Text":"which is minus 10."},{"Start":"01:50.540 ","End":"01:56.885","Text":"All I have to do is just write the solution in an organized way,"},{"Start":"01:56.885 ","End":"02:00.215","Text":"what x equals and what y equals,"},{"Start":"02:00.215 ","End":"02:02.965","Text":"so this is minus 7 from here,"},{"Start":"02:02.965 ","End":"02:06.240","Text":"minus 10 from here."},{"Start":"02:06.240 ","End":"02:09.105","Text":"I like to highlight it."},{"Start":"02:09.105 ","End":"02:13.180","Text":"There we go. We\u0027re done."}],"ID":4966},{"Watched":false,"Name":"Exercise 6","Duration":"5m 15s","ChapterTopicVideoID":4962,"CourseChapterTopicPlaylistID":3865,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"Here we have a system of 2 equations and 2 unknowns; x and y,"},{"Start":"00:05.280 ","End":"00:11.280","Text":"and they made it a little bit harder for us this time because we have x\u0027s,"},{"Start":"00:11.280 ","End":"00:13.610","Text":"and y\u0027s, and numbers on both sides."},{"Start":"00:13.610 ","End":"00:15.434","Text":"Let\u0027s just collect everything."},{"Start":"00:15.434 ","End":"00:18.465","Text":"I want x\u0027s and y\u0027s on the left,"},{"Start":"00:18.465 ","End":"00:20.292","Text":"numbers on the right,"},{"Start":"00:20.292 ","End":"00:24.705","Text":"and I\u0027ll do each 1 of them separately."},{"Start":"00:24.705 ","End":"00:26.235","Text":"Let\u0027s start with the first 1."},{"Start":"00:26.235 ","End":"00:28.915","Text":"You know what? I\u0027m going to give the equations numbers."},{"Start":"00:28.915 ","End":"00:34.275","Text":"I\u0027ll call this equation 1 and this equation 2."},{"Start":"00:34.275 ","End":"00:37.070","Text":"Let\u0027s work on equation 1 for now,"},{"Start":"00:37.070 ","End":"00:38.870","Text":"and we\u0027re collecting like terms."},{"Start":"00:38.870 ","End":"00:41.540","Text":"Let\u0027s collect the x\u0027s first on the left."},{"Start":"00:41.540 ","End":"00:44.510","Text":"On the left I already have minus x,"},{"Start":"00:44.510 ","End":"00:50.490","Text":"and I\u0027m going to bring the other 4x over so it\u0027s minus 4x."},{"Start":"00:50.490 ","End":"00:52.950","Text":"That\u0027s all the x\u0027s, now y\u0027s."},{"Start":"00:52.950 ","End":"00:56.235","Text":"I have 3y, it\u0027s a plus."},{"Start":"00:56.235 ","End":"00:58.890","Text":"I\u0027m going to bring the other 3y over,"},{"Start":"00:58.890 ","End":"01:01.335","Text":"so that\u0027s another plus 3y."},{"Start":"01:01.335 ","End":"01:04.080","Text":"Now the numbers I\u0027m throwing on the right."},{"Start":"01:04.080 ","End":"01:07.920","Text":"I had this 2 already and I\u0027m going to throw this to 2,"},{"Start":"01:07.920 ","End":"01:10.465","Text":"so it becomes minus 2."},{"Start":"01:10.465 ","End":"01:15.230","Text":"I\u0027m going to continue with this first equation,"},{"Start":"01:15.230 ","End":"01:20.450","Text":"so let\u0027s see minus x minus 4x is minus"},{"Start":"01:20.450 ","End":"01:29.380","Text":"5x plus 3y plus 3y plus 6y and here 2 minus 2 is 0."},{"Start":"01:29.380 ","End":"01:33.030","Text":"This is a simplified equation 1,"},{"Start":"01:33.030 ","End":"01:35.150","Text":"so I\u0027ll just write that."},{"Start":"01:35.150 ","End":"01:37.280","Text":"That\u0027s like equation 1, we just simplify it."},{"Start":"01:37.280 ","End":"01:38.915","Text":"Let\u0027s go to number 2."},{"Start":"01:38.915 ","End":"01:41.180","Text":"I\u0027ll do the same thing."},{"Start":"01:41.180 ","End":"01:44.480","Text":"Let\u0027s see, I\u0027ll do it over here this time."},{"Start":"01:44.480 ","End":"01:46.160","Text":"I want the x\u0027s first;"},{"Start":"01:46.160 ","End":"01:48.685","Text":"so it\u0027s 2x minus 4x,"},{"Start":"01:48.685 ","End":"01:52.110","Text":"but the minus 4x comes over as plus 4x."},{"Start":"01:52.110 ","End":"01:55.170","Text":"Then y, I have a minus y here."},{"Start":"01:55.170 ","End":"01:58.490","Text":"This 1 when I bring it over is minus 5y,"},{"Start":"01:58.490 ","End":"02:01.640","Text":"then here are the numbers I already had 3."},{"Start":"02:01.640 ","End":"02:04.540","Text":"I bring this 3 over and it\u0027s another plus 3,"},{"Start":"02:04.540 ","End":"02:05.750","Text":"so what do I have?"},{"Start":"02:05.750 ","End":"02:07.880","Text":"2 and 4 is 6x."},{"Start":"02:07.880 ","End":"02:10.610","Text":"Minus 1 minus 5 is minus 6y,"},{"Start":"02:10.610 ","End":"02:14.060","Text":"and this equals 6."},{"Start":"02:14.060 ","End":"02:18.230","Text":"Another thing I\u0027d like to mention is that if you see that everything divides by"},{"Start":"02:18.230 ","End":"02:22.490","Text":"a common factor and it\u0027s obvious in this case that everything\u0027s divisible by 6,"},{"Start":"02:22.490 ","End":"02:27.245","Text":"then just to get the numbers smaller I would divide by that 6."},{"Start":"02:27.245 ","End":"02:32.835","Text":"We\u0027ll get x minus y equals 1."},{"Start":"02:32.835 ","End":"02:35.190","Text":"This is a simplified number 2,"},{"Start":"02:35.190 ","End":"02:38.405","Text":"so I\u0027ll just indicate that."},{"Start":"02:38.405 ","End":"02:41.270","Text":"What I\u0027d like to do is take this and this together and"},{"Start":"02:41.270 ","End":"02:46.040","Text":"just put them as another set of equations."},{"Start":"02:46.040 ","End":"02:48.440","Text":"Actually the word is system of equations,"},{"Start":"02:48.440 ","End":"02:50.690","Text":"but people also say sets of equations."},{"Start":"02:50.690 ","End":"02:54.530","Text":"I\u0027ll keep this 1 here and I\u0027ll just write this 1 here."},{"Start":"02:54.530 ","End":"02:58.530","Text":"X minus y equals 1,"},{"Start":"02:58.530 ","End":"03:01.810","Text":"and I\u0027ll just copy number 2."},{"Start":"03:02.720 ","End":"03:08.480","Text":"I have much nicer system of 2 equations and 2 unknowns."},{"Start":"03:08.480 ","End":"03:10.871","Text":"Look at this and look at the original,"},{"Start":"03:10.871 ","End":"03:12.500","Text":"I think this is better."},{"Start":"03:12.500 ","End":"03:16.930","Text":"We have a choice of either elimination or substitution."},{"Start":"03:16.930 ","End":"03:20.840","Text":"In this case I\u0027m going to go for substitution because I can see that I"},{"Start":"03:20.840 ","End":"03:24.740","Text":"can easily take x and get it in terms of y,"},{"Start":"03:24.740 ","End":"03:26.750","Text":"but just throw this y over to the other side."},{"Start":"03:26.750 ","End":"03:34.340","Text":"Let\u0027s do that. I\u0027ll take the second equation and write it as x=y plus 1,"},{"Start":"03:34.340 ","End":"03:36.155","Text":"I just threw this over the other side."},{"Start":"03:36.155 ","End":"03:39.020","Text":"Now that I have what x equals substituted in"},{"Start":"03:39.020 ","End":"03:43.675","Text":"the other equation in number 1 and I get minus 5."},{"Start":"03:43.675 ","End":"03:45.920","Text":"It will look the same except instead of x,"},{"Start":"03:45.920 ","End":"03:53.720","Text":"I put y plus 1 in brackets plus 6y=0."},{"Start":"03:53.720 ","End":"03:55.925","Text":"Let\u0027s simplify this."},{"Start":"03:55.925 ","End":"03:58.650","Text":"I get minus 5y,"},{"Start":"03:59.170 ","End":"04:05.580","Text":"and then minus 5 plus 6y=0."},{"Start":"04:06.760 ","End":"04:10.780","Text":"Let me continue over here."},{"Start":"04:10.820 ","End":"04:20.068","Text":"I leave the y\u0027s on the left and have minus 5y plus 6y and numbers on the right,"},{"Start":"04:20.068 ","End":"04:22.185","Text":"so it becomes plus 5."},{"Start":"04:22.185 ","End":"04:24.915","Text":"What is minus 5y plus 6y?"},{"Start":"04:24.915 ","End":"04:27.500","Text":"Isn\u0027t it just 1y? In other words,"},{"Start":"04:27.500 ","End":"04:30.005","Text":"y is equal to 5."},{"Start":"04:30.005 ","End":"04:38.970","Text":"I have y=5 I can put that in here and x=y plus 1."},{"Start":"04:39.880 ","End":"04:42.915","Text":"X=5; parenthesis is just parenthesis,"},{"Start":"04:42.915 ","End":"04:47.365","Text":"plus 1 which is equal to 6."},{"Start":"04:47.365 ","End":"04:51.325","Text":"Now that I\u0027ve got this and this let me just collect it together,"},{"Start":"04:51.325 ","End":"04:53.120","Text":"what x is and y is here."},{"Start":"04:53.120 ","End":"04:54.263","Text":"Over here I\u0027ll do that,"},{"Start":"04:54.263 ","End":"05:00.080","Text":"it\u0027s always x equals something and y equals something."},{"Start":"05:00.080 ","End":"05:03.367","Text":"Answer is 2 inequalities,"},{"Start":"05:03.367 ","End":"05:05.105","Text":"what x is and what y is."},{"Start":"05:05.105 ","End":"05:10.720","Text":"X is 6 and y is 5."},{"Start":"05:10.720 ","End":"05:15.430","Text":"I\u0027ll just highlight that and we\u0027re done."}],"ID":4967},{"Watched":false,"Name":"Exercise 7","Duration":"5m 55s","ChapterTopicVideoID":4963,"CourseChapterTopicPlaylistID":3865,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.965","Text":"Here I have once again a system of 2 equations in 2 unknowns, x and y."},{"Start":"00:04.965 ","End":"00:10.425","Text":"This time I see there are fractions and I want to get rid of the fractions."},{"Start":"00:10.425 ","End":"00:13.545","Text":"Just like with addition and subtraction of fractions,"},{"Start":"00:13.545 ","End":"00:18.525","Text":"we want to use the technique of common denominator. I\u0027ll work over here."},{"Start":"00:18.525 ","End":"00:20.820","Text":"I look at 8, 16,"},{"Start":"00:20.820 ","End":"00:25.785","Text":"and 4 and I see what is the common denominator."},{"Start":"00:25.785 ","End":"00:29.205","Text":"Clear that 16 will do for all of them."},{"Start":"00:29.205 ","End":"00:35.035","Text":"I\u0027m going to multiply this equation, multiply by 16."},{"Start":"00:35.035 ","End":"00:36.740","Text":"Now because of the fractions,"},{"Start":"00:36.740 ","End":"00:41.640","Text":"what I have to do is just multiply this numerator by 2."},{"Start":"00:41.640 ","End":"00:43.110","Text":"Why 2?"},{"Start":"00:43.110 ","End":"00:45.330","Text":"Because that\u0027s 16 over 8."},{"Start":"00:45.330 ","End":"00:51.620","Text":"Here, 16 into 16 goes once so I have to multiply this one by 1."},{"Start":"00:51.620 ","End":"00:54.050","Text":"The last one, 16 over 4 is 4,"},{"Start":"00:54.050 ","End":"00:56.350","Text":"I have to multiply it by 4."},{"Start":"00:56.350 ","End":"01:00.990","Text":"What I get is, I\u0027m going to write it just below this one."},{"Start":"01:00.990 ","End":"01:03.060","Text":"2x minus 3,"},{"Start":"01:03.060 ","End":"01:10.740","Text":"I\u0027ll just leave it as 2x minus 3 minus 1 times"},{"Start":"01:10.740 ","End":"01:19.390","Text":"x plus y and then equals 4 times y minus 1."},{"Start":"01:19.390 ","End":"01:24.740","Text":"This is what I get when I multiply both sides of this equation by 16 and of course,"},{"Start":"01:24.740 ","End":"01:29.190","Text":"we can do the same thing to both sides of an equation or equality."},{"Start":"01:29.500 ","End":"01:33.650","Text":"Now, I have this equation and this equation,"},{"Start":"01:33.650 ","End":"01:39.580","Text":"It\u0027s like this was equation number 1 and this was equation number 2."},{"Start":"01:39.580 ","End":"01:44.300","Text":"But now I can look at this as my new number 1 after simplification"},{"Start":"01:44.300 ","End":"01:50.400","Text":"so these are the 2 that I have to solve now."},{"Start":"01:51.910 ","End":"01:55.055","Text":"Let\u0027s simplify each one."},{"Start":"01:55.055 ","End":"01:59.275","Text":"I\u0027ll start with this one."},{"Start":"01:59.275 ","End":"02:02.010","Text":"Let\u0027s see, just multiplying brackets,"},{"Start":"02:02.010 ","End":"02:05.250","Text":"2x minus twice 3."},{"Start":"02:05.250 ","End":"02:11.240","Text":"Then here is a minus 1 so I reverse each of them so it\u0027s minus x and minus y."},{"Start":"02:11.240 ","End":"02:15.660","Text":"Here, 4 times y minus 4 times 1."},{"Start":"02:15.660 ","End":"02:18.135","Text":"Bring the x\u0027s and y\u0027s to the left."},{"Start":"02:18.135 ","End":"02:21.575","Text":"I have 2x and I have minus x."},{"Start":"02:21.575 ","End":"02:23.640","Text":"Next, collect y\u0027s."},{"Start":"02:23.640 ","End":"02:28.045","Text":"Minus y, and from here I go minus 4y."},{"Start":"02:28.045 ","End":"02:30.515","Text":"Throw the numbers onto the right."},{"Start":"02:30.515 ","End":"02:34.235","Text":"I have 6 becomes a plus 6,"},{"Start":"02:34.235 ","End":"02:41.430","Text":"and then a minus 4 and so let\u0027s see what I get."},{"Start":"02:41.430 ","End":"02:47.700","Text":"I get 2x minus x is x minus y minus"},{"Start":"02:47.700 ","End":"02:54.600","Text":"4y is minus 5y and 6 minus 4 is 2."},{"Start":"02:54.600 ","End":"02:58.000","Text":"So that simplified the first."},{"Start":"03:00.050 ","End":"03:05.135","Text":"I\u0027ll write a note to myself that this was number 1 after further simplification."},{"Start":"03:05.135 ","End":"03:08.450","Text":"Now let\u0027s go to number 2 and let\u0027s do that over here."},{"Start":"03:08.450 ","End":"03:11.675","Text":"Let\u0027s put a little arrow to show where I\u0027m coming from."},{"Start":"03:11.675 ","End":"03:13.790","Text":"No, not this one."},{"Start":"03:13.790 ","End":"03:16.135","Text":"Sorry, I meant this one."},{"Start":"03:16.135 ","End":"03:21.720","Text":"3 times 2x is 6x minus 3 times"},{"Start":"03:21.720 ","End":"03:30.190","Text":"y minus 4x minus 11 equals 0."},{"Start":"03:30.190 ","End":"03:34.145","Text":"Let\u0027s see, x\u0027s here."},{"Start":"03:34.145 ","End":"03:36.455","Text":"Let\u0027s just do it in 1 step."},{"Start":"03:36.455 ","End":"03:42.115","Text":"6x minus 4x is plus 2x and"},{"Start":"03:42.115 ","End":"03:48.200","Text":"then the minus 3y stays and the 11 goes over to the other side."},{"Start":"03:48.200 ","End":"03:51.030","Text":"This is equal to 11."},{"Start":"03:51.830 ","End":"03:57.555","Text":"This is my number 2 and I just label it here."},{"Start":"03:57.555 ","End":"04:00.630","Text":"Look, this I have and this I have."},{"Start":"04:00.630 ","End":"04:02.710","Text":"What I suggest,"},{"Start":"04:02.710 ","End":"04:05.885","Text":"I\u0027m thinking about either substitution or elimination,"},{"Start":"04:05.885 ","End":"04:11.320","Text":"and here it\u0027s easy to get x in terms of y so I\u0027m going to go for the substitution method."},{"Start":"04:11.320 ","End":"04:18.700","Text":"From here, I can get that x equals plus 5y plus 2."},{"Start":"04:18.700 ","End":"04:23.070","Text":"Now I\u0027m going to substitute that up here."},{"Start":"04:23.070 ","End":"04:29.645","Text":"I get 2 and instead of x I write what x is equal to in brackets,"},{"Start":"04:29.645 ","End":"04:32.480","Text":"5y plus 2,"},{"Start":"04:32.480 ","End":"04:36.730","Text":"and then minus 3y equals 11."},{"Start":"04:36.730 ","End":"04:41.595","Text":"Now I want to simplify some more so 2 5y is 10,"},{"Start":"04:41.595 ","End":"04:44.280","Text":"plus 2 2 is 4,"},{"Start":"04:44.280 ","End":"04:47.130","Text":"minus 3y equals 11."},{"Start":"04:47.130 ","End":"04:52.805","Text":"Because we\u0027re working here with 1 equation with 1 unknown which is y,"},{"Start":"04:52.805 ","End":"04:56.700","Text":"let\u0027s leave the y on the left so that\u0027s"},{"Start":"04:56.700 ","End":"05:01.490","Text":"10y minus 3y is 7y and throw the 4 over to the other side."},{"Start":"05:01.490 ","End":"05:02.660","Text":"We can mentally do this."},{"Start":"05:02.660 ","End":"05:04.885","Text":"11 minus 4 is 7."},{"Start":"05:04.885 ","End":"05:11.030","Text":"Now divide by 7 and we have that y is equal to 1."},{"Start":"05:11.030 ","End":"05:12.605","Text":"That\u0027s the first one."},{"Start":"05:12.605 ","End":"05:14.030","Text":"When you have a substitution,"},{"Start":"05:14.030 ","End":"05:18.560","Text":"you just plug this in where it was substituted so now x"},{"Start":"05:18.560 ","End":"05:26.155","Text":"equals 5y is 5 times 1 plus 2."},{"Start":"05:26.155 ","End":"05:32.805","Text":"That\u0027s 5 plus 2 and that is equal to 7."},{"Start":"05:32.805 ","End":"05:35.560","Text":"Now I have y is 1, x is 7."},{"Start":"05:35.560 ","End":"05:40.115","Text":"Let me just write that down at the side somewhere maybe over here is good place."},{"Start":"05:40.115 ","End":"05:49.355","Text":"I write x equals 7 and y equals 1 and that\u0027s the solution."},{"Start":"05:49.355 ","End":"05:54.360","Text":"You know me, I like to highlight the solution so there we are."}],"ID":4968},{"Watched":false,"Name":"Exercise 8","Duration":"6m 47s","ChapterTopicVideoID":4964,"CourseChapterTopicPlaylistID":3865,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"Once again, we have a system of equations,"},{"Start":"00:03.120 ","End":"00:06.389","Text":"2 equations and 2 unknowns, x and y."},{"Start":"00:06.389 ","End":"00:09.180","Text":"Each time they make it a little bit harder,"},{"Start":"00:09.180 ","End":"00:12.120","Text":"this time both of them have fractions in."},{"Start":"00:12.120 ","End":"00:14.940","Text":"Let\u0027s see how we get rid of the fractions."},{"Start":"00:14.940 ","End":"00:17.460","Text":"What\u0027s the common denominator going to be?"},{"Start":"00:17.460 ","End":"00:19.410","Text":"In the case of the first equation,"},{"Start":"00:19.410 ","End":"00:21.765","Text":"I have to look at 4, 5, and 10,"},{"Start":"00:21.765 ","End":"00:23.220","Text":"and after thinking a little bit,"},{"Start":"00:23.220 ","End":"00:27.705","Text":"I see that 20 is the best I can do for a common denominator."},{"Start":"00:27.705 ","End":"00:30.420","Text":"I\u0027m going to multiply this one by 20,"},{"Start":"00:30.420 ","End":"00:32.085","Text":"and the second one,"},{"Start":"00:32.085 ","End":"00:33.945","Text":"4 will do for both,"},{"Start":"00:33.945 ","End":"00:37.320","Text":"so I\u0027m going to multiply the second one by 4."},{"Start":"00:38.150 ","End":"00:40.530","Text":"For this one,"},{"Start":"00:40.530 ","End":"00:43.110","Text":"I multiply by 20."},{"Start":"00:43.110 ","End":"00:44.940","Text":"Let\u0027s see what I need to do,"},{"Start":"00:44.940 ","End":"00:46.530","Text":"20/4 is 5,"},{"Start":"00:46.530 ","End":"00:50.435","Text":"so I take the numerator and multiply that by 5."},{"Start":"00:50.435 ","End":"00:53.535","Text":"Here, 20/5 is 4,"},{"Start":"00:53.535 ","End":"00:57.425","Text":"so I need to take this numerator and multiply it by 4,"},{"Start":"00:57.425 ","End":"00:59.795","Text":"and 20/10 is 2."},{"Start":"00:59.795 ","End":"01:02.885","Text":"Similarly here, 4/4 is 1,"},{"Start":"01:02.885 ","End":"01:08.430","Text":"and 4/2 is 2."},{"Start":"01:08.430 ","End":"01:11.330","Text":"The way this works is we just take each time"},{"Start":"01:11.330 ","End":"01:15.740","Text":"the thing that I\u0027ve marked in orange here and times the numerator,"},{"Start":"01:15.740 ","End":"01:20.880","Text":"so it\u0027s 5 times 3x minus 1."},{"Start":"01:20.880 ","End":"01:22.965","Text":"There is a minus here,"},{"Start":"01:22.965 ","End":"01:28.950","Text":"and then 4 times still the 2,"},{"Start":"01:28.950 ","End":"01:30.570","Text":"the 2 has to stay,"},{"Start":"01:30.570 ","End":"01:33.880","Text":"and then the x minus y."},{"Start":"01:34.430 ","End":"01:42.190","Text":"Here 2 times 3 times x plus 3."},{"Start":"01:43.310 ","End":"01:46.440","Text":"I can either simplify this."},{"Start":"01:46.440 ","End":"01:52.175","Text":"I\u0027ll continue with this just to be in parallel, in sync."},{"Start":"01:52.175 ","End":"01:55.790","Text":"This equation becomes, let\u0027s see,"},{"Start":"01:55.790 ","End":"02:01.405","Text":"1 times x plus 1 is just x plus 1."},{"Start":"02:01.405 ","End":"02:06.185","Text":"I\u0027ll put it as 1 times x plus 1 so you can see where I\u0027m coming from,"},{"Start":"02:06.185 ","End":"02:12.405","Text":"and then minus 2 times y."},{"Start":"02:12.405 ","End":"02:17.050","Text":"Here I forgot to say that we have to multiply the right-hand side by 4,"},{"Start":"02:17.050 ","End":"02:18.400","Text":"also there\u0027s no fractions,"},{"Start":"02:18.400 ","End":"02:20.450","Text":"it\u0027s just the full 4."},{"Start":"02:20.450 ","End":"02:25.065","Text":"Here this is equal is 4 times 1 is 4."},{"Start":"02:25.065 ","End":"02:28.750","Text":"Now, let\u0027s see."},{"Start":"02:28.880 ","End":"02:31.140","Text":"I\u0027m going to work this."},{"Start":"02:31.140 ","End":"02:34.020","Text":"I\u0027ll tell you what,"},{"Start":"02:34.020 ","End":"02:37.655","Text":"I\u0027ll simplify this one over here,"},{"Start":"02:37.655 ","End":"02:40.325","Text":"and then I\u0027ll simplify this one over there."},{"Start":"02:40.325 ","End":"02:42.695","Text":"Let\u0027s start with this one,"},{"Start":"02:42.695 ","End":"02:47.405","Text":"1 times x plus 1 is x plus 1,"},{"Start":"02:47.405 ","End":"02:51.625","Text":"then minus 2y equals 4."},{"Start":"02:51.625 ","End":"02:55.280","Text":"If I want to bring it into the standard form,"},{"Start":"02:55.280 ","End":"03:01.670","Text":"then already I have the x and then I take the y\u0027s minus 2y,"},{"Start":"03:01.670 ","End":"03:02.780","Text":"numbers on the right,"},{"Start":"03:02.780 ","End":"03:05.385","Text":"so it\u0027s 1 comes over as minus,"},{"Start":"03:05.385 ","End":"03:07.770","Text":"so this is 4 minus 1 is 3."},{"Start":"03:07.770 ","End":"03:09.360","Text":"That\u0027s this one simplified."},{"Start":"03:09.360 ","End":"03:11.585","Text":"Now this one a little bit more work,"},{"Start":"03:11.585 ","End":"03:18.960","Text":"5 times 3x is 15x minus 5."},{"Start":"03:19.000 ","End":"03:23.760","Text":"Let me just think of the 4 times 2."},{"Start":"03:23.760 ","End":"03:25.620","Text":"I\u0027ll just think of it as 8 to save a step,"},{"Start":"03:25.620 ","End":"03:29.620","Text":"and I\u0027ll think if 2 times 3 is 6 and save a step."},{"Start":"03:29.660 ","End":"03:33.390","Text":"What I\u0027ll get is minus 8 times x minus y,"},{"Start":"03:33.390 ","End":"03:35.100","Text":"so it\u0027s minus 8x,"},{"Start":"03:35.100 ","End":"03:39.435","Text":"but minus 8 minus y is plus 8y."},{"Start":"03:39.435 ","End":"03:41.145","Text":"Watch out for that."},{"Start":"03:41.145 ","End":"03:50.355","Text":"Then we have 6 times x is 6x,and 6 times 3 is 18."},{"Start":"03:50.355 ","End":"03:52.320","Text":"Let\u0027s see now. X is on the left,"},{"Start":"03:52.320 ","End":"03:58.365","Text":"so I have 15x minus 8x and minus 6x."},{"Start":"03:58.365 ","End":"03:59.670","Text":"That\u0027s taking care of the x."},{"Start":"03:59.670 ","End":"04:04.875","Text":"Now the y\u0027s, all I have is one place, 8y."},{"Start":"04:04.875 ","End":"04:06.960","Text":"Now numbers on the right,"},{"Start":"04:06.960 ","End":"04:08.955","Text":"I already had an 18 here."},{"Start":"04:08.955 ","End":"04:12.045","Text":"The 5 comes over as plus 5."},{"Start":"04:12.045 ","End":"04:16.215","Text":"Let\u0027s see, 15 minus 8 minus 6,"},{"Start":"04:16.215 ","End":"04:18.990","Text":"15 minus 14 is 1, so it\u0027s 1x."},{"Start":"04:18.990 ","End":"04:25.365","Text":"1x plus 8y equals,"},{"Start":"04:25.365 ","End":"04:28.320","Text":"15 and 5 is 23."},{"Start":"04:28.320 ","End":"04:35.500","Text":"Altogether, what\u0027s happened is this was my equation number 1,"},{"Start":"04:35.500 ","End":"04:37.960","Text":"and I have to do a lot of simplification, let\u0027s see."},{"Start":"04:37.960 ","End":"04:42.620","Text":"We ended up here and this is going to be my equation 1."},{"Start":"04:43.060 ","End":"04:46.895","Text":"This one through here and here and here,"},{"Start":"04:46.895 ","End":"04:48.620","Text":"this was my number 2,"},{"Start":"04:48.620 ","End":"04:52.400","Text":"and this is here going to be my equation number 2."},{"Start":"04:52.400 ","End":"04:55.610","Text":"I could put them together nicely with curly braces,"},{"Start":"04:55.610 ","End":"04:56.945","Text":"but I won\u0027t bother."},{"Start":"04:56.945 ","End":"05:02.825","Text":"What we\u0027ll do is take a look and see if we want to use elimination or substitution."},{"Start":"05:02.825 ","End":"05:04.685","Text":"Both would be good."},{"Start":"05:04.685 ","End":"05:07.820","Text":"But I think we can easily get"},{"Start":"05:07.820 ","End":"05:11.830","Text":"x in terms of y from this or from this, as a matter of fact."},{"Start":"05:11.830 ","End":"05:13.630","Text":"I\u0027ll tell you what, I\u0027ll go with this."},{"Start":"05:13.630 ","End":"05:15.115","Text":"I\u0027ll go with substitution,"},{"Start":"05:15.115 ","End":"05:24.045","Text":"and I\u0027ll write this one as x equals plus 2y plus 3,"},{"Start":"05:24.045 ","End":"05:27.125","Text":"and substitute that in here."},{"Start":"05:27.125 ","End":"05:29.290","Text":"Let\u0027s just scroll down a bit,"},{"Start":"05:29.290 ","End":"05:31.640","Text":"get some more room."},{"Start":"05:33.200 ","End":"05:36.735","Text":"Putting x is 2y plus 3 here,"},{"Start":"05:36.735 ","End":"05:41.670","Text":"I get 2y plus 3."},{"Start":"05:41.670 ","End":"05:44.550","Text":"I just put it in brackets for emphasis,"},{"Start":"05:44.550 ","End":"05:51.685","Text":"plus 8y equals 23. Then let\u0027s see."},{"Start":"05:51.685 ","End":"05:53.650","Text":"We have an equation and just y,"},{"Start":"05:53.650 ","End":"05:57.430","Text":"2y and 8y is 10y,"},{"Start":"05:57.430 ","End":"06:01.055","Text":"and the 3 goes over to the other side is minus 3,"},{"Start":"06:01.055 ","End":"06:03.810","Text":"23 minus 3 is 20."},{"Start":"06:03.810 ","End":"06:05.865","Text":"Now dividing by 10,"},{"Start":"06:05.865 ","End":"06:10.020","Text":"we see that y equals 2."},{"Start":"06:10.020 ","End":"06:11.800","Text":"If y equals 2,"},{"Start":"06:11.800 ","End":"06:16.160","Text":"I can now substitute that over here because I\u0027ve got x in terms of y."},{"Start":"06:16.160 ","End":"06:17.720","Text":"So x equals twice."},{"Start":"06:17.720 ","End":"06:19.655","Text":"Now y is 2,"},{"Start":"06:19.655 ","End":"06:23.755","Text":"so I\u0027ll put that 2 here, plus 3."},{"Start":"06:23.755 ","End":"06:25.770","Text":"This equals 2 times 2 is 4,"},{"Start":"06:25.770 ","End":"06:27.975","Text":"plus 3 is 7."},{"Start":"06:27.975 ","End":"06:30.270","Text":"Now I have what everything is."},{"Start":"06:30.270 ","End":"06:33.270","Text":"I\u0027ll just write that somewhere."},{"Start":"06:33.270 ","End":"06:34.715","Text":"Here\u0027s a nice spot."},{"Start":"06:34.715 ","End":"06:41.850","Text":"I write that as x equals 7 and y equals 2,"},{"Start":"06:41.850 ","End":"06:46.980","Text":"and I\u0027ll highlight that and we are done."}],"ID":4969},{"Watched":false,"Name":"Exercise 9","Duration":"1m 45s","ChapterTopicVideoID":4965,"CourseChapterTopicPlaylistID":3865,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"Here we have yet another system of equations,"},{"Start":"00:03.060 ","End":"00:05.880","Text":"2 equations and 2 unknowns, x, y."},{"Start":"00:05.880 ","End":"00:08.175","Text":"Let\u0027s see if we can solve it."},{"Start":"00:08.175 ","End":"00:11.715","Text":"We have to decide to go for substitution or elimination."},{"Start":"00:11.715 ","End":"00:14.610","Text":"But look, we have a plus 3 and minus 3,"},{"Start":"00:14.610 ","End":"00:16.980","Text":"so it begs for the elimination methods."},{"Start":"00:16.980 ","End":"00:20.084","Text":"Let\u0027s add these 2 equations together."},{"Start":"00:20.084 ","End":"00:23.840","Text":"If 2 things are equal and another 2 things are equal,"},{"Start":"00:23.840 ","End":"00:25.955","Text":"we can add the equals to the equals."},{"Start":"00:25.955 ","End":"00:29.690","Text":"So x plus x is 2x,"},{"Start":"00:29.690 ","End":"00:33.930","Text":"3y and minus 3y is 0."},{"Start":"00:33.930 ","End":"00:35.175","Text":"We don\u0027t write anything,"},{"Start":"00:35.175 ","End":"00:38.595","Text":"and 5 plus 3=8,"},{"Start":"00:38.595 ","End":"00:44.260","Text":"2x=8, just divide by 2, x=4."},{"Start":"00:44.510 ","End":"00:48.305","Text":"We\u0027ve solved the x now we need to get y."},{"Start":"00:48.305 ","End":"00:52.670","Text":"We just have to substitute in either one of these."},{"Start":"00:52.670 ","End":"00:55.760","Text":"Let\u0027s say the first because it\u0027s first."},{"Start":"00:55.760 ","End":"00:57.935","Text":"I\u0027ll put that into here."},{"Start":"00:57.935 ","End":"01:00.965","Text":"Then what I get is x is 4,"},{"Start":"01:00.965 ","End":"01:08.780","Text":"so 4 plus 3y=5."},{"Start":"01:08.780 ","End":"01:15.290","Text":"Now subtract 4 from both sides or bring the numbers over to the right."},{"Start":"01:15.290 ","End":"01:20.335","Text":"Eventually I get 3y=5 minus 4, which is 1."},{"Start":"01:20.335 ","End":"01:30.485","Text":"Now I divide by 3 both sides or bring the 3 to the denominator, y=1/3."},{"Start":"01:30.485 ","End":"01:33.530","Text":"Just to write it nicely,"},{"Start":"01:33.530 ","End":"01:41.225","Text":"we summarize and say the solution is x=4, y=1/3."},{"Start":"01:41.225 ","End":"01:45.090","Text":"That\u0027s the answer. We are done."}],"ID":4970},{"Watched":false,"Name":"Exercise 10","Duration":"2m 21s","ChapterTopicVideoID":4966,"CourseChapterTopicPlaylistID":3865,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.850","Text":"Here we\u0027re given a system of two equations and two unknowns which we have to solve."},{"Start":"00:05.850 ","End":"00:07.545","Text":"I\u0027ll solve it over here."},{"Start":"00:07.545 ","End":"00:11.100","Text":"I look at this, and I\u0027m thinking substitution or elimination."},{"Start":"00:11.100 ","End":"00:12.510","Text":"I notice that there\u0027s a 5,"},{"Start":"00:12.510 ","End":"00:14.400","Text":"and a 5 here."},{"Start":"00:14.400 ","End":"00:17.370","Text":"If I just multiply one of them,"},{"Start":"00:17.370 ","End":"00:20.710","Text":"let\u0027s say I multiply the second by minus 1,"},{"Start":"00:20.710 ","End":"00:23.640","Text":"then I\u0027ll get a 5, and a minus 5,"},{"Start":"00:23.640 ","End":"00:25.140","Text":"not a 5, and a 5, anyway,"},{"Start":"00:25.140 ","End":"00:27.660","Text":"let\u0027s just multiply this by minus 1,"},{"Start":"00:27.660 ","End":"00:37.140","Text":"which means reversing the sign, so -5x-3y=-23."},{"Start":"00:37.140 ","End":"00:42.615","Text":"Now, let me take this one and this one,"},{"Start":"00:42.615 ","End":"00:44.135","Text":"and add the 2,"},{"Start":"00:44.135 ","End":"00:46.370","Text":"just ignore the middle one."},{"Start":"00:46.370 ","End":"00:50.815","Text":"5 and -5 is nothing,"},{"Start":"00:50.815 ","End":"00:53.940","Text":"2+(-3) is -1,"},{"Start":"00:53.940 ","End":"00:56.070","Text":"so we have minus y,"},{"Start":"00:56.070 ","End":"01:02.310","Text":"and 14 and -23 gives us minus 9."},{"Start":"01:02.310 ","End":"01:05.690","Text":"Basically, I can cancel the minuses."},{"Start":"01:05.690 ","End":"01:08.240","Text":"Just divide both sides by -1,"},{"Start":"01:08.240 ","End":"01:10.985","Text":"and you get that y=9."},{"Start":"01:10.985 ","End":"01:14.270","Text":"Now I just have to find what x is."},{"Start":"01:14.270 ","End":"01:17.195","Text":"I\u0027ll substitute it, let\u0027s say,"},{"Start":"01:17.195 ","End":"01:18.830","Text":"in this one,"},{"Start":"01:18.830 ","End":"01:22.579","Text":"I\u0027ll put the y=9 here."},{"Start":"01:22.579 ","End":"01:32.815","Text":"What I get is 5x+2(9)=14."},{"Start":"01:32.815 ","End":"01:41.325","Text":"5x equals, let\u0027s bring the numbers to the other side, 14 minus,"},{"Start":"01:41.325 ","End":"01:46.320","Text":"twice 9 is 18,"},{"Start":"01:46.320 ","End":"01:48.810","Text":"so it\u0027s 14 minus 18,"},{"Start":"01:48.810 ","End":"01:52.210","Text":"which is minus 4."},{"Start":"01:52.210 ","End":"02:01.080","Text":"Now if I divide both sides by 5, I get x=-4/5."},{"Start":"02:01.610 ","End":"02:04.220","Text":"This and this together give the answer,"},{"Start":"02:04.220 ","End":"02:07.070","Text":"but I like to write it properly."},{"Start":"02:07.070 ","End":"02:12.635","Text":"The solution is x=-4/5,"},{"Start":"02:12.635 ","End":"02:17.190","Text":"and y=9."},{"Start":"02:17.500 ","End":"02:20.700","Text":"We are done."}],"ID":4971}],"Thumbnail":null,"ID":3865},{"Name":"Linear Two Unknowns, 0 Or Infinite Solutions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Linear in Two Unknowns - Degenerate","Duration":"6m 36s","ChapterTopicVideoID":4974,"CourseChapterTopicPlaylistID":3866,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.190","Text":"In this clip, I\u0027m going to also be talking about linear equations in 2 unknowns still,"},{"Start":"00:05.190 ","End":"00:06.885","Text":"but a different topic."},{"Start":"00:06.885 ","End":"00:10.050","Text":"Let me call it unusual cases,"},{"Start":"00:10.050 ","End":"00:13.358","Text":"although I have to explain what I mean by unusual."},{"Start":"00:13.358 ","End":"00:20.920","Text":"Let me just write it first unusual cases and put that in quotes and highlight it there."},{"Start":"00:20.920 ","End":"00:23.540","Text":"I\u0027ll tell you what I mean by usual."},{"Start":"00:23.540 ","End":"00:26.030","Text":"In the usual case,"},{"Start":"00:26.030 ","End":"00:31.640","Text":"what we have is 1 solution only."},{"Start":"00:31.640 ","End":"00:38.510","Text":"We have a system of 2 equations with 2 unknowns and there\u0027s only one solution."},{"Start":"00:38.510 ","End":"00:41.690","Text":"I know when I write a solution I say x equals and y equals,"},{"Start":"00:41.690 ","End":"00:44.435","Text":"but that\u0027s still part of 1 solution."},{"Start":"00:44.435 ","End":"00:52.340","Text":"What I call unusual is that there are other possibilities."},{"Start":"00:52.340 ","End":"00:56.730","Text":"We could have no solution and"},{"Start":"00:56.730 ","End":"01:03.420","Text":"the other possibility there could be an infinite number of solutions."},{"Start":"01:04.470 ","End":"01:08.075","Text":"We can also have an extreme case where"},{"Start":"01:08.075 ","End":"01:15.685","Text":"all values of x and y are solutions, doesn\u0027t happen often."},{"Start":"01:15.685 ","End":"01:18.145","Text":"I\u0027ll give you an example of each."},{"Start":"01:18.145 ","End":"01:20.510","Text":"First example I\u0027ll give,"},{"Start":"01:20.510 ","End":"01:25.865","Text":"let\u0027s take x plus y=3"},{"Start":"01:25.865 ","End":"01:33.345","Text":"and 2x plus 2y=7."},{"Start":"01:33.345 ","End":"01:37.315","Text":"We could solve it in either method, substitution or elimination."},{"Start":"01:37.315 ","End":"01:39.415","Text":"Let\u0027s say I go for substitution."},{"Start":"01:39.415 ","End":"01:43.928","Text":"I take the first one and say from here let\u0027s say I"},{"Start":"01:43.928 ","End":"01:50.635","Text":"isolate x or extract x or whatever."},{"Start":"01:50.635 ","End":"01:55.760","Text":"x equals minus y plus 3,"},{"Start":"01:55.760 ","End":"01:58.265","Text":"I bring the y over to the other side."},{"Start":"01:58.265 ","End":"02:02.630","Text":"Now I have x equals and I plug that in here,"},{"Start":"02:02.630 ","End":"02:08.060","Text":"just have to put x there and so I get twice minus y"},{"Start":"02:08.060 ","End":"02:13.875","Text":"plus 3 plus 2y=7,"},{"Start":"02:13.875 ","End":"02:17.190","Text":"open the brackets, minus 2y,"},{"Start":"02:17.190 ","End":"02:22.515","Text":"twice 3 is 6 plus 2y=7"},{"Start":"02:22.515 ","End":"02:29.490","Text":"and minus 2y and plus 2y cancel each other out."},{"Start":"02:29.490 ","End":"02:35.525","Text":"What I\u0027m left with is that 6=7 which is not possible."},{"Start":"02:35.525 ","End":"02:38.105","Text":"Whenever you reach an impossibility,"},{"Start":"02:38.105 ","End":"02:41.100","Text":"it means there is no solution."},{"Start":"02:42.500 ","End":"02:46.070","Text":"You can see by looking at it,"},{"Start":"02:46.070 ","End":"02:47.240","Text":"why it\u0027s so, I mean,"},{"Start":"02:47.240 ","End":"02:51.965","Text":"what I did is I took this x plus y=3 in my head and doubled it and said well,"},{"Start":"02:51.965 ","End":"02:53.405","Text":"if x plus y is 3,"},{"Start":"02:53.405 ","End":"02:54.950","Text":"then twice this,"},{"Start":"02:54.950 ","End":"02:56.840","Text":"2x plus 2y must be 6."},{"Start":"02:56.840 ","End":"02:57.860","Text":"It can\u0027t be 7,"},{"Start":"02:57.860 ","End":"03:01.040","Text":"so I cooked it up so it wouldn\u0027t have an answer,"},{"Start":"03:01.040 ","End":"03:03.530","Text":"but it can happen."},{"Start":"03:03.530 ","End":"03:06.530","Text":"That\u0027s an example of no solution."},{"Start":"03:06.530 ","End":"03:12.005","Text":"Let\u0027s see what happens or if I can find an example of this case."},{"Start":"03:12.005 ","End":"03:13.775","Text":"I\u0027ll use the same idea."},{"Start":"03:13.775 ","End":"03:16.550","Text":"Suppose x plus y=3,"},{"Start":"03:16.550 ","End":"03:22.330","Text":"and I say that 2x plus 2y=6."},{"Start":"03:22.330 ","End":"03:24.615","Text":"Now, what would happen?"},{"Start":"03:24.615 ","End":"03:27.440","Text":"The same thing would happen."},{"Start":"03:27.440 ","End":"03:30.350","Text":"I would get from the first equation,"},{"Start":"03:30.350 ","End":"03:32.059","Text":"bring the y over,"},{"Start":"03:32.059 ","End":"03:37.325","Text":"get that x equals minus y plus 3,"},{"Start":"03:37.325 ","End":"03:42.950","Text":"and then I would substitute that in here and get twice"},{"Start":"03:42.950 ","End":"03:48.540","Text":"minus y plus 3 plus 2y equals,"},{"Start":"03:48.540 ","End":"03:50.130","Text":"but this time not 7,"},{"Start":"03:50.130 ","End":"03:54.240","Text":"6 and cutting out the middle step."},{"Start":"03:54.240 ","End":"03:56.160","Text":"In the end, what you will get is that"},{"Start":"03:56.160 ","End":"04:05.720","Text":"6=6 and that\u0027s a tautology or it\u0027s something that\u0027s evidently true."},{"Start":"04:05.720 ","End":"04:07.460","Text":"When you get something like this,"},{"Start":"04:07.460 ","End":"04:13.110","Text":"it\u0027s a sign that you have an infinite number of solutions and in this case,"},{"Start":"04:13.110 ","End":"04:14.600","Text":"just like I said before,"},{"Start":"04:14.600 ","End":"04:18.770","Text":"the second equation is really the same as the first equation."},{"Start":"04:18.770 ","End":"04:21.140","Text":"Why do we have an infinite number of solutions?"},{"Start":"04:21.140 ","End":"04:23.960","Text":"Because essentially we only have 1 equation."},{"Start":"04:23.960 ","End":"04:26.375","Text":"I could say that the first one,"},{"Start":"04:26.375 ","End":"04:33.070","Text":"this one is really the only equation because the second one can be deduced from it."},{"Start":"04:33.070 ","End":"04:34.825","Text":"If x plus y is 3 and I double it,"},{"Start":"04:34.825 ","End":"04:36.955","Text":"so I only have 1 equation,"},{"Start":"04:36.955 ","End":"04:44.520","Text":"and this way there\u0027s obviously many solutions for example, 0 plus 3=3."},{"Start":"04:44.520 ","End":"04:45.780","Text":"I could take x is 0,"},{"Start":"04:45.780 ","End":"04:48.690","Text":"y is 3, 1 plus 2 is 3,"},{"Start":"04:48.690 ","End":"04:50.730","Text":"2 plus 1 is 3,"},{"Start":"04:50.730 ","End":"04:53.535","Text":"3 plus 0 is 3,"},{"Start":"04:53.535 ","End":"04:56.850","Text":"4 and minus 1 is 3."},{"Start":"04:56.850 ","End":"04:59.130","Text":"I could take x=4,"},{"Start":"04:59.130 ","End":"05:01.270","Text":"y equals minus 1, and so on."},{"Start":"05:01.270 ","End":"05:04.075","Text":"This could be each time my x,"},{"Start":"05:04.075 ","End":"05:07.760","Text":"each time this will be the corresponding y."},{"Start":"05:08.040 ","End":"05:11.050","Text":"Obviously, this goes on forever,"},{"Start":"05:11.050 ","End":"05:12.790","Text":"an infinite number of solutions."},{"Start":"05:12.790 ","End":"05:17.395","Text":"But notice though that even though there\u0027s an infinite number of solutions,"},{"Start":"05:17.395 ","End":"05:19.450","Text":"it doesn\u0027t mean that every x,"},{"Start":"05:19.450 ","End":"05:21.684","Text":"y I take will be a solution."},{"Start":"05:21.684 ","End":"05:25.771","Text":"For example, now I\u0027ll use a different color to say it\u0027s a bad example,"},{"Start":"05:25.771 ","End":"05:33.575","Text":"suppose I took x=4 and y=3."},{"Start":"05:33.575 ","End":"05:35.410","Text":"Is that a solution?"},{"Start":"05:35.410 ","End":"05:41.275","Text":"No, because x plus y=7 which is not the same as 3."},{"Start":"05:41.275 ","End":"05:44.695","Text":"Although there are many solutions,"},{"Start":"05:44.695 ","End":"05:48.775","Text":"not every pair of values is a solution."},{"Start":"05:48.775 ","End":"05:55.770","Text":"Rare but not impossible cases to get that everything is a solution."},{"Start":"05:55.770 ","End":"05:58.025","Text":"I\u0027m deliberately giving a silly example."},{"Start":"05:58.025 ","End":"05:59.570","Text":"I mean, all the examples are going to be silly."},{"Start":"05:59.570 ","End":"06:02.930","Text":"Suppose I said x=x and y=y."},{"Start":"06:02.930 ","End":"06:09.005","Text":"I mean, that\u0027s 2 equations and 2 unknowns but obviously whatever x and y I gave you,"},{"Start":"06:09.005 ","End":"06:11.640","Text":"it\u0027s going to be true,"},{"Start":"06:11.640 ","End":"06:15.835","Text":"so this is all x and y."},{"Start":"06:15.835 ","End":"06:18.845","Text":"It\u0027s not just infinite but everything."},{"Start":"06:18.845 ","End":"06:22.680","Text":"Because here we had infinite but not everything."},{"Start":"06:23.780 ","End":"06:30.020","Text":"Anyway, those are the 3 cases and you won\u0027t find this in the solved example,"},{"Start":"06:30.020 ","End":"06:32.810","Text":"but you\u0027ll find examples of this and this."},{"Start":"06:32.810 ","End":"06:35.220","Text":"That\u0027s it."}],"ID":4972},{"Watched":false,"Name":"Exercise 1","Duration":"2m 5s","ChapterTopicVideoID":4972,"CourseChapterTopicPlaylistID":3866,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.715","Text":"Here we have a system of equations."},{"Start":"00:02.715 ","End":"00:05.340","Text":"Precisely it\u0027s 2 equations and 2 unknowns,"},{"Start":"00:05.340 ","End":"00:09.419","Text":"x and y, so let\u0027s do the solution."},{"Start":"00:09.419 ","End":"00:11.100","Text":"I\u0027d like to label them,"},{"Start":"00:11.100 ","End":"00:15.145","Text":"Number 1 and Number 2."},{"Start":"00:15.145 ","End":"00:19.250","Text":"We have to decide if we want to use the method of substitution or elimination,"},{"Start":"00:19.250 ","End":"00:21.485","Text":"I think I\u0027ll go for elimination."},{"Start":"00:21.485 ","End":"00:24.380","Text":"I look at this because there\u0027s only one, x here."},{"Start":"00:24.380 ","End":"00:27.140","Text":"So I could multiply this whole equation by minus"},{"Start":"00:27.140 ","End":"00:30.530","Text":"4 and then when I add, I\u0027ll get rid of x."},{"Start":"00:30.530 ","End":"00:34.670","Text":"So I\u0027ll just write a little minus 4 here to remind myself"},{"Start":"00:34.670 ","End":"00:39.900","Text":"I\u0027m multiplying this by minus 4 and I\u0027m going to do it down here."},{"Start":"00:39.910 ","End":"00:57.045","Text":"What I get is everything times so"},{"Start":"00:57.045 ","End":"01:04.550","Text":"minus 4x and minus 4 times this is minus 8y equals minus 4."},{"Start":"01:04.550 ","End":"01:09.530","Text":"Now we want to look at these 2 and add them."},{"Start":"01:09.530 ","End":"01:13.670","Text":"I\u0027m going to do an addition of these 2 so what I get is"},{"Start":"01:13.670 ","End":"01:18.570","Text":"4x and minus 4x cancel each other out, nothing."},{"Start":"01:18.570 ","End":"01:21.150","Text":"8y minus 8y,"},{"Start":"01:21.150 ","End":"01:23.490","Text":"they also cancel each other out."},{"Start":"01:23.490 ","End":"01:26.285","Text":"I can\u0027t just not write anything, I have to write 0."},{"Start":"01:26.285 ","End":"01:31.355","Text":"I mean this plus this is 0, and then 5 and minus 4 is 1."},{"Start":"01:31.355 ","End":"01:36.650","Text":"What I get is 0=1. How can that be?"},{"Start":"01:36.650 ","End":"01:40.280","Text":"The answer is it cannot be and the fact that we reach"},{"Start":"01:40.280 ","End":"01:44.180","Text":"something impossible means that there\u0027s no solution to the equation."},{"Start":"01:44.180 ","End":"01:46.460","Text":"If there was a solution to the equation,"},{"Start":"01:46.460 ","End":"01:49.595","Text":"it wouldn\u0027t imply that 0 equals 1 and that\u0027s never going to be,"},{"Start":"01:49.595 ","End":"01:54.725","Text":"so we just write that there is no solution or you can write a full sentence."},{"Start":"01:54.725 ","End":"02:00.960","Text":"The system of equations above has no solution."},{"Start":"02:01.240 ","End":"02:05.430","Text":"There\u0027s nothing more to be done. Well, that is it."}],"ID":4973},{"Watched":false,"Name":"Exercise 2","Duration":"5m 11s","ChapterTopicVideoID":4973,"CourseChapterTopicPlaylistID":3866,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.705","Text":"Here we have to solve the system of equations,"},{"Start":"00:03.705 ","End":"00:08.205","Text":"this 1, 2 equations in 2 unknowns. Solve it over here."},{"Start":"00:08.205 ","End":"00:10.020","Text":"I\u0027d like to label them,"},{"Start":"00:10.020 ","End":"00:14.370","Text":"equation number 1 and equation Number 2."},{"Start":"00:14.370 ","End":"00:17.760","Text":"It looked like they really need some simplification,"},{"Start":"00:17.760 ","End":"00:18.915","Text":"so let\u0027s do that."},{"Start":"00:18.915 ","End":"00:23.430","Text":"Let me work with equation Number 1 here,"},{"Start":"00:23.430 ","End":"00:27.135","Text":"and maybe over here I\u0027ll work with equation Number 2."},{"Start":"00:27.135 ","End":"00:30.890","Text":"So let\u0027s see, expanding the brackets for Number 1,"},{"Start":"00:30.890 ","End":"00:36.090","Text":"I get 2x minus 2y plus 4y=1 plus x."},{"Start":"00:40.160 ","End":"00:45.090","Text":"Now the usual, we bring the x\u0027s to the left and the y, let\u0027s see, why do we have x?"},{"Start":"00:45.090 ","End":"00:51.260","Text":"We have 2x here and we have an x here which is minus x,"},{"Start":"00:51.260 ","End":"00:58.620","Text":"and then the y, I have minus 2y plus 4y and on the right,"},{"Start":"00:58.620 ","End":"01:01.020","Text":"I have a Number 1."},{"Start":"01:01.020 ","End":"01:03.885","Text":"Just collect together the x\u0027s,"},{"Start":"01:03.885 ","End":"01:05.730","Text":"and then the y\u0027s,"},{"Start":"01:05.730 ","End":"01:08.730","Text":"2 minus 1 is 1 so I have 1x left,"},{"Start":"01:08.730 ","End":"01:11.040","Text":"minus 2y plus 4y,"},{"Start":"01:11.040 ","End":"01:16.485","Text":"minus 2 plus 4 is 2, so it\u0027s 2y=1."},{"Start":"01:16.485 ","End":"01:18.975","Text":"This is my simplified number 1."},{"Start":"01:18.975 ","End":"01:26.460","Text":"Now, let me do Number 2, so left-hand side,"},{"Start":"01:26.460 ","End":"01:28.810","Text":"nothing to expand,"},{"Start":"01:29.780 ","End":"01:35.100","Text":"2 minus 7y plus x is equal on the right-hand side,"},{"Start":"01:35.100 ","End":"01:36.900","Text":"I expand 3 times x,"},{"Start":"01:36.900 ","End":"01:39.070","Text":"so on 3x minus 3y."},{"Start":"01:41.690 ","End":"01:44.990","Text":"Once again, we bring the x\u0027s to the left."},{"Start":"01:44.990 ","End":"01:51.455","Text":"Let\u0027s see x\u0027s, we have a plus x from here and a minus 3x here."},{"Start":"01:51.455 ","End":"01:55.370","Text":"Then we have minus 7y."},{"Start":"01:55.370 ","End":"02:00.115","Text":"But the 3y becomes plus 3y."},{"Start":"02:00.115 ","End":"02:03.315","Text":"On the right, we bring the 2 over,"},{"Start":"02:03.315 ","End":"02:06.135","Text":"so we have minus 2,"},{"Start":"02:06.135 ","End":"02:08.760","Text":"and now let\u0027s see, collecting the x\u0027s,"},{"Start":"02:08.760 ","End":"02:11.600","Text":"1 minus 3 is minus 2,"},{"Start":"02:11.600 ","End":"02:19.271","Text":"for x\u0027s."},{"Start":"02:19.271 ","End":"02:25.185","Text":"minus 7 plus 3 is minus 4y, and this is equal to minus 2, just as is."},{"Start":"02:25.185 ","End":"02:29.985","Text":"I mentioned that if we have something that"},{"Start":"02:29.985 ","End":"02:35.980","Text":"divides everything a common divisor in this case 2 or even minus 2,"},{"Start":"02:35.980 ","End":"02:39.010","Text":"I can divide by it to simplify and it\u0027s worth doing,"},{"Start":"02:39.010 ","End":"02:42.850","Text":"so let\u0027s divide everything by minus 2."},{"Start":"02:42.850 ","End":"02:46.960","Text":"This becomes minus 2x over minus 2 is just x,"},{"Start":"02:46.960 ","End":"02:52.960","Text":"minus 4 over minus 2 is plus 2,"},{"Start":"02:52.960 ","End":"02:58.250","Text":"and minus 2 over minus 2 is 1."},{"Start":"02:58.250 ","End":"03:02.490","Text":"They look remarkably similar Number 1 and 2."},{"Start":"03:02.780 ","End":"03:07.520","Text":"It turns out that both equations are basically"},{"Start":"03:07.520 ","End":"03:12.540","Text":"equivalent to the same equation and we really only have 1 equation."},{"Start":"03:12.710 ","End":"03:15.590","Text":"If we have just 1 equation,"},{"Start":"03:15.590 ","End":"03:17.990","Text":"and I\u0027ll highlight it,"},{"Start":"03:17.990 ","End":"03:22.910","Text":"and if you get to this point where you only have 1 effective equation, I mean we had 2,"},{"Start":"03:22.910 ","End":"03:25.675","Text":"but they were the same, then,"},{"Start":"03:25.675 ","End":"03:29.950","Text":"what you say is that there are infinite number of solutions."},{"Start":"03:29.950 ","End":"03:35.595","Text":"You have infinite number of solutions."},{"Start":"03:35.595 ","End":"03:40.520","Text":"But that doesn\u0027t mean that every x and y are a solution."},{"Start":"03:40.520 ","End":"03:42.605","Text":"Let me give you an example."},{"Start":"03:42.605 ","End":"03:45.935","Text":"First of all, I\u0027ll show you that it has several solutions."},{"Start":"03:45.935 ","End":"03:53.415","Text":"For example, you can follow my thinking,"},{"Start":"03:53.415 ","End":"03:58.485","Text":"I\u0027m thinking if I put y=0, then x=1."},{"Start":"03:58.485 ","End":"04:03.150","Text":"So I can say that x=1,"},{"Start":"04:03.150 ","End":"04:06.255","Text":"Y=0 will be a solution,"},{"Start":"04:06.255 ","End":"04:08.300","Text":"and indeed if you plug it in,"},{"Start":"04:08.300 ","End":"04:14.195","Text":"I\u0027m also thinking how about if I took y=2,"},{"Start":"04:14.195 ","End":"04:16.520","Text":"and then have 4,"},{"Start":"04:16.520 ","End":"04:18.200","Text":"so let x is minus 3,"},{"Start":"04:18.200 ","End":"04:25.145","Text":"so I say let x is minus 3 and y=2."},{"Start":"04:25.145 ","End":"04:31.490","Text":"That will also be a solution because minus 3 plus twice 2 is1, and so on."},{"Start":"04:31.490 ","End":"04:33.560","Text":"There are infinite number of solutions,"},{"Start":"04:33.560 ","End":"04:36.440","Text":"but not everything is a solution."},{"Start":"04:36.440 ","End":"04:40.990","Text":"For example, if I take x=0,"},{"Start":"04:40.990 ","End":"04:47.700","Text":"y=0, then that will not work because 0 plus twice 0 is not equal to 1."},{"Start":"04:47.700 ","End":"04:53.194","Text":"So not everything is a solution but there still are an infinite number of solutions,"},{"Start":"04:53.194 ","End":"04:55.130","Text":"that could be a whole bunch of these."},{"Start":"04:55.130 ","End":"05:00.890","Text":"The way you could generate 1 is just take y to be any value and then compute its x."},{"Start":"05:00.890 ","End":"05:02.900","Text":"Wherever I put for y, I can find an x."},{"Start":"05:02.900 ","End":"05:05.045","Text":"There are an infinite number of solutions,"},{"Start":"05:05.045 ","End":"05:07.600","Text":"but not everything is a solution."},{"Start":"05:07.600 ","End":"05:11.470","Text":"That\u0027s all I\u0027m going to say here. We\u0027re done."}],"ID":4974}],"Thumbnail":null,"ID":3866},{"Name":"Applications of Linear Equations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"3m 4s","ChapterTopicVideoID":8102,"CourseChapterTopicPlaylistID":3867,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.028","Text":"Here we have a word problem where Jim has his plumbing service and he charges $40"},{"Start":"00:08.028 ","End":"00:11.850","Text":"for a service call and then"},{"Start":"00:11.850 ","End":"00:16.755","Text":"an additional $25 an hour depending on how long it takes him to repair it."},{"Start":"00:16.755 ","End":"00:22.590","Text":"What we\u0027re asked to do is for a general time t in hours,"},{"Start":"00:22.590 ","End":"00:28.059","Text":"to write the total cost C as an equation in terms of t."},{"Start":"00:28.059 ","End":"00:36.675","Text":"Given t to figure out what is C in terms of t. This is part a."},{"Start":"00:36.675 ","End":"00:40.160","Text":"Now, I think it\u0027s clear that C is made up of 2 parts,"},{"Start":"00:40.160 ","End":"00:41.975","Text":"there\u0027s the fixed charge,"},{"Start":"00:41.975 ","End":"00:43.340","Text":"let\u0027s work in dollars,"},{"Start":"00:43.340 ","End":"00:45.560","Text":"by the way, I won\u0027t keep writing dollars."},{"Start":"00:45.560 ","End":"00:54.185","Text":"The charge C or a total cost C in dollars is the $40 flat rate just for the house call,"},{"Start":"00:54.185 ","End":"00:58.500","Text":"and then $25 times the number of hours,"},{"Start":"00:58.500 ","End":"00:59.850","Text":"like if you work 2 hours,"},{"Start":"00:59.850 ","End":"01:02.445","Text":"he\u0027d take $50, 25 times 2."},{"Start":"01:02.445 ","End":"01:09.700","Text":"In general, it\u0027s going to be 25 times t. That\u0027s the cost of the repair time."},{"Start":"01:09.700 ","End":"01:11.930","Text":"That\u0027s it for a."},{"Start":"01:11.930 ","End":"01:17.420","Text":"Now in b, what we\u0027re asked to do and it says solve explicitly for t in terms of C,"},{"Start":"01:17.420 ","End":"01:19.550","Text":"it just means to extract t,"},{"Start":"01:19.550 ","End":"01:21.590","Text":"just t is on 1 side,"},{"Start":"01:21.590 ","End":"01:24.050","Text":"and the rest of it is on the other side."},{"Start":"01:24.050 ","End":"01:26.495","Text":"What we could do is,"},{"Start":"01:26.495 ","End":"01:28.080","Text":"let\u0027s say we switch sides,"},{"Start":"01:28.080 ","End":"01:33.560","Text":"C can go over to the right and the 25t goes over to the left,"},{"Start":"01:33.560 ","End":"01:37.490","Text":"that goes to the left, this goes to the right and then they change signs,"},{"Start":"01:37.490 ","End":"01:47.555","Text":"so what we have is minus 25t equals 40 minus C. Now,"},{"Start":"01:47.555 ","End":"01:52.760","Text":"what I suggest doing is dividing by minus 25,"},{"Start":"01:52.760 ","End":"01:55.250","Text":"which is the coefficient of t,"},{"Start":"01:55.250 ","End":"02:04.115","Text":"and then we\u0027ll get just t equals 40 minus C divided by minus 25."},{"Start":"02:04.115 ","End":"02:07.160","Text":"I\u0027d like to tidy this up a little bit."},{"Start":"02:07.160 ","End":"02:08.730","Text":"I want to get rid of this minus,"},{"Start":"02:08.730 ","End":"02:10.020","Text":"there\u0027s too many minuses,"},{"Start":"02:10.020 ","End":"02:13.460","Text":"so I can on the bottom put just 25 and"},{"Start":"02:13.460 ","End":"02:17.465","Text":"now I have to multiply the numerator by minus, or minus 1."},{"Start":"02:17.465 ","End":"02:20.795","Text":"One way to do this with a difference is just to reverse the order,"},{"Start":"02:20.795 ","End":"02:22.850","Text":"instead of writing minus 40 plus C,"},{"Start":"02:22.850 ","End":"02:25.880","Text":"I can write C minus 40,"},{"Start":"02:25.880 ","End":"02:28.925","Text":"so I would like to leave it this way."},{"Start":"02:28.925 ","End":"02:31.460","Text":"I\u0027ll just highlight the answers of part a,"},{"Start":"02:31.460 ","End":"02:33.290","Text":"C in terms of t is this,"},{"Start":"02:33.290 ","End":"02:37.865","Text":"and for part b, t in terms of C is this."},{"Start":"02:37.865 ","End":"02:42.769","Text":"We\u0027re done, but I\u0027ll just like to throw in a bit of intuition why this makes sense."},{"Start":"02:42.769 ","End":"02:45.470","Text":"If you know how much the total cost was"},{"Start":"02:45.470 ","End":"02:48.365","Text":"and you wanted to figure out how many hours you work, you say, Okay,"},{"Start":"02:48.365 ","End":"02:51.050","Text":"let\u0027s subtract the $40 flat rate for"},{"Start":"02:51.050 ","End":"02:56.300","Text":"the service charge and everything else is the stuff he did in an hourly rate,"},{"Start":"02:56.300 ","End":"02:58.430","Text":"and since there were $25 per hour,"},{"Start":"02:58.430 ","End":"03:01.100","Text":"you divide by 25 to get the number of hours,"},{"Start":"03:01.100 ","End":"03:05.250","Text":"so it makes sense. That\u0027s it."}],"ID":8218},{"Watched":false,"Name":"Exercise 2","Duration":"3m 22s","ChapterTopicVideoID":8103,"CourseChapterTopicPlaylistID":3867,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.488","Text":"We have here a word problem where social group bought 200 tickets for a concert."},{"Start":"00:06.488 ","End":"00:08.759","Text":"There\u0027s 2 kinds of tickets."},{"Start":"00:08.759 ","End":"00:11.160","Text":"The orchestra tickets which are more expensive,"},{"Start":"00:11.160 ","End":"00:15.975","Text":"that $12, and the balcony tickets, just $8."},{"Start":"00:15.975 ","End":"00:19.650","Text":"Now, we know that the total amount was"},{"Start":"00:19.650 ","End":"00:25.155","Text":"$2,080 and we\u0027ve got to figure out how many of each type,"},{"Start":"00:25.155 ","End":"00:29.640","Text":"ie, how many orchestra tickets and how many balcony tickets."},{"Start":"00:29.640 ","End":"00:32.675","Text":"Now, we don\u0027t need 2 variables, we could say,"},{"Start":"00:32.675 ","End":"00:38.163","Text":"let X be the number of balcony tickets and Y be the number of orchestral tickets."},{"Start":"00:38.163 ","End":"00:42.650","Text":"But we don\u0027t need that because we know that together there were 200,"},{"Start":"00:42.650 ","End":"00:45.980","Text":"so as soon as we know 1 kind we can figure the other."},{"Start":"00:45.980 ","End":"00:49.070","Text":"I\u0027m not sure which to give the letter X to,"},{"Start":"00:49.070 ","End":"00:54.146","Text":"so let\u0027s go with the expensive ones and call that X and say X="},{"Start":"00:54.146 ","End":"01:00.815","Text":"the number of orchestra tickets."},{"Start":"01:00.815 ","End":"01:03.905","Text":"Like I said, we know how many balcony tickets to wear"},{"Start":"01:03.905 ","End":"01:08.260","Text":"because I just have to subtract from 200 that\u0027s the remainder."},{"Start":"01:08.260 ","End":"01:15.885","Text":"I did copy paste change that to balcony tickets."},{"Start":"01:15.885 ","End":"01:18.065","Text":"Now we\u0027ve got to figure out the cost."},{"Start":"01:18.065 ","End":"01:23.105","Text":"These are the $12 tickets and these are the $8 tickets."},{"Start":"01:23.105 ","End":"01:31.890","Text":"I can figure out that there were $12X spent on the orchestra tickets."},{"Start":"01:31.890 ","End":"01:35.835","Text":"For the balcony I\u0027ve got this number at $8 each,"},{"Start":"01:35.835 ","End":"01:41.465","Text":"so it\u0027s 8 times 200 minus X was spent on balcony tickets."},{"Start":"01:41.465 ","End":"01:47.750","Text":"The equation comes from saying that I know the total cost was 2,080."},{"Start":"01:47.750 ","End":"01:51.545","Text":"This is our equation I want to just solve it."},{"Start":"01:51.545 ","End":"01:54.685","Text":"Let\u0027s start by opening the brackets."},{"Start":"01:54.685 ","End":"01:56.685","Text":"We\u0027ve got 12X,"},{"Start":"01:56.685 ","End":"02:05.190","Text":"8 times 200 is 1,600 minus 8X=2,080."},{"Start":"02:05.190 ","End":"02:09.795","Text":"What I\u0027m going do is leave the Xs on the left and the numbers on the right."},{"Start":"02:09.795 ","End":"02:13.180","Text":"X is 12X minus 8X,"},{"Start":"02:13.180 ","End":"02:14.645","Text":"that\u0027s already on the left."},{"Start":"02:14.645 ","End":"02:23.000","Text":"On the right I have 2,080 and that bring the 1,600 over it becomes minus 1,600."},{"Start":"02:23.000 ","End":"02:25.490","Text":"Now collecting like terms 12 minus 8,"},{"Start":"02:25.490 ","End":"02:33.850","Text":"so we have 4X is equal to this minus this is 480."},{"Start":"02:33.850 ","End":"02:38.715","Text":"X equals 120. Now,"},{"Start":"02:38.715 ","End":"02:44.460","Text":"you can\u0027t leave your answer as X=120 you have to interpret it."},{"Start":"02:44.460 ","End":"02:48.480","Text":"What we would say that there would be"},{"Start":"02:48.480 ","End":"02:54.275","Text":"120 orchestra tickets and just did a quick copy paste there,"},{"Start":"02:54.275 ","End":"02:57.320","Text":"and number of balcony tickets is 200 minus X so I do"},{"Start":"02:57.320 ","End":"03:00.560","Text":"a little calculation on the side though I can do it in our heads."},{"Start":"03:00.560 ","End":"03:05.660","Text":"200 minus 120 is 80."},{"Start":"03:05.660 ","End":"03:10.700","Text":"I\u0027d say that there were 80 balcony tickets."},{"Start":"03:10.700 ","End":"03:13.190","Text":"That should do it."},{"Start":"03:13.190 ","End":"03:16.310","Text":"You might want to highlight the important bits."},{"Start":"03:16.310 ","End":"03:22.770","Text":"There were 120 orchestra tickets and 80 balcony tickets. That\u0027s it."}],"ID":8219},{"Watched":false,"Name":"Exercise 3","Duration":"2m 58s","ChapterTopicVideoID":8104,"CourseChapterTopicPlaylistID":3867,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"This exercise is one of those word problems involving distance,"},{"Start":"00:04.830 ","End":"00:06.540","Text":"speed, and time."},{"Start":"00:06.540 ","End":"00:10.890","Text":"The cyclist cycles at a certain speed for 4 hours."},{"Start":"00:10.890 ","End":"00:17.393","Text":"His name is Joe, and he realizes that if he had driven 5 miles an hour faster,"},{"Start":"00:17.393 ","End":"00:21.945","Text":"he would have covered it in 3 hours instead of 4 hours and saved himself an hour."},{"Start":"00:21.945 ","End":"00:26.670","Text":"The question is, what distance does he travel during the 4-hour trip?"},{"Start":"00:26.670 ","End":"00:30.645","Text":"Meaning in reality, not the hypothetical."},{"Start":"00:30.645 ","End":"00:35.480","Text":"Will you set up an equation and what is the variable?"},{"Start":"00:35.480 ","End":"00:36.965","Text":"What don\u0027t we know?"},{"Start":"00:36.965 ","End":"00:39.110","Text":"Well, I think if you read it,"},{"Start":"00:39.110 ","End":"00:42.440","Text":"it\u0027s fairly clear that the key lines here it says,"},{"Start":"00:42.440 ","End":"00:44.135","Text":"at a certain speed,"},{"Start":"00:44.135 ","End":"00:46.220","Text":"and we don\u0027t know that certain speed."},{"Start":"00:46.220 ","End":"00:48.050","Text":"Why don\u0027t we give it a name?"},{"Start":"00:48.050 ","End":"00:49.700","Text":"I don\u0027t like to use s for speed,"},{"Start":"00:49.700 ","End":"00:51.620","Text":"they like to use v for velocity."},{"Start":"00:51.620 ","End":"00:54.965","Text":"I\u0027ll call that certain speed, v,"},{"Start":"00:54.965 ","End":"00:59.360","Text":"and I want to remind you of the most basic formula in"},{"Start":"00:59.360 ","End":"01:06.283","Text":"these problems with distance and speed or velocity, times time."},{"Start":"01:06.283 ","End":"01:08.405","Text":"You have to make sure the units are right."},{"Start":"01:08.405 ","End":"01:10.880","Text":"Like if you\u0027re using miles and hours,"},{"Start":"01:10.880 ","End":"01:12.680","Text":"you need miles per hour and so on."},{"Start":"01:12.680 ","End":"01:15.096","Text":"Let\u0027s do 2 scenarios."},{"Start":"01:15.096 ","End":"01:16.910","Text":"One is the actual scenario,"},{"Start":"01:16.910 ","End":"01:19.145","Text":"and the other is the hypothetical scenario,"},{"Start":"01:19.145 ","End":"01:21.065","Text":"but the distance is the same."},{"Start":"01:21.065 ","End":"01:25.430","Text":"Now, the distance, what he actually did,"},{"Start":"01:25.430 ","End":"01:27.815","Text":"I\u0027ll just write it as distance."},{"Start":"01:27.815 ","End":"01:29.565","Text":"On the 1 hand,"},{"Start":"01:29.565 ","End":"01:36.830","Text":"he drove at speed v for 4 hours so it\u0027s 4 times v. This is the actual,"},{"Start":"01:36.830 ","End":"01:38.998","Text":"and in the hypothetical,"},{"Start":"01:38.998 ","End":"01:40.850","Text":"it\u0027s the same distance,"},{"Start":"01:40.850 ","End":"01:44.080","Text":"but what it is is that it\u0027s only 3 hours."},{"Start":"01:44.080 ","End":"01:45.360","Text":"On the other hand,"},{"Start":"01:45.360 ","End":"01:46.515","Text":"it\u0027s not v,"},{"Start":"01:46.515 ","End":"01:51.670","Text":"it\u0027s 5 miles an hour faster so it\u0027s v plus 5."},{"Start":"01:51.670 ","End":"01:53.330","Text":"Now, the distance is the same,"},{"Start":"01:53.330 ","End":"01:55.085","Text":"so I can equate these 2."},{"Start":"01:55.085 ","End":"02:02.770","Text":"This gives me the equation that 4v equals 3 times v plus 5."},{"Start":"02:02.770 ","End":"02:05.075","Text":"Now it\u0027s straightforward algebra."},{"Start":"02:05.075 ","End":"02:10.940","Text":"4 v equals 3v plus 3 times 5 is 15."},{"Start":"02:10.940 ","End":"02:15.865","Text":"Bring this to the other side and we\u0027ve got v= 15."},{"Start":"02:15.865 ","End":"02:18.616","Text":"We can\u0027t say the answer is v=15,"},{"Start":"02:18.616 ","End":"02:20.690","Text":"you have to see what did they ask?"},{"Start":"02:20.690 ","End":"02:22.430","Text":"What did I ask? What\u0027s the question?"},{"Start":"02:22.430 ","End":"02:24.590","Text":"What is the distance traveled?"},{"Start":"02:24.590 ","End":"02:29.330","Text":"Even if you\u0027d say the velocity initially was 15 miles per hour,"},{"Start":"02:29.330 ","End":"02:30.943","Text":"still wouldn\u0027t be the answer,"},{"Start":"02:30.943 ","End":"02:32.569","Text":"we want this distance."},{"Start":"02:32.569 ","End":"02:36.900","Text":"All we have to do is say that the distance that he traveled,"},{"Start":"02:36.900 ","End":"02:38.150","Text":"I write the word distance again,"},{"Start":"02:38.150 ","End":"02:46.040","Text":"I could\u0027ve just said d is 4 times v is 15, which is 60."},{"Start":"02:46.040 ","End":"02:53.290","Text":"At the end, just remember to write the units it\u0027s recommended and it\u0027s miles."},{"Start":"02:53.290 ","End":"02:58.150","Text":"That\u0027s our answer and we\u0027re done."}],"ID":8220},{"Watched":false,"Name":"Exercise 4","Duration":"2m 25s","ChapterTopicVideoID":4975,"CourseChapterTopicPlaylistID":3867,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"Let\u0027s read the problem carefully."},{"Start":"00:02.970 ","End":"00:11.580","Text":"The price of an electrical appliance is $142.60 after adding a sales tax of 6 percent."},{"Start":"00:11.580 ","End":"00:14.670","Text":"What\u0027s the price of the appliance before tax?"},{"Start":"00:14.670 ","End":"00:19.238","Text":"One of the first things you want to do is assign a variable,"},{"Start":"00:19.238 ","End":"00:25.170","Text":"and sometimes the most obvious the best is to assign a name to what we\u0027re looking for,"},{"Start":"00:25.170 ","End":"00:26.820","Text":"the price before tax."},{"Start":"00:26.820 ","End":"00:32.280","Text":"Let\u0027s do that. Let\u0027s say that the price before tax,"},{"Start":"00:32.280 ","End":"00:34.380","Text":"I don\u0027t want to use X all the time,"},{"Start":"00:34.380 ","End":"00:36.105","Text":"P for price. How about that?"},{"Start":"00:36.105 ","End":"00:42.755","Text":"Is P. Now we have to find an equation involving P and solve for it."},{"Start":"00:42.755 ","End":"00:45.380","Text":"What we can say is build up to it."},{"Start":"00:45.380 ","End":"00:46.790","Text":"Let\u0027s say P is the price."},{"Start":"00:46.790 ","End":"00:48.695","Text":"Let\u0027s see what the sales tax is."},{"Start":"00:48.695 ","End":"00:52.680","Text":"The sales tax on that is 6 percent of it,"},{"Start":"00:52.680 ","End":"00:58.760","Text":"which is 6 percent of P. What is 6 percent? Is 0.06."},{"Start":"00:58.760 ","End":"01:03.635","Text":"We want to work in decimals, so 0.06P."},{"Start":"01:03.635 ","End":"01:07.460","Text":"After tax, we have to add the two together."},{"Start":"01:07.460 ","End":"01:13.745","Text":"We get P plus the tax which is 0.06P."},{"Start":"01:13.745 ","End":"01:15.985","Text":"If we simplify this, this is 1P,"},{"Start":"01:15.985 ","End":"01:23.645","Text":"this is 1.06P; 1 plus 0.06."},{"Start":"01:23.645 ","End":"01:31.790","Text":"Now we have an equation because we know that the price after tax is 142.60,"},{"Start":"01:31.790 ","End":"01:33.785","Text":"assuming we\u0027re working in dollars,"},{"Start":"01:33.785 ","End":"01:36.350","Text":"which is 142.6 as a decimal,"},{"Start":"01:36.350 ","End":"01:38.120","Text":"we don\u0027t need the trailing 0."},{"Start":"01:38.120 ","End":"01:44.330","Text":"All that remains to do now is to divide both sides by 1.06,"},{"Start":"01:44.330 ","End":"01:52.670","Text":"and so we get P equals 142.6 divided by 1.06."},{"Start":"01:52.670 ","End":"01:55.385","Text":"This we\u0027ll do on the calculator."},{"Start":"01:55.385 ","End":"02:02.720","Text":"My calculator says 134.528 something."},{"Start":"02:02.720 ","End":"02:05.910","Text":"Of course we want to round it off to the nearest cent,"},{"Start":"02:05.910 ","End":"02:11.650","Text":"so the answer will be $134."},{"Start":"02:12.350 ","End":"02:16.274","Text":"In other words, it\u0027s $0.53 is the closest,"},{"Start":"02:16.274 ","End":"02:18.185","Text":"and that\u0027s the answer, the price."},{"Start":"02:18.185 ","End":"02:25.950","Text":"You\u0027d write the full sentence, the price before tax is $134.53. Well done."}],"ID":4975},{"Watched":false,"Name":"Exercise 5","Duration":"2m 13s","ChapterTopicVideoID":4976,"CourseChapterTopicPlaylistID":3867,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.670","Text":"First thing we want to do is read the problem."},{"Start":"00:02.670 ","End":"00:07.635","Text":"A clothes store is having a 30 percent off end of season sale on all items."},{"Start":"00:07.635 ","End":"00:11.235","Text":"The sale price of the shirt is 19.95."},{"Start":"00:11.235 ","End":"00:14.175","Text":"What was the original price of the shirt?"},{"Start":"00:14.175 ","End":"00:19.905","Text":"First thing we want to do is assign a variable or more than 1 variable."},{"Start":"00:19.905 ","End":"00:23.280","Text":"Usually the obvious thing is to assign what we\u0027re looking for,"},{"Start":"00:23.280 ","End":"00:25.425","Text":"which is the original price of the shirt."},{"Start":"00:25.425 ","End":"00:27.450","Text":"Let\u0027s choose a variable name,"},{"Start":"00:27.450 ","End":"00:29.160","Text":"I\u0027ll choose p for price."},{"Start":"00:29.160 ","End":"00:30.780","Text":"I don\u0027t always use x."},{"Start":"00:30.780 ","End":"00:32.190","Text":"Usually in word problems,"},{"Start":"00:32.190 ","End":"00:33.854","Text":"you choose a meaningful letter,"},{"Start":"00:33.854 ","End":"00:39.575","Text":"p is the original price of the shirt,"},{"Start":"00:39.575 ","End":"00:44.390","Text":"and of course we\u0027re assuming that it says all items of the shirt is also,"},{"Start":"00:44.390 ","End":"00:46.955","Text":"this is after the 30 percent discount."},{"Start":"00:46.955 ","End":"00:53.880","Text":"Now, a 30 percent"},{"Start":"00:53.880 ","End":"00:57.300","Text":"of p is p times 30 percent,"},{"Start":"00:57.300 ","End":"01:00.283","Text":"but we need that in decimal or as a fraction."},{"Start":"01:00.283 ","End":"01:04.516","Text":"I\u0027ll choose a decimal that\u0027s 30 over a 100 is 0.3,"},{"Start":"01:04.516 ","End":"01:07.790","Text":"so this is 0.3p."},{"Start":"01:07.790 ","End":"01:09.784","Text":"This is the discount."},{"Start":"01:09.784 ","End":"01:15.180","Text":"The final price, which is p minus 0.3p,"},{"Start":"01:15.180 ","End":"01:21.150","Text":"that\u0027s going to equal this 19.95, that\u0027s our equation."},{"Start":"01:21.150 ","End":"01:26.010","Text":"The left hand side simplifies p minus 0.3p then is,"},{"Start":"01:26.010 ","End":"01:29.270","Text":"1 minus 0.3 is 0.7."},{"Start":"01:29.270 ","End":"01:34.430","Text":"We can do that in our heads, p is 19.95."},{"Start":"01:34.430 ","End":"01:40.875","Text":"Now, what we want to do to get p is just divide by 0.7."},{"Start":"01:40.875 ","End":"01:48.710","Text":"We get that p equals 19.95 over 0.7."},{"Start":"01:48.710 ","End":"01:51.590","Text":"For this, we need a calculator,"},{"Start":"01:51.590 ","End":"01:56.390","Text":"and my calculator says 28.5,"},{"Start":"01:56.390 ","End":"01:58.805","Text":"but we want that in dollars."},{"Start":"01:58.805 ","End":"02:06.290","Text":"We would write it as $28.50 and usually it puts it in a sentence."},{"Start":"02:06.290 ","End":"02:11.030","Text":"The original price of the shirt was $28.50."},{"Start":"02:11.030 ","End":"02:13.380","Text":"That\u0027s it."}],"ID":4976},{"Watched":false,"Name":"Exercise 6","Duration":"7m 8s","ChapterTopicVideoID":4977,"CourseChapterTopicPlaylistID":3867,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"Here we have a word problem, let\u0027s read it."},{"Start":"00:02.820 ","End":"00:07.979","Text":"Involves 2 trains which start out 280 miles apart."},{"Start":"00:07.979 ","End":"00:10.830","Text":"If you live in a country where they don\u0027t use miles,"},{"Start":"00:10.830 ","End":"00:14.370","Text":"just put the word kilometers instead of miles."},{"Start":"00:14.370 ","End":"00:20.190","Text":"280 miles apart and they\u0027re traveling towards each other and they meet after 3.5 hours."},{"Start":"00:20.190 ","End":"00:23.015","Text":"But one of the trains is faster,"},{"Start":"00:23.015 ","End":"00:26.205","Text":"travels 10 miles an hour faster than the other."},{"Start":"00:26.205 ","End":"00:29.221","Text":"I guess we should have said they move at constant speeds."},{"Start":"00:29.221 ","End":"00:32.400","Text":"Anyway, how fast did each of them travel?"},{"Start":"00:32.400 ","End":"00:35.415","Text":"You might want to take time to read this again."},{"Start":"00:35.415 ","End":"00:39.314","Text":"Then you possibly would want to draw a little sketch."},{"Start":"00:39.314 ","End":"00:42.055","Text":"It\u0027s not essential but it often helps."},{"Start":"00:42.055 ","End":"00:47.210","Text":"Let\u0027s say, one train starts out from this point and the other train"},{"Start":"00:47.210 ","End":"00:52.730","Text":"starts out from this point and they traveled towards each other."},{"Start":"00:52.730 ","End":"00:55.585","Text":"Let\u0027s say, this one is the faster one."},{"Start":"00:55.585 ","End":"00:58.520","Text":"I\u0027ll put a double arrow to show it\u0027s the faster one and"},{"Start":"00:58.520 ","End":"01:01.310","Text":"the other train travels in this direction,"},{"Start":"01:01.310 ","End":"01:03.155","Text":"and that\u0027s the slower one."},{"Start":"01:03.155 ","End":"01:05.180","Text":"I connected this with a line."},{"Start":"01:05.180 ","End":"01:07.070","Text":"This is like the railroad track,"},{"Start":"01:07.070 ","End":"01:08.869","Text":"I just give them names."},{"Start":"01:08.869 ","End":"01:10.190","Text":"This is a plane A,"},{"Start":"01:10.190 ","End":"01:12.335","Text":"the fast one, plane B."},{"Start":"01:12.335 ","End":"01:16.100","Text":"What I do know is that the distance as I put an arrow,"},{"Start":"01:16.100 ","End":"01:21.545","Text":"distance from here to here is 280 miles."},{"Start":"01:21.545 ","End":"01:24.515","Text":"Let\u0027s also mark the place where they meet."},{"Start":"01:24.515 ","End":"01:27.035","Text":"They meet somewhere here."},{"Start":"01:27.035 ","End":"01:32.030","Text":"They\u0027re obviously going to meet right of the middle because this one is the faster one."},{"Start":"01:32.030 ","End":"01:34.940","Text":"This doesn\u0027t have to be to scale but I would think that"},{"Start":"01:34.940 ","End":"01:38.795","Text":"this point should be seen to be closer to B than to A,"},{"Start":"01:38.795 ","End":"01:42.605","Text":"because this is the faster one it covers more distance."},{"Start":"01:42.605 ","End":"01:48.200","Text":"Now, before we assign variables sometimes we write what are called word equations."},{"Start":"01:48.200 ","End":"01:54.950","Text":"For example, I might want to say that the distance from here to here."},{"Start":"01:54.950 ","End":"01:56.330","Text":"Let me just join that up."},{"Start":"01:56.330 ","End":"01:59.585","Text":"This is the distance traveled by train A."},{"Start":"01:59.585 ","End":"02:03.740","Text":"Let\u0027s just write distance of A,"},{"Start":"02:03.740 ","End":"02:06.965","Text":"just for short and from here to here,"},{"Start":"02:06.965 ","End":"02:14.460","Text":"this would be the distance traveled by train B. I\u0027ll just write it as that."},{"Start":"02:14.460 ","End":"02:16.790","Text":"These are corresponding here,"},{"Start":"02:16.790 ","End":"02:18.320","Text":"this corresponds to this,"},{"Start":"02:18.320 ","End":"02:19.790","Text":"this is the middle."},{"Start":"02:19.790 ","End":"02:22.805","Text":"We can write two things at least."},{"Start":"02:22.805 ","End":"02:28.225","Text":"We can write that the distance of A plus the distance of B is 280."},{"Start":"02:28.225 ","End":"02:31.770","Text":"Just even write distance A,"},{"Start":"02:31.770 ","End":"02:38.630","Text":"just to abbreviate, plus distance B equals 280."},{"Start":"02:38.630 ","End":"02:40.355","Text":"This plus this is this."},{"Start":"02:40.355 ","End":"02:43.820","Text":"The other thing we know is something involving their speeds,"},{"Start":"02:43.820 ","End":"02:46.360","Text":"I like to use the word velocity rather than speed."},{"Start":"02:46.360 ","End":"02:55.148","Text":"Let\u0027s say that B travels at velocity of v. Now we\u0027ve assigned a variable,"},{"Start":"02:55.148 ","End":"03:04.475","Text":"and this one would be therefore v plus 10 for the velocity."},{"Start":"03:04.475 ","End":"03:09.065","Text":"I guess I meant to first of all say it as a word equation that"},{"Start":"03:09.065 ","End":"03:14.340","Text":"the velocity of A is 10 more than the others,"},{"Start":"03:14.340 ","End":"03:16.530","Text":"means that it\u0027s the other plus 10,"},{"Start":"03:16.530 ","End":"03:21.865","Text":"is velocity of B plus 10."},{"Start":"03:21.865 ","End":"03:25.580","Text":"I guess I should have said this first and then labeled the velocity of"},{"Start":"03:25.580 ","End":"03:30.950","Text":"B to call that v. This is v,"},{"Start":"03:30.950 ","End":"03:32.390","Text":"this is v plus 10."},{"Start":"03:32.390 ","End":"03:35.405","Text":"You could have done it the other way around and said,"},{"Start":"03:35.405 ","End":"03:41.305","Text":"use some latter for the velocity of A and then say that this is that minus 10."},{"Start":"03:41.305 ","End":"03:43.030","Text":"I prefer a plus than a minus,"},{"Start":"03:43.030 ","End":"03:47.090","Text":"so the smaller one is the one I labeled v but you could\u0027ve done it the other way around."},{"Start":"03:47.090 ","End":"03:50.635","Text":"This is v and then this is v minus 10."},{"Start":"03:50.635 ","End":"03:56.765","Text":"Now there\u0027s something you have to remember from distance and the speed and time problems."},{"Start":"03:56.765 ","End":"03:58.550","Text":"That is that in general,"},{"Start":"03:58.550 ","End":"04:03.440","Text":"that the distance covered by something moving at a constant speed,"},{"Start":"04:03.440 ","End":"04:09.970","Text":"is speed or velocity multiplied by time."},{"Start":"04:09.970 ","End":"04:14.930","Text":"If we use this general property in each of these,"},{"Start":"04:14.930 ","End":"04:17.270","Text":"let\u0027s take the first equation."},{"Start":"04:17.270 ","End":"04:19.040","Text":"We can rewrite it,"},{"Start":"04:19.040 ","End":"04:28.385","Text":"that the distance of A is the velocity of A multiplied by,"},{"Start":"04:28.385 ","End":"04:31.610","Text":"now the time is the same for both."},{"Start":"04:31.610 ","End":"04:35.600","Text":"They both travel the same 3.5 hours when they meet."},{"Start":"04:35.600 ","End":"04:37.265","Text":"Otherwise it doesn\u0027t count as meeting,"},{"Start":"04:37.265 ","End":"04:38.975","Text":"meeting means they traveled for the same time,"},{"Start":"04:38.975 ","End":"04:44.270","Text":"time is 3/1/2 plus the distance of B is"},{"Start":"04:44.270 ","End":"04:51.140","Text":"the velocity of B times 3/1/2,"},{"Start":"04:51.140 ","End":"04:53.470","Text":"and this is going to be equal to 280."},{"Start":"04:53.470 ","End":"04:56.330","Text":"Now, the other thing we can do is substitute now because we\u0027ve already"},{"Start":"04:56.330 ","End":"04:59.435","Text":"said the velocity of A is v plus 10"},{"Start":"04:59.435 ","End":"05:05.730","Text":"and the velocity of B is just v. What we get is that,"},{"Start":"05:05.730 ","End":"05:10.095","Text":"v plus 10 times"},{"Start":"05:10.095 ","End":"05:20.190","Text":"3/1/2 plus v times 3/1/2 is equal to 280."},{"Start":"05:20.190 ","End":"05:27.033","Text":"Now we\u0027ve got an equation involving v. Let\u0027s see if we can collect terms,"},{"Start":"05:27.033 ","End":"05:28.310","Text":"if we open the brackets."},{"Start":"05:28.310 ","End":"05:33.013","Text":"First of all expand, we\u0027ve got 3/1/2v plus,"},{"Start":"05:33.013 ","End":"05:42.960","Text":"3/1/2 times 10 is 35 plus 3/1/2v is equal to 280."},{"Start":"05:42.960 ","End":"05:46.857","Text":"We can combine 3/1/2v with 3/1/2v,"},{"Start":"05:46.857 ","End":"05:49.938","Text":"so we\u0027ve got 7v."},{"Start":"05:49.938 ","End":"06:01.385","Text":"We can move the 35 to the other side and get 280 minus 35 is 245."},{"Start":"06:01.385 ","End":"06:06.185","Text":"Then if we divide both sides by 7,"},{"Start":"06:06.185 ","End":"06:09.440","Text":"then we get v equals, let\u0027s see,"},{"Start":"06:09.440 ","End":"06:11.914","Text":"245 over 7"},{"Start":"06:11.914 ","End":"06:21.140","Text":"is 35."},{"Start":"06:21.140 ","End":"06:25.715","Text":"Now, you can\u0027t leave the answer as v equals 35."},{"Start":"06:25.715 ","End":"06:27.005","Text":"To remember what they are,"},{"Start":"06:27.005 ","End":"06:28.490","Text":"let\u0027s go back and see."},{"Start":"06:28.490 ","End":"06:32.440","Text":"The question asked, how fast did each of them travel?"},{"Start":"06:32.440 ","End":"06:42.604","Text":"You would write train B traveled at 35 and our speeds are in miles per hour."},{"Start":"06:42.604 ","End":"06:46.785","Text":"Then A traveled at,"},{"Start":"06:46.785 ","End":"06:48.940","Text":"it was 10 miles an hour more."},{"Start":"06:48.940 ","End":"06:53.990","Text":"Remember we said that the velocity of A is v plus 10, so 45."},{"Start":"06:53.990 ","End":"06:58.910","Text":"At the side I would do rather 35 plus 10 equals 45."},{"Start":"06:58.910 ","End":"07:00.830","Text":"But it\u0027s how easy we did it in our heads,"},{"Start":"07:00.830 ","End":"07:08.310","Text":"it\u0027s 45 miles per hour and that then is the answer. We\u0027re done."}],"ID":4977},{"Watched":false,"Name":"Exercise 7","Duration":"4m 47s","ChapterTopicVideoID":4978,"CourseChapterTopicPlaylistID":3867,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.489","Text":"Let\u0027s read this word problem."},{"Start":"00:02.489 ","End":"00:07.845","Text":"A car and a truck start traveling north simultaneously on the same highway."},{"Start":"00:07.845 ","End":"00:12.345","Text":"At the start, the truck is 35 kilometers north of the car."},{"Start":"00:12.345 ","End":"00:13.913","Text":"If you don\u0027t like kilometers,"},{"Start":"00:13.913 ","End":"00:19.530","Text":"you can just replace the word kilometer with mile throughout and same mathematics."},{"Start":"00:19.530 ","End":"00:23.175","Text":"It\u0027s only in the United States do they still not use kilometers."},{"Start":"00:23.175 ","End":"00:28.875","Text":"The car travels at 90 kilometers per hour and the truck at 70 kilometers per hour."},{"Start":"00:28.875 ","End":"00:31.020","Text":"The car is faster than the truck naturally."},{"Start":"00:31.020 ","End":"00:34.320","Text":"After how long will the car catch up with the truck?"},{"Start":"00:34.320 ","End":"00:36.600","Text":"I think a diagram would help."},{"Start":"00:36.600 ","End":"00:39.570","Text":"Most people find a sketch or diagram helpful."},{"Start":"00:39.570 ","End":"00:41.665","Text":"Some people prefer to imagine it."},{"Start":"00:41.665 ","End":"00:44.330","Text":"Let\u0027s say this is the highway."},{"Start":"00:44.330 ","End":"00:47.930","Text":"Let\u0027s say that north is in this direction."},{"Start":"00:47.930 ","End":"00:51.150","Text":"This is north, although maybe you could change"},{"Start":"00:51.150 ","End":"00:54.950","Text":"the word to East if you like thinking of north as upwards on the page,"},{"Start":"00:54.950 ","End":"00:56.045","Text":"but this is North."},{"Start":"00:56.045 ","End":"00:58.655","Text":"The three important points that we have to mark."},{"Start":"00:58.655 ","End":"01:01.790","Text":"There\u0027s the place where the car starts from,"},{"Start":"01:01.790 ","End":"01:05.160","Text":"and it\u0027s moving in this direction."},{"Start":"01:05.160 ","End":"01:09.995","Text":"I could even write the speed of the car as 90 kilometers per hour."},{"Start":"01:09.995 ","End":"01:13.440","Text":"Then there\u0027s the place where the truck starts from,"},{"Start":"01:13.440 ","End":"01:14.720","Text":"it starts from here,"},{"Start":"01:14.720 ","End":"01:16.920","Text":"and it\u0027s also going north,"},{"Start":"01:16.920 ","End":"01:19.920","Text":"but it\u0027s going at 70 kilometers per hour."},{"Start":"01:19.920 ","End":"01:23.270","Text":"At some point, because the car is going faster it catches up with"},{"Start":"01:23.270 ","End":"01:28.445","Text":"the truck at someplace here and this is the meeting place."},{"Start":"01:28.445 ","End":"01:31.879","Text":"Now, there are three different distances involved."},{"Start":"01:31.879 ","End":"01:33.275","Text":"There is the distance,"},{"Start":"01:33.275 ","End":"01:35.285","Text":"actually mark it right on the highway,"},{"Start":"01:35.285 ","End":"01:40.460","Text":"from here to here which is 35 kilometers."},{"Start":"01:40.460 ","End":"01:43.925","Text":"Then we have the distance from here to here,"},{"Start":"01:43.925 ","End":"01:47.285","Text":"which is the distance traveled by the truck."},{"Start":"01:47.285 ","End":"01:50.135","Text":"We\u0027ll just call it distance of truck."},{"Start":"01:50.135 ","End":"01:52.000","Text":"Then from here to here,"},{"Start":"01:52.000 ","End":"01:55.740","Text":"put a little arrow here and here."},{"Start":"01:55.740 ","End":"02:00.210","Text":"From here to here is the distance traveled by the car."},{"Start":"02:00.210 ","End":"02:03.230","Text":"Just call it dist of car."},{"Start":"02:03.230 ","End":"02:08.060","Text":"Now, what we know is that this is equal to this plus this."},{"Start":"02:08.060 ","End":"02:15.890","Text":"We can have a word equation that the distance traveled by the car is equal to"},{"Start":"02:15.890 ","End":"02:24.815","Text":"35 plus the distance traveled by the truck, that\u0027s one-word equation."},{"Start":"02:24.815 ","End":"02:29.780","Text":"Now, you also have to remember in all these distance speed time problems that"},{"Start":"02:29.780 ","End":"02:36.950","Text":"the basic formula is that the distance is equal to the speed or velocity I prefer,"},{"Start":"02:36.950 ","End":"02:41.810","Text":"multiplied by the time in the appropriate units."},{"Start":"02:41.810 ","End":"02:44.293","Text":"This is miles, this is miles per hour,"},{"Start":"02:44.293 ","End":"02:46.580","Text":"and this is hours or kilometers,"},{"Start":"02:46.580 ","End":"02:48.545","Text":"kilometers per hour of hours."},{"Start":"02:48.545 ","End":"02:53.450","Text":"Also, this assumes a constant velocity that everything goes at a constant speed."},{"Start":"02:53.450 ","End":"03:01.430","Text":"If we use this formula on each of the three pieces or rather on 35 we know,"},{"Start":"03:01.430 ","End":"03:02.620","Text":"but the distance,"},{"Start":"03:02.620 ","End":"03:06.130","Text":"we can write as speed times time."},{"Start":"03:06.130 ","End":"03:10.505","Text":"I think this is a good time to assign a variable at the time."},{"Start":"03:10.505 ","End":"03:14.600","Text":"Let that equals t. The time is the same for both."},{"Start":"03:14.600 ","End":"03:17.075","Text":"They both started simultaneously."},{"Start":"03:17.075 ","End":"03:20.720","Text":"When they meet, they both traveled for the same time."},{"Start":"03:20.720 ","End":"03:22.250","Text":"That\u0027s the how long."},{"Start":"03:22.250 ","End":"03:25.280","Text":"We\u0027ll let that time equals t. This time is"},{"Start":"03:25.280 ","End":"03:28.850","Text":"exactly the how long that we\u0027re asked for in the question."},{"Start":"03:28.850 ","End":"03:30.595","Text":"Now, we can use this formula,"},{"Start":"03:30.595 ","End":"03:36.665","Text":"the distance of the car is the velocity times time the velocity is 90."},{"Start":"03:36.665 ","End":"03:43.260","Text":"We have 90 times t is equal to 35,"},{"Start":"03:43.260 ","End":"03:44.520","Text":"I\u0027m just copying,"},{"Start":"03:44.520 ","End":"03:48.376","Text":"and the distance of the truck also is velocity times time,"},{"Start":"03:48.376 ","End":"03:54.485","Text":"it\u0027s 70 times the same t. This is a straightforward linear equation."},{"Start":"03:54.485 ","End":"03:56.510","Text":"Just bring the variables to the left."},{"Start":"03:56.510 ","End":"03:58.010","Text":"We have 20t,"},{"Start":"03:58.010 ","End":"04:03.420","Text":"the 20 is from 90 minus 70 is equal to 35."},{"Start":"04:03.420 ","End":"04:09.170","Text":"We get that t equals 35 over 20."},{"Start":"04:09.170 ","End":"04:17.969","Text":"If we do that on the calculator it\u0027s 1.75 and that is in hours."},{"Start":"04:17.969 ","End":"04:23.480","Text":"So we could write the answer as the how long is the car will catch up"},{"Start":"04:23.480 ","End":"04:29.585","Text":"with the truck after 1.75 hours."},{"Start":"04:29.585 ","End":"04:35.390","Text":"Or we could be nice and say 0.75 of an hour is 3/4 of an hour,"},{"Start":"04:35.390 ","End":"04:40.835","Text":"so we could say 1 hour and 45 minutes."},{"Start":"04:40.835 ","End":"04:43.070","Text":"This is totally optional."},{"Start":"04:43.070 ","End":"04:47.940","Text":"This could be perfectly good as an answer, and that\u0027s it."}],"ID":4978},{"Watched":false,"Name":"Exercise 8","Duration":"6m 20s","ChapterTopicVideoID":4979,"CourseChapterTopicPlaylistID":3867,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.255","Text":"We have here a word problem, let\u0027s read it."},{"Start":"00:03.255 ","End":"00:10.935","Text":"A water pump can fill a reservoir or a pool in 8 hours and there\u0027s another water pump,"},{"Start":"00:10.935 ","End":"00:16.710","Text":"and working on its own it would fill the same reservoir in 13 hours,"},{"Start":"00:16.710 ","End":"00:18.989","Text":"so the second pump is less efficient."},{"Start":"00:18.989 ","End":"00:22.200","Text":"Now the question is, how long would it take for both pumps working"},{"Start":"00:22.200 ","End":"00:25.980","Text":"together simultaneously to fill that reservoir?"},{"Start":"00:25.980 ","End":"00:29.610","Text":"Obviously, it\u0027s not going to be 8 plus 13 which is 21,"},{"Start":"00:29.610 ","End":"00:31.770","Text":"it\u0027s going to be less than each of them,"},{"Start":"00:31.770 ","End":"00:34.410","Text":"going to be less than 8 hours because they\u0027re working together."},{"Start":"00:34.410 ","End":"00:36.060","Text":"It\u0027s only going to reduce the time."},{"Start":"00:36.060 ","End":"00:40.965","Text":"Now there\u0027s a standard approach for this problem is to introduce the concept of"},{"Start":"00:40.965 ","End":"00:46.830","Text":"a job that the whole filling the reservoir is 1 job."},{"Start":"00:46.830 ","End":"00:50.780","Text":"Then we talk about the rates of each pump."},{"Start":"00:50.780 ","End":"00:54.315","Text":"The rate of pump,"},{"Start":"00:54.315 ","End":"00:59.595","Text":"this one is called pump A and this is pump B so the rate of A,"},{"Start":"00:59.595 ","End":"01:03.030","Text":"if it does the whole job in 8 hours,"},{"Start":"01:03.030 ","End":"01:08.000","Text":"it\u0027s 1/8 of the job per hour and"},{"Start":"01:08.000 ","End":"01:16.100","Text":"the rate of B is 1/13 of a job per hour."},{"Start":"01:16.100 ","End":"01:19.325","Text":"If you have difficulty with the concept of a job,"},{"Start":"01:19.325 ","End":"01:23.465","Text":"you could actually invent a number of gallons that you want."},{"Start":"01:23.465 ","End":"01:27.515","Text":"You could say let the reservoir be 1,000 gallons and then do"},{"Start":"01:27.515 ","End":"01:32.555","Text":"1,000 over 8 gallons per hour and 1,000 over 13 gallons per hour."},{"Start":"01:32.555 ","End":"01:35.720","Text":"The 1,000 is actually cancels out so you could use any number."},{"Start":"01:35.720 ","End":"01:38.720","Text":"If you don\u0027t like the concept of the job,"},{"Start":"01:38.720 ","End":"01:43.340","Text":"then you could convert it to another unit or you could work in percentages,"},{"Start":"01:43.340 ","End":"01:46.550","Text":"say that the whole job is 100 percent and so on."},{"Start":"01:46.550 ","End":"01:52.235","Text":"But I think you can understand the concept of the whole task is 1 job."},{"Start":"01:52.235 ","End":"01:56.615","Text":"Now, what we do when the both working together is we add the rates."},{"Start":"01:56.615 ","End":"02:00.530","Text":"We say that the combined rate or the simultaneous or together rate,"},{"Start":"02:00.530 ","End":"02:05.805","Text":"combined rate means both pumps working together we add the rate."},{"Start":"02:05.805 ","End":"02:07.860","Text":"This does 1/8 of a job per hour,"},{"Start":"02:07.860 ","End":"02:13.745","Text":"and this does 1/13 of a job per hour together when they\u0027re working simultaneously,"},{"Start":"02:13.745 ","End":"02:19.325","Text":"they do 1/8 plus 1/13 jobs per hour, so to speak."},{"Start":"02:19.325 ","End":"02:22.940","Text":"Now, in all these questions with jobs and rates,"},{"Start":"02:22.940 ","End":"02:29.180","Text":"it\u0027s always just like with distance that we have a standard formula that the work done,"},{"Start":"02:29.180 ","End":"02:33.800","Text":"let\u0027s say the portion of the job that each one does is"},{"Start":"02:33.800 ","End":"02:39.230","Text":"the rate in jobs per hour or whatever it is,"},{"Start":"02:39.230 ","End":"02:42.530","Text":"times the time multiplied by,"},{"Start":"02:42.530 ","End":"02:45.485","Text":"let me emphasize it and write the multiplication this way."},{"Start":"02:45.485 ","End":"02:48.240","Text":"This has to be if it is in hours,"},{"Start":"02:48.240 ","End":"02:49.730","Text":"this is in jobs per hour."},{"Start":"02:49.730 ","End":"02:53.045","Text":"If this is in seconds, this is in jobs per second and so on."},{"Start":"02:53.045 ","End":"02:56.750","Text":"What we do is we do this for"},{"Start":"02:56.750 ","End":"03:04.350","Text":"the combined rate and we say the combined rate multiplied by the time,"},{"Start":"03:04.350 ","End":"03:06.710","Text":"the time that they were combined is equal to"},{"Start":"03:06.710 ","End":"03:09.650","Text":"portion of the job would be the whole job is 1."},{"Start":"03:09.650 ","End":"03:16.135","Text":"What I\u0027m saying is that the whole thing is 1 job equals the rate,"},{"Start":"03:16.135 ","End":"03:20.125","Text":"which is 1/8 plus 1/13"},{"Start":"03:20.125 ","End":"03:25.580","Text":"multiplied by the time we haven\u0027t assigned the variable yet but that time I\u0027ll call it"},{"Start":"03:25.580 ","End":"03:30.435","Text":"t. That\u0027s the how long is in t and I\u0027m measuring it"},{"Start":"03:30.435 ","End":"03:33.270","Text":"in hours cause that\u0027s the unit I have chosen"},{"Start":"03:33.270 ","End":"03:36.240","Text":"for the rate also that\u0027s the only unit that is mentioned here,"},{"Start":"03:36.240 ","End":"03:44.750","Text":"so times t. Now that\u0027s a pretty straightforward equation because I now get that t is"},{"Start":"03:44.750 ","End":"03:53.930","Text":"equal to 1 over 1/8 plus 1/13 and push you could do this as a decimal on the calculator,"},{"Start":"03:53.930 ","End":"03:56.720","Text":"but let\u0027s practice fractions."},{"Start":"03:56.720 ","End":"04:00.065","Text":"No. Let\u0027s just continue developing it."},{"Start":"04:00.065 ","End":"04:02.450","Text":"Let\u0027s do the denominator first,"},{"Start":"04:02.450 ","End":"04:04.295","Text":"1/8 plus 1/13,"},{"Start":"04:04.295 ","End":"04:08.490","Text":"we put a common denominator and let it be 8 times"},{"Start":"04:08.490 ","End":"04:14.645","Text":"13 that\u0027s the least common denominator cause they have nothing in common so 8 times 13."},{"Start":"04:14.645 ","End":"04:18.290","Text":"Well, I\u0027ll just write it as 8 times 13 and multiply it afterwards,"},{"Start":"04:18.290 ","End":"04:20.905","Text":"so,8 goes into 8 times 13,"},{"Start":"04:20.905 ","End":"04:25.260","Text":"13 times, so this is 13 over 13 times 8."},{"Start":"04:25.260 ","End":"04:27.780","Text":"Here I multiply by 8,"},{"Start":"04:27.780 ","End":"04:30.215","Text":"so it\u0027s 8 over 8 times 13."},{"Start":"04:30.215 ","End":"04:34.640","Text":"Basically, what is often done is if you choose the product to be the common denominator,"},{"Start":"04:34.640 ","End":"04:37.265","Text":"it\u0027s this diagonal plus this diagonal,"},{"Start":"04:37.265 ","End":"04:39.335","Text":"13 plus 8 over this."},{"Start":"04:39.335 ","End":"04:42.875","Text":"Now remember when we take the reciprocal of a fraction,"},{"Start":"04:42.875 ","End":"04:50.010","Text":"we just invert the fraction so I can write it as 8 times 13 over 13 plus 8."},{"Start":"04:50.010 ","End":"04:52.445","Text":"Now time to actually do some calculations."},{"Start":"04:52.445 ","End":"04:54.380","Text":"I think we can do these in our head."},{"Start":"04:54.380 ","End":"05:00.225","Text":"Certainly the 13 plus 8 is 21 is easy enough and let\u0027s see, 8 times 13."},{"Start":"05:00.225 ","End":"05:05.640","Text":"Well, another 4 times 13 is 52 from playing cards or anyway."},{"Start":"05:05.640 ","End":"05:09.410","Text":"8 times 13 is going to be double, 52 is 104,"},{"Start":"05:09.410 ","End":"05:13.905","Text":"or use the calculator or use multiplication pencil and paper."},{"Start":"05:13.905 ","End":"05:16.260","Text":"This is what we get."},{"Start":"05:16.260 ","End":"05:22.445","Text":"Then we do need a calculator for the final answer and this comes out."},{"Start":"05:22.445 ","End":"05:26.645","Text":"My calculator says 4.952,"},{"Start":"05:26.645 ","End":"05:30.170","Text":"then rounding off to 4 places, this is what I make it."},{"Start":"05:30.170 ","End":"05:33.890","Text":"That\u0027s approximately, of course and that is hours."},{"Start":"05:33.890 ","End":"05:36.305","Text":"This is fine as the solution,"},{"Start":"05:36.305 ","End":"05:39.290","Text":"but I want to continue because sometimes they say,"},{"Start":"05:39.290 ","End":"05:42.965","Text":"oh, find the answer in hours and minutes maybe."},{"Start":"05:42.965 ","End":"05:46.330","Text":"If it was in hours and minutes and this is the optional part,"},{"Start":"05:46.330 ","End":"05:48.030","Text":"so I\u0027ll use another color, what I would do,"},{"Start":"05:48.030 ","End":"05:51.890","Text":"I would say 4 hours and lets see,"},{"Start":"05:51.890 ","End":"05:54.110","Text":"how many minutes and something minutes."},{"Start":"05:54.110 ","End":"05:56.030","Text":"Now how do I get the minutes?"},{"Start":"05:56.030 ","End":"06:02.885","Text":"I can just take this part here and multiply this by 60 minutes and if I do that,"},{"Start":"06:02.885 ","End":"06:06.860","Text":"then we get to the nearest minute, 57 minutes."},{"Start":"06:06.860 ","End":"06:10.100","Text":"Again, this is approximately and I also say this is optional."},{"Start":"06:10.100 ","End":"06:13.100","Text":"I\u0027m just doing it cause sometimes they ask in hours and minutes,"},{"Start":"06:13.100 ","End":"06:15.320","Text":"so though usually not."},{"Start":"06:15.320 ","End":"06:20.370","Text":"That\u0027s it. 4.95-4 hours and we\u0027re done."}],"ID":4979},{"Watched":false,"Name":"Exercise 9","Duration":"3m 55s","ChapterTopicVideoID":4980,"CourseChapterTopicPlaylistID":3867,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.800","Text":"Here we have a word problem."},{"Start":"00:01.800 ","End":"00:04.545","Text":"Tom can paint a fence in 14 hours."},{"Start":"00:04.545 ","End":"00:08.685","Text":"Tom and Becky can paint together in only 9 hours."},{"Start":"00:08.685 ","End":"00:13.950","Text":"The question is, how long would it take Becky to paint the fence on her own?"},{"Start":"00:13.950 ","End":"00:18.330","Text":"This is the problem that uses the concept of job and rate."},{"Start":"00:18.330 ","End":"00:20.175","Text":"We\u0027ve done a similar one before,"},{"Start":"00:20.175 ","End":"00:23.655","Text":"but I\u0027ll mention that we use the concept of"},{"Start":"00:23.655 ","End":"00:27.870","Text":"a job to indicate the whole amount of work that has to be done."},{"Start":"00:27.870 ","End":"00:31.155","Text":"The whole painting of the fence Is 1 job."},{"Start":"00:31.155 ","End":"00:38.585","Text":"For each 1 we talk about a rate of how many jobs per hour does each one do,"},{"Start":"00:38.585 ","End":"00:39.920","Text":"and when they do it together,"},{"Start":"00:39.920 ","End":"00:41.875","Text":"we add the rates."},{"Start":"00:41.875 ","End":"00:45.435","Text":"We say that Tom\u0027s rate"},{"Start":"00:45.435 ","End":"00:53.100","Text":"equals 1/14 of a job per hour,"},{"Start":"00:53.100 ","End":"00:56.549","Text":"and Becky\u0027s rate, we don\u0027t know."},{"Start":"00:56.549 ","End":"00:58.214","Text":"I\u0027ll leave that for the moment,"},{"Start":"00:58.214 ","End":"01:00.605","Text":"so leave a place holder for Becky."},{"Start":"01:00.605 ","End":"01:05.470","Text":"But the combined rate is 1/9."},{"Start":"01:05.470 ","End":"01:10.845","Text":"If we do the job in 9 hours then they do 1/9 of the job per hour."},{"Start":"01:10.845 ","End":"01:12.650","Text":"Now, Becky is blank for"},{"Start":"01:12.650 ","End":"01:15.800","Text":"the moment because we don\u0027t know how long it takes her to paint the fence."},{"Start":"01:15.800 ","End":"01:17.690","Text":"That\u0027s what we are asking."},{"Start":"01:17.690 ","End":"01:21.595","Text":"If I let this how long as a time be,"},{"Start":"01:21.595 ","End":"01:24.725","Text":"say t for time in hours,"},{"Start":"01:24.725 ","End":"01:32.215","Text":"then Becky\u0027s rate would be 1 over t job per hour."},{"Start":"01:32.215 ","End":"01:34.280","Text":"Her whole job takes t hours."},{"Start":"01:34.280 ","End":"01:36.920","Text":"When each hour she does went over t of the job."},{"Start":"01:36.920 ","End":"01:39.930","Text":"Then let me say that this is in hours,"},{"Start":"01:39.930 ","End":"01:42.095","Text":"I like to write the unit."},{"Start":"01:42.095 ","End":"01:47.400","Text":"T is the variable which says how long would Becky take,"},{"Start":"01:47.400 ","End":"01:49.065","Text":"and that\u0027s what we\u0027re looking for."},{"Start":"01:49.065 ","End":"01:52.790","Text":"Now, there is an addition here because when they\u0027re working together,"},{"Start":"01:52.790 ","End":"01:53.960","Text":"we add the rates."},{"Start":"01:53.960 ","End":"01:59.900","Text":"Each hour Tom does a 14th of the job and each hour Becky does 1/t of the job."},{"Start":"01:59.900 ","End":"02:01.940","Text":"This plus this equals this."},{"Start":"02:01.940 ","End":"02:09.735","Text":"In other words, 1/14 plus 1/t equals 1/9,"},{"Start":"02:09.735 ","End":"02:12.680","Text":"and we\u0027re looking for t. That is the variable."},{"Start":"02:12.680 ","End":"02:17.075","Text":"Let\u0027s solve it. Let\u0027s bring the number over to the other side and say that"},{"Start":"02:17.075 ","End":"02:23.315","Text":"1/t equals 1/9 minus 1/14."},{"Start":"02:23.315 ","End":"02:24.905","Text":"Let\u0027s see what this equals."},{"Start":"02:24.905 ","End":"02:30.220","Text":"If we let the common denominator be 9 times 14,"},{"Start":"02:30.220 ","End":"02:32.275","Text":"there\u0027s nothing lower that will do it."},{"Start":"02:32.275 ","End":"02:34.310","Text":"Turns out if they don\u0027t have any common factor then the"},{"Start":"02:34.310 ","End":"02:36.210","Text":"best you can do is to multiply them,"},{"Start":"02:36.210 ","End":"02:40.240","Text":"and actually you can almost blindly use the method of a diagonal."},{"Start":"02:40.240 ","End":"02:42.715","Text":"This times this is 14,"},{"Start":"02:42.715 ","End":"02:46.555","Text":"this times this 9 times 1 is 9 over this times this."},{"Start":"02:46.555 ","End":"02:51.015","Text":"In other words, we get 5 over, let\u0027s see,"},{"Start":"02:51.015 ","End":"02:56.715","Text":"9 times 14 would be 126."},{"Start":"02:56.715 ","End":"02:59.760","Text":"That\u0027s 1/t we\u0027re now working in the algebra."},{"Start":"02:59.760 ","End":"03:02.400","Text":"T is the reciprocal of that,"},{"Start":"03:02.400 ","End":"03:04.200","Text":"is 1 over this."},{"Start":"03:04.200 ","End":"03:05.820","Text":"We invert it."},{"Start":"03:05.820 ","End":"03:10.485","Text":"Well, I\u0027ll write it first as 1/5/126,"},{"Start":"03:10.485 ","End":"03:12.480","Text":"and we take 1 over a fraction,"},{"Start":"03:12.480 ","End":"03:17.295","Text":"the reciprocal we just invert it,126/5."},{"Start":"03:17.295 ","End":"03:24.170","Text":"We do the division, 5 goes into 125 times and 1/5,"},{"Start":"03:24.170 ","End":"03:29.410","Text":"and that\u0027s in hours and that\u0027s our answer."},{"Start":"03:31.160 ","End":"03:33.635","Text":"That\u0027s fine as an answer."},{"Start":"03:33.635 ","End":"03:37.915","Text":"If sometimes they ask for what it is in hours and minutes,"},{"Start":"03:37.915 ","End":"03:39.200","Text":"and in that case,"},{"Start":"03:39.200 ","End":"03:40.820","Text":"if they asked in hours and minutes,"},{"Start":"03:40.820 ","End":"03:43.325","Text":"I would say 25 hours."},{"Start":"03:43.325 ","End":"03:44.840","Text":"What is the fifth of an hour?"},{"Start":"03:44.840 ","End":"03:49.175","Text":"Well, 60/5 is 12 minutes."},{"Start":"03:49.175 ","End":"03:52.790","Text":"But I would say that this is the answer."},{"Start":"03:52.790 ","End":"03:56.130","Text":"This is just in case. We are done."}],"ID":4980},{"Watched":false,"Name":"Exercise 10","Duration":"5m 4s","ChapterTopicVideoID":4981,"CourseChapterTopicPlaylistID":3867,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"Here we have a word problem,"},{"Start":"00:02.040 ","End":"00:07.965","Text":"and actually I got the idea from a cocktail mixed drink called a martini,"},{"Start":"00:07.965 ","End":"00:11.565","Text":"which is made from vermouth and gin,"},{"Start":"00:11.565 ","End":"00:15.990","Text":"and typically vermouth has around 17 percent alcohol,"},{"Start":"00:15.990 ","End":"00:19.005","Text":"whereas gin has 40 percent alcohol."},{"Start":"00:19.005 ","End":"00:24.420","Text":"The idea here is how much vermouth to mix with how much gin, I\u0027ll just read it,"},{"Start":"00:24.420 ","End":"00:27.765","Text":"how much vermouth should be mixed with 50 milliliters of"},{"Start":"00:27.765 ","End":"00:32.460","Text":"gin to get a cocktail which is 30 percent alcohol,"},{"Start":"00:32.460 ","End":"00:36.810","Text":"and I\u0027ve given here the 17 and the 40 or the percentages."},{"Start":"00:36.810 ","End":"00:40.480","Text":"What we\u0027ll do is first of all assign a variable name,"},{"Start":"00:40.480 ","End":"00:45.365","Text":"and we\u0027ll use the quantity that we don\u0027t know which is the amount of vermouth,"},{"Start":"00:45.365 ","End":"00:49.855","Text":"how much vermouth, this we\u0027ll call x,"},{"Start":"00:49.855 ","End":"00:51.960","Text":"so x equals amount,"},{"Start":"00:51.960 ","End":"00:57.530","Text":"we\u0027ll abbreviate amount to amt of the vermouth."},{"Start":"00:57.530 ","End":"00:59.855","Text":"The general tactic will be,"},{"Start":"00:59.855 ","End":"01:03.800","Text":"first of all to say that the volume of vermouth"},{"Start":"01:03.800 ","End":"01:08.135","Text":"plus the volume of gin will be the volume of the cocktail."},{"Start":"01:08.135 ","End":"01:11.780","Text":"I might also say the amount of alcohol in the vermouth plus the amount"},{"Start":"01:11.780 ","End":"01:15.625","Text":"of alcohol in the gin is the amount of alcohol in the cocktail."},{"Start":"01:15.625 ","End":"01:17.145","Text":"That\u0027s for amounts,"},{"Start":"01:17.145 ","End":"01:19.085","Text":"x is the amount of vermouth."},{"Start":"01:19.085 ","End":"01:22.010","Text":"We know that the amount of gin is 50 milliliters,"},{"Start":"01:22.010 ","End":"01:29.915","Text":"but x plus 50 will equal the total amount of cocktail of mixed drink."},{"Start":"01:29.915 ","End":"01:32.270","Text":"Now as for the alcohol,"},{"Start":"01:32.270 ","End":"01:39.425","Text":"we have the amount of alcohol in the vermouth."},{"Start":"01:39.425 ","End":"01:42.260","Text":"How do we compute the amount of alcohol?"},{"Start":"01:42.260 ","End":"01:45.590","Text":"Well, in general, not just for the vermouth, in general,"},{"Start":"01:45.590 ","End":"01:48.860","Text":"the amount of alcohol,"},{"Start":"01:48.860 ","End":"01:50.645","Text":"the volume of alcohol,"},{"Start":"01:50.645 ","End":"01:58.100","Text":"is going to equal the amount of drink times the percentage of alcohol."},{"Start":"01:58.100 ","End":"02:00.635","Text":"It\u0027s just a percentage question."},{"Start":"02:00.635 ","End":"02:08.283","Text":"For example, here, the amount of alcohol in the vermouth would equal 17 percent,"},{"Start":"02:08.283 ","End":"02:10.465","Text":"I\u0027ll take the percentage first actually,"},{"Start":"02:10.465 ","End":"02:15.070","Text":"is 17 percent of the x,"},{"Start":"02:15.070 ","End":"02:24.825","Text":"and the amount of alcohol in the gin is equal to 40 percent of 50,"},{"Start":"02:24.825 ","End":"02:26.910","Text":"we\u0027re working in milliliters let\u0027s say,"},{"Start":"02:26.910 ","End":"02:35.460","Text":"and the amount of alcohol in the cocktail as a whole will equal this plus this,"},{"Start":"02:35.460 ","End":"02:41.970","Text":"so it\u0027s 17 percent of x is 0.17x,"},{"Start":"02:41.970 ","End":"02:43.620","Text":"let\u0027s work in decimals,"},{"Start":"02:43.620 ","End":"02:48.470","Text":"and here we have 0.4 times 50."},{"Start":"02:48.470 ","End":"02:50.720","Text":"I\u0027ll use the multiplication sign like this,"},{"Start":"02:50.720 ","End":"02:52.910","Text":"so it won\u0027t be confused with the decimal point."},{"Start":"02:52.910 ","End":"02:54.755","Text":"This is x, this is multiplication."},{"Start":"02:54.755 ","End":"02:56.690","Text":"Well, let\u0027s just not get confused."},{"Start":"02:56.690 ","End":"03:00.685","Text":"This will equal the sum of these two."},{"Start":"03:00.685 ","End":"03:03.020","Text":"Well, let\u0027s just write this in 2 ways."},{"Start":"03:03.020 ","End":"03:05.300","Text":"This is what gives us the equation."},{"Start":"03:05.300 ","End":"03:07.055","Text":"I\u0027ll write it over here."},{"Start":"03:07.055 ","End":"03:08.570","Text":"On the one hand,"},{"Start":"03:08.570 ","End":"03:17.085","Text":"it\u0027s 0.17x plus 0.4 times 50 is 20,"},{"Start":"03:17.085 ","End":"03:18.935","Text":"but on the other hand,"},{"Start":"03:18.935 ","End":"03:27.075","Text":"it\u0027s also equal to 30 percent using this formula of the total volume,"},{"Start":"03:27.075 ","End":"03:29.940","Text":"and the total volume is x plus 50."},{"Start":"03:29.940 ","End":"03:31.310","Text":"Let\u0027s modify this slightly."},{"Start":"03:31.310 ","End":"03:33.100","Text":"I want to put this over here."},{"Start":"03:33.100 ","End":"03:34.950","Text":"I\u0027ll just move this over here,"},{"Start":"03:34.950 ","End":"03:36.720","Text":"it will be more convenient,"},{"Start":"03:36.720 ","End":"03:41.930","Text":"and then to say that this 30 percent of x plus 50,"},{"Start":"03:41.930 ","End":"03:47.235","Text":"I\u0027ll write it over here as 0.3 times x plus 50."},{"Start":"03:47.235 ","End":"03:49.830","Text":"This line is without decimals."},{"Start":"03:49.830 ","End":"03:54.210","Text":"Now it\u0027s a straightforward linear equation. Let\u0027s just solve it."},{"Start":"03:54.210 ","End":"03:58.380","Text":"0.17x plus 20 equals"},{"Start":"03:58.380 ","End":"04:05.820","Text":"0.3x plus 0.3 times 50 is 15."},{"Start":"04:05.820 ","End":"04:11.000","Text":"Let\u0027s move the numbers to the left and the variables on the right,"},{"Start":"04:11.000 ","End":"04:12.230","Text":"so it\u0027ll be positive,"},{"Start":"04:12.230 ","End":"04:15.740","Text":"so 20 minus 15 is 5,"},{"Start":"04:15.740 ","End":"04:19.775","Text":"and 0.3 minus 0.17 is"},{"Start":"04:19.775 ","End":"04:26.640","Text":"0.13x because this is like 0.30 and 30 minus 17 is 13,"},{"Start":"04:26.640 ","End":"04:33.465","Text":"and so x is equal to 5/0.13,"},{"Start":"04:33.465 ","End":"04:40.870","Text":"and this will equal approximately 38.46,"},{"Start":"04:40.870 ","End":"04:44.010","Text":"and so this is our answer."},{"Start":"04:45.610 ","End":"04:49.360","Text":"How much vermouth? Let\u0027s round it off to the nearest milliliter,"},{"Start":"04:49.360 ","End":"04:51.515","Text":"or I could leave it like this, doesn\u0027t matter."},{"Start":"04:51.515 ","End":"05:00.785","Text":"38.46 milliliters of vermouth in the cocktail, etc."},{"Start":"05:00.785 ","End":"05:04.170","Text":"This is the answer."}],"ID":4981},{"Watched":false,"Name":"Exercise 11","Duration":"7m 2s","ChapterTopicVideoID":4982,"CourseChapterTopicPlaylistID":3867,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.670","Text":"Here we have a word problem let\u0027s read it."},{"Start":"00:02.670 ","End":"00:07.095","Text":"We have 2 concentrations of sulfuric acid solutions:"},{"Start":"00:07.095 ","End":"00:12.540","Text":"15 percent and 40 percent the solutions are in water, I should say."},{"Start":"00:12.540 ","End":"00:20.189","Text":"How much of each should we mix together to get 100 liters of a 25 percent solution?"},{"Start":"00:20.189 ","End":"00:25.335","Text":"The last sentence just says that you have enough or any amount that you need of"},{"Start":"00:25.335 ","End":"00:30.375","Text":"each of these 2 concentrations they\u0027re 15 and the 40 percent."},{"Start":"00:30.375 ","End":"00:32.760","Text":"Let me just explain this in other words."},{"Start":"00:32.760 ","End":"00:35.340","Text":"Basically, you can order from the supplier"},{"Start":"00:35.340 ","End":"00:38.030","Text":"only 2 kinds of sulfuric acid and as much as you need it."},{"Start":"00:38.030 ","End":"00:40.400","Text":"There\u0027s a 15 percent concentration of"},{"Start":"00:40.400 ","End":"00:43.355","Text":"acid in water and there\u0027s a 40 percent concentration."},{"Start":"00:43.355 ","End":"00:50.030","Text":"But what you need for some chemical experiment is a 100 liters of 25 percent solution."},{"Start":"00:50.030 ","End":"00:54.350","Text":"By mixing the 15 and the 40 in the right quantities,"},{"Start":"00:54.350 ","End":"00:58.060","Text":"you\u0027re going to get a 25 percent solution and this is how much you\u0027re going to get of it."},{"Start":"00:58.060 ","End":"01:02.630","Text":"Let\u0027s see how much of each and do we need"},{"Start":"01:02.630 ","End":"01:07.550","Text":"2 variables because we have to know how much of each concentration,"},{"Start":"01:07.550 ","End":"01:09.080","Text":"how much of each solution."},{"Start":"01:09.080 ","End":"01:14.750","Text":"We don\u0027t need 2 variables because if we take a variable for one of them,"},{"Start":"01:14.750 ","End":"01:16.520","Text":"since we have a 100 liters together,"},{"Start":"01:16.520 ","End":"01:19.610","Text":"we can get the other variable in terms of that."},{"Start":"01:19.610 ","End":"01:24.560","Text":"Well, let\u0027s start. We\u0027ll work in liters."},{"Start":"01:24.560 ","End":"01:32.300","Text":"Let\u0027s say we have x liters of the 15 percent solution and then we"},{"Start":"01:32.300 ","End":"01:39.950","Text":"automatically have 100 minus x liters of the other solution,"},{"Start":"01:39.950 ","End":"01:42.995","Text":"which is the 40 percent solution."},{"Start":"01:42.995 ","End":"01:49.910","Text":"Right away, I\u0027ve used the fact which I could\u0027ve said in words that the amount of"},{"Start":"01:49.910 ","End":"01:52.655","Text":"this solution plus the amount of this solution is"},{"Start":"01:52.655 ","End":"01:57.020","Text":"100 but it\u0027s obvious so I just went ahead and said,"},{"Start":"01:57.020 ","End":"01:59.780","Text":"okay, then we have 100 minus x of the other."},{"Start":"01:59.780 ","End":"02:05.140","Text":"Now, the other word equation is the total amount of"},{"Start":"02:05.140 ","End":"02:09.800","Text":"the actual sulfuric acid in each solution is"},{"Start":"02:09.800 ","End":"02:14.420","Text":"going to equal the amount of acid in the mixture solution."},{"Start":"02:14.420 ","End":"02:17.570","Text":"In general, the amount,"},{"Start":"02:17.570 ","End":"02:18.680","Text":"I\u0027ll write that as Amt,"},{"Start":"02:18.680 ","End":"02:23.570","Text":"amount of acid in the solution,"},{"Start":"02:23.570 ","End":"02:25.640","Text":"I\u0027ll abbreviate that also,"},{"Start":"02:25.640 ","End":"02:32.044","Text":"is equal to the percentage of acid, how come?"},{"Start":"02:32.044 ","End":"02:36.950","Text":"That\u0027s the concentration whether it\u0027s a 15 or the 40 or the 25 or whatever,"},{"Start":"02:36.950 ","End":"02:44.945","Text":"the percentage of acid times the amount of the solution,"},{"Start":"02:44.945 ","End":"02:47.200","Text":"the volume of the solution."},{"Start":"02:47.200 ","End":"02:52.655","Text":"I\u0027m not going to do a computation with the amounts of acid in the 3 solutions."},{"Start":"02:52.655 ","End":"02:57.080","Text":"I have, basically the 15 percent solution,"},{"Start":"02:57.080 ","End":"03:04.655","Text":"I have the 40 percent solution and I have the mixture which is the 25 percent solution."},{"Start":"03:04.655 ","End":"03:09.200","Text":"Let\u0027s compute the amount of acid in each of them."},{"Start":"03:09.200 ","End":"03:12.665","Text":"Amount of acid."},{"Start":"03:12.665 ","End":"03:16.470","Text":"Here I have 15 percent,"},{"Start":"03:16.470 ","End":"03:25.760","Text":"which is 0.15 times the amount of 15 percent solution, which is x."},{"Start":"03:25.760 ","End":"03:28.200","Text":"Here I have 40 percent,"},{"Start":"03:28.200 ","End":"03:35.055","Text":"which is 0.4 times this amount which is 100 minus x."},{"Start":"03:35.055 ","End":"03:39.965","Text":"Here I get the equation because on the one hand,"},{"Start":"03:39.965 ","End":"03:46.925","Text":"I have 25 percent of 100 because of the concentration equation."},{"Start":"03:46.925 ","End":"03:48.335","Text":"On the other hand,"},{"Start":"03:48.335 ","End":"03:53.000","Text":"the amount of acid in the mixture is equal to the amount of acid here plus this."},{"Start":"03:53.000 ","End":"03:56.655","Text":"There was this, plus this equals this,"},{"Start":"03:56.655 ","End":"03:58.460","Text":"I get that on the other hand,"},{"Start":"03:58.460 ","End":"04:06.415","Text":"is equal to 0.15x plus 0.4 times 100 minus x."},{"Start":"04:06.415 ","End":"04:07.805","Text":"Let\u0027s go over that again."},{"Start":"04:07.805 ","End":"04:11.299","Text":"I\u0027m computing the amount of acid in the mixture in 2 ways."},{"Start":"04:11.299 ","End":"04:14.765","Text":"One way is taking the concentration 25 percent or"},{"Start":"04:14.765 ","End":"04:18.950","Text":"0.25 of the 100 liters, on the other hand,"},{"Start":"04:18.950 ","End":"04:21.175","Text":"it\u0027s equal to the amount of acid in"},{"Start":"04:21.175 ","End":"04:25.010","Text":"one solution plus the amount of acid in the other and in each case,"},{"Start":"04:25.010 ","End":"04:31.055","Text":"I multiplied the percentage by the amount to get the quantity of just sulfuric acid."},{"Start":"04:31.055 ","End":"04:35.025","Text":"Now it\u0027s a linear equation let\u0027s just work on it."},{"Start":"04:35.025 ","End":"04:37.745","Text":"I\u0027ll copy the left-hand side."},{"Start":"04:37.745 ","End":"04:40.655","Text":"Well, actually I can multiply it out, why not?"},{"Start":"04:40.655 ","End":"04:42.305","Text":"So I\u0027ll erase that."},{"Start":"04:42.305 ","End":"04:47.480","Text":"0.25 times 100 is just 25 that\u0027s straightforward,"},{"Start":"04:47.480 ","End":"04:54.565","Text":"is equal to 0.15x plus 0.4 times 100 is 40."},{"Start":"04:54.565 ","End":"04:59.693","Text":"Multiplying by 100 is relatively easy and then minus 0.4x."},{"Start":"04:59.693 ","End":"05:05.075","Text":"Now let\u0027s put xs on the left and numbers on the right."},{"Start":"05:05.075 ","End":"05:10.240","Text":"On the left, I have 0.4x"},{"Start":"05:10.240 ","End":"05:16.335","Text":"minus 0.15x is equal to,"},{"Start":"05:16.335 ","End":"05:20.670","Text":"on the right-hand side 40 minus 25,"},{"Start":"05:20.670 ","End":"05:23.130","Text":"0.4 is the same as 0.40,"},{"Start":"05:23.130 ","End":"05:30.824","Text":"so 40 minus 15 is 25 so I\u0027ve got 0.25x is equal to"},{"Start":"05:30.824 ","End":"05:40.060","Text":"40 minus 25 is 15. x is equal to 15 divided by,"},{"Start":"05:40.060 ","End":"05:44.380","Text":"we just dividing both sides by 0.25 and"},{"Start":"05:44.380 ","End":"05:48.945","Text":"this point we can use a calculator or I can try without a calculator."},{"Start":"05:48.945 ","End":"05:51.170","Text":"Let\u0027s see. To get rid of the decimal,"},{"Start":"05:51.170 ","End":"05:59.230","Text":"I can multiply top and bottom by a 100 and then I get 1500 over just 25,"},{"Start":"05:59.230 ","End":"06:03.765","Text":"25 goes into 1500 if I divide it,"},{"Start":"06:03.765 ","End":"06:06.090","Text":"I can see that 25 goes into 100,"},{"Start":"06:06.090 ","End":"06:09.995","Text":"4 times, times the 15 is 60."},{"Start":"06:09.995 ","End":"06:12.110","Text":"Better to do it on the calculator and check."},{"Start":"06:12.110 ","End":"06:19.390","Text":"I make the answer as 60 liters of 15 percent solution,"},{"Start":"06:19.390 ","End":"06:21.540","Text":"but the question asked,"},{"Start":"06:21.540 ","End":"06:24.220","Text":"how much of each?"},{"Start":"06:24.220 ","End":"06:29.060","Text":"I also have to say how much of the 40 percent solution,"},{"Start":"06:29.060 ","End":"06:31.175","Text":"that\u0027s going to be 100 minus x."},{"Start":"06:31.175 ","End":"06:33.930","Text":"Let me just scroll down."},{"Start":"06:35.060 ","End":"06:37.200","Text":"Do a side exercise,"},{"Start":"06:37.200 ","End":"06:41.580","Text":"a 100 minus 60 equals 40,"},{"Start":"06:41.580 ","End":"06:47.985","Text":"so it\u0027s 40 liters of the 40 percent,"},{"Start":"06:47.985 ","End":"06:52.555","Text":"I should write the word solution, abbreviated solution."},{"Start":"06:52.555 ","End":"06:54.195","Text":"There\u0027s 40 liters,"},{"Start":"06:54.195 ","End":"06:55.830","Text":"these are both 2 answers,"},{"Start":"06:55.830 ","End":"06:57.855","Text":"the question asked how much of each."},{"Start":"06:57.855 ","End":"07:02.800","Text":"Okay. We are done."}],"ID":4982},{"Watched":false,"Name":"Exercise 12","Duration":"2m 52s","ChapterTopicVideoID":4983,"CourseChapterTopicPlaylistID":3867,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.225","Text":"This exercise is a word problem. Let\u0027s read it."},{"Start":"00:03.225 ","End":"00:07.815","Text":"A rectangular field is surrounded by 90 meters of fence."},{"Start":"00:07.815 ","End":"00:10.935","Text":"The field is twice as long as it is wide."},{"Start":"00:10.935 ","End":"00:13.410","Text":"What are the dimensions of the field?"},{"Start":"00:13.410 ","End":"00:16.905","Text":"I think a sketch would probably help here."},{"Start":"00:16.905 ","End":"00:20.080","Text":"Let\u0027s say we want to draw a rectangular field."},{"Start":"00:20.080 ","End":"00:25.350","Text":"It has 4 sides, 1,2,3,4."},{"Start":"00:27.050 ","End":"00:29.765","Text":"I straightened it out a bit."},{"Start":"00:29.765 ","End":"00:32.885","Text":"Let\u0027s decide on a variable."},{"Start":"00:32.885 ","End":"00:34.520","Text":"Well, we need 2 things?"},{"Start":"00:34.520 ","End":"00:38.225","Text":"We need both the length and the width, that\u0027s the dimensions."},{"Start":"00:38.225 ","End":"00:40.565","Text":"This is the width and that\u0027s the length."},{"Start":"00:40.565 ","End":"00:44.270","Text":"But we really only need to decide on a name for one of them."},{"Start":"00:44.270 ","End":"00:46.370","Text":"Because one\u0027s, twice the other."},{"Start":"00:46.370 ","End":"00:48.770","Text":"I either say that this is half of this,"},{"Start":"00:48.770 ","End":"00:49.820","Text":"or this is twice this."},{"Start":"00:49.820 ","End":"00:57.395","Text":"I prefer to say that this is x and that this would be twice x."},{"Start":"00:57.395 ","End":"01:06.365","Text":"That\u0027s that and the equation that we have is that we have 90 meters of fence."},{"Start":"01:06.365 ","End":"01:09.755","Text":"Now, the fence has 4 parts."},{"Start":"01:09.755 ","End":"01:11.735","Text":"We have x here, 2x here,"},{"Start":"01:11.735 ","End":"01:13.610","Text":"this is also 2x,"},{"Start":"01:13.610 ","End":"01:15.950","Text":"this is also x."},{"Start":"01:15.950 ","End":"01:19.699","Text":"Basically what we have is the perimeter,"},{"Start":"01:19.699 ","End":"01:20.975","Text":"that\u0027s what it\u0027s called,"},{"Start":"01:20.975 ","End":"01:23.495","Text":"the length going around,"},{"Start":"01:23.495 ","End":"01:28.645","Text":"we have x plus 2x plus x again,"},{"Start":"01:28.645 ","End":"01:34.625","Text":"plus 2x again, all this is equal to 90."},{"Start":"01:34.625 ","End":"01:36.740","Text":"Now if we add up the x\u0027s here,"},{"Start":"01:36.740 ","End":"01:39.605","Text":"1 and 2 and 1 and 2 is 6."},{"Start":"01:39.605 ","End":"01:43.950","Text":"So we get that 6x is equal to 90."},{"Start":"01:43.950 ","End":"01:46.555","Text":"Then dividing both sides by 6,"},{"Start":"01:46.555 ","End":"01:50.750","Text":"we get that x is equal to 15."},{"Start":"01:50.750 ","End":"01:55.520","Text":"Now the question asks for the dimensions,"},{"Start":"01:55.520 ","End":"02:00.305","Text":"so you could say the field is"},{"Start":"02:00.305 ","End":"02:08.040","Text":"15 meters by 2x is 30 meters."},{"Start":"02:08.040 ","End":"02:09.930","Text":"Or you could just say,"},{"Start":"02:09.930 ","End":"02:16.910","Text":"something like the length equals 30, width equals 15."},{"Start":"02:16.910 ","End":"02:18.260","Text":"Many ways to phrase it,"},{"Start":"02:18.260 ","End":"02:19.760","Text":"I\u0027ll just leave it like that."},{"Start":"02:19.760 ","End":"02:21.845","Text":"It\u0027s 15 by 30."},{"Start":"02:21.845 ","End":"02:24.860","Text":"In fact, we could even check that everything works here,"},{"Start":"02:24.860 ","End":"02:26.900","Text":"because if this is the width,"},{"Start":"02:26.900 ","End":"02:30.230","Text":"I\u0027ll just write w here and this is the length l. We"},{"Start":"02:30.230 ","End":"02:33.725","Text":"see that the length is indeed twice the width."},{"Start":"02:33.725 ","End":"02:37.260","Text":"So that satisfies that condition."},{"Start":"02:37.260 ","End":"02:39.830","Text":"Is it 90 meters of fence?"},{"Start":"02:39.830 ","End":"02:41.650","Text":"Well, if this is 30,"},{"Start":"02:41.650 ","End":"02:46.700","Text":"we have twice 30 is 60 and twice 15 is another 30,"},{"Start":"02:46.700 ","End":"02:47.990","Text":"so it really is 90,"},{"Start":"02:47.990 ","End":"02:49.550","Text":"so it all works out."},{"Start":"02:49.550 ","End":"02:52.860","Text":"And this was a short problem. We are done."}],"ID":4983}],"Thumbnail":null,"ID":3867},{"Name":"Quadratic Equations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Quadratic Equations, Formula","Duration":"7m 48s","ChapterTopicVideoID":4993,"CourseChapterTopicPlaylistID":3868,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.310","Text":"Now we come to quadratic equations which are not linear."},{"Start":"00:05.310 ","End":"00:07.170","Text":"Let me show you an example of"},{"Start":"00:07.170 ","End":"00:17.415","Text":"1(x)^2 plus 2x minus 8=0."},{"Start":"00:17.415 ","End":"00:19.005","Text":"This is an example."},{"Start":"00:19.005 ","End":"00:23.200","Text":"What makes it quadratic is the presence of (x)^2."},{"Start":"00:23.200 ","End":"00:29.010","Text":"I think the word quadratic comes from the word square of squared in Latin."},{"Start":"00:29.010 ","End":"00:36.465","Text":"I know that things with this prefix like quadrilateral and quadrangle it means full."},{"Start":"00:36.465 ","End":"00:45.200","Text":"Anyway, quadratic just means having an (x)^2 in it and also an x and a number."},{"Start":"00:45.200 ","End":"00:48.330","Text":"This is a typical quadratic equation."},{"Start":"00:48.880 ","End":"00:54.965","Text":"There\u0027s not an easy way to solve it unless you know the trick or the formula."},{"Start":"00:54.965 ","End":"00:56.990","Text":"If you try just messing with it,"},{"Start":"00:56.990 ","End":"00:59.450","Text":"you won\u0027t get very far."},{"Start":"00:59.450 ","End":"01:05.030","Text":"Let me give you the formula for solving such a thing."},{"Start":"01:05.030 ","End":"01:11.630","Text":"Here is the formula and I need to explain what\u0027s going on here."},{"Start":"01:11.630 ","End":"01:14.750","Text":"First of all, it assumes that we start out"},{"Start":"01:14.750 ","End":"01:18.260","Text":"with an equation that looks something like this."},{"Start":"01:18.260 ","End":"01:21.005","Text":"This is in fact true."},{"Start":"01:21.005 ","End":"01:23.463","Text":"This does look like this,"},{"Start":"01:23.463 ","End":"01:33.270","Text":"provided that if we take a=1 because this is"},{"Start":"01:33.270 ","End":"01:37.020","Text":"really (1x)^2 and if we take"},{"Start":"01:37.020 ","End":"01:45.070","Text":"b=2 and if we take c=-8,"},{"Start":"01:45.070 ","End":"01:47.840","Text":"if you put these values in here for a,"},{"Start":"01:47.840 ","End":"01:51.655","Text":"b and c then you will get what we have here."},{"Start":"01:51.655 ","End":"01:54.920","Text":"Before the actual formula path,"},{"Start":"01:54.920 ","End":"01:56.495","Text":"which is this is presented,"},{"Start":"01:56.495 ","End":"02:00.050","Text":"we\u0027ll assume that we\u0027re in this situation that we have something (x)^2"},{"Start":"02:00.050 ","End":"02:04.460","Text":"plus something x plus something equals 0, which we do."},{"Start":"02:04.460 ","End":"02:09.980","Text":"It\u0027s also an important remark that a not equal to 0,"},{"Start":"02:09.980 ","End":"02:13.480","Text":"meaning that there is something before the (x)^2."},{"Start":"02:13.480 ","End":"02:14.870","Text":"There is an x squared term."},{"Start":"02:14.870 ","End":"02:18.745","Text":"If a was 0, there wouldn\u0027t be an x^2 and it wouldn\u0027t be a quadratic equation."},{"Start":"02:18.745 ","End":"02:21.780","Text":"This only makes sense."},{"Start":"02:21.780 ","End":"02:25.250","Text":"Now let\u0027s come to the actual formula part."},{"Start":"02:25.250 ","End":"02:33.560","Text":"This is the setup and we see that it gives us what x is but notice this peculiar thing,"},{"Start":"02:33.560 ","End":"02:37.010","Text":"this plus or minus sign."},{"Start":"02:37.010 ","End":"02:42.800","Text":"What this actually means is that there could be 2 solutions and they\u0027re often,"},{"Start":"02:42.800 ","End":"02:47.360","Text":"1 is obtained by using the plus sign and 1"},{"Start":"02:47.360 ","End":"02:52.490","Text":"by using the minus sign so we actually have a 2-in-1 here."},{"Start":"02:52.490 ","End":"02:58.435","Text":"It\u0027s best explained if we just go ahead and solve this equation using this formula."},{"Start":"02:58.435 ","End":"03:04.490","Text":"We get that x equals I\u0027m just doing it step by step,"},{"Start":"03:04.490 ","End":"03:11.905","Text":"minus b, b is 2 so minus b means minus 2."},{"Start":"03:11.905 ","End":"03:14.510","Text":"Then I put this plus or minus sign,"},{"Start":"03:14.510 ","End":"03:19.055","Text":"which is just a plus with an extra line here is the plus or minus."},{"Start":"03:19.055 ","End":"03:28.860","Text":"Then we have the square root of (b)^2 b is 2 so I\u0027ll just write it as"},{"Start":"03:28.860 ","End":"03:35.910","Text":"2^2 minus 4ac,4 is"},{"Start":"03:35.910 ","End":"03:40.375","Text":"just 4 and then a is 1."},{"Start":"03:40.375 ","End":"03:44.600","Text":"You have to put a dot or brackets because it\u0027s multiplication."},{"Start":"03:44.600 ","End":"03:48.005","Text":"I mean, when you don\u0027t write anything in algebra,"},{"Start":"03:48.005 ","End":"03:50.600","Text":"when you put things next to each other it\u0027s a multiplication so I need"},{"Start":"03:50.600 ","End":"03:53.690","Text":"to indicate 4 times,"},{"Start":"03:53.690 ","End":"03:58.075","Text":"1 times and c is minus 8."},{"Start":"03:58.075 ","End":"04:02.225","Text":"Here I\u0027ll need to put the minus 8 parenthesis,"},{"Start":"04:02.225 ","End":"04:05.520","Text":"separate the 2 operations times and"},{"Start":"04:05.520 ","End":"04:12.855","Text":"minus and then over 2a,"},{"Start":"04:12.855 ","End":"04:16.515","Text":"2 is 2 and a we already said is 1."},{"Start":"04:16.515 ","End":"04:22.010","Text":"This is very common for a to be 1 very often the (x)^2 doesn\u0027t have anything here,"},{"Start":"04:22.010 ","End":"04:23.525","Text":"which means that it\u0027s like a 1."},{"Start":"04:23.525 ","End":"04:26.270","Text":"Anyway, 2a is not 21,"},{"Start":"04:26.270 ","End":"04:28.800","Text":"it\u0027s 2 times 1."},{"Start":"04:30.370 ","End":"04:32.840","Text":"We just have to evaluate this."},{"Start":"04:32.840 ","End":"04:37.400","Text":"Now. Remember the order of operations brackets first."},{"Start":"04:37.400 ","End":"04:41.135","Text":"Well, what\u0027s under the square root sign counts like it was a bracket."},{"Start":"04:41.135 ","End":"04:45.725","Text":"I have to do what\u0027s under the square root sign first and often."},{"Start":"04:45.725 ","End":"04:49.650","Text":"In fact, I\u0027ll start off right away by doing it at the side."},{"Start":"04:49.690 ","End":"04:54.605","Text":"This part here, I like doing as a side exercise."},{"Start":"04:54.605 ","End":"05:04.320","Text":"What I get is 2^2 is 4 and minus,"},{"Start":"05:04.320 ","End":"05:12.610","Text":"now 4 times 1 times minus 8 is minus 32."},{"Start":"05:13.400 ","End":"05:23.140","Text":"That makes it 4 plus 32, which is 36."},{"Start":"05:23.570 ","End":"05:33.230","Text":"We have minus 2 plus or minus the square root of."},{"Start":"05:33.230 ","End":"05:36.890","Text":"Now this is what we did at the side, that\u0027s 36."},{"Start":"05:36.890 ","End":"05:47.070","Text":"Then over 2 the square root of 36 is 6 so I get minus 2 plus or"},{"Start":"05:47.070 ","End":"05:52.380","Text":"minus 6 over 2 and then"},{"Start":"05:52.380 ","End":"05:58.220","Text":"I like to branch off and on one end of the branch,"},{"Start":"05:58.220 ","End":"06:02.015","Text":"I\u0027ll do a plus and the other hand I\u0027ll do a minus."},{"Start":"06:02.015 ","End":"06:03.695","Text":"If we take the plus,"},{"Start":"06:03.695 ","End":"06:11.825","Text":"we have minus 2 plus 6 over 2 and if you take the other fork in the road,"},{"Start":"06:11.825 ","End":"06:14.445","Text":"then it\u0027s minus 2,"},{"Start":"06:14.445 ","End":"06:20.090","Text":"minus 6 over 2 and then we just evaluate minus 2"},{"Start":"06:20.090 ","End":"06:26.810","Text":"plus 6 plus 4 plus 4 over 2 is 2."},{"Start":"06:26.810 ","End":"06:28.795","Text":"Then the other case,"},{"Start":"06:28.795 ","End":"06:30.350","Text":"we took the minus sign,"},{"Start":"06:30.350 ","End":"06:32.150","Text":"we have minus 2 minus 6 over 2."},{"Start":"06:32.150 ","End":"06:34.490","Text":"This is minus 8 over 2,"},{"Start":"06:34.490 ","End":"06:40.825","Text":"minus over plus is minus 8 over 2 is 4 so we get minus 4."},{"Start":"06:40.825 ","End":"06:43.530","Text":"These are the 2 solutions,"},{"Start":"06:43.530 ","End":"06:47.590","Text":"so x could be 2 or it could be minus 4."},{"Start":"06:47.590 ","End":"06:54.240","Text":"I often write the answer as 2 equals 2 or minus 4."},{"Start":"06:54.240 ","End":"06:56.900","Text":"When I say, I mean both could be."},{"Start":"06:56.900 ","End":"06:59.870","Text":"Let\u0027s check that these are really solutions."},{"Start":"06:59.870 ","End":"07:01.640","Text":"Let\u0027s try plugging in 2."},{"Start":"07:01.640 ","End":"07:06.235","Text":"I\u0027m going to try and do it mentally: 2^2 is 4"},{"Start":"07:06.235 ","End":"07:12.394","Text":"twice 2 is another 4 so we\u0027re up to 8 minus 8 is 0."},{"Start":"07:12.394 ","End":"07:15.425","Text":"Great 2 works as being checked."},{"Start":"07:15.425 ","End":"07:17.735","Text":"Let\u0027s try the minus 4,"},{"Start":"07:17.735 ","End":"07:23.820","Text":"minus 4^2 is plus 16,"},{"Start":"07:23.820 ","End":"07:30.975","Text":"2 times minus 4 is minus 8 so we got 16 minus 8 minus 8 and I believe that is 0."},{"Start":"07:30.975 ","End":"07:34.615","Text":"This is checked too so 2 solutions."},{"Start":"07:34.615 ","End":"07:39.080","Text":"Later on we\u0027ll see that it\u0027s not always 2 solutions wondering how can that be?"},{"Start":"07:39.080 ","End":"07:40.295","Text":"We have a plus and a minus."},{"Start":"07:40.295 ","End":"07:43.085","Text":"Well, we will see in due course,"},{"Start":"07:43.085 ","End":"07:48.990","Text":"looking at the time, I think we\u0027ll just take a break now and continue in the next clip."}],"ID":4984},{"Watched":false,"Name":"Quadratic Equations, Formula (continued)","Duration":"7m 13s","ChapterTopicVideoID":4992,"CourseChapterTopicPlaylistID":3868,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"Here we are continuing from the previous part of the clip."},{"Start":"00:04.440 ","End":"00:06.375","Text":"This is our next example."},{"Start":"00:06.375 ","End":"00:09.450","Text":"The first thing we have to do is to identify a,"},{"Start":"00:09.450 ","End":"00:11.729","Text":"b, and c from the formula."},{"Start":"00:11.729 ","End":"00:15.570","Text":"When you see minus x^2 is like minus 1x^2,"},{"Start":"00:15.570 ","End":"00:18.420","Text":"so a is minus 1,"},{"Start":"00:18.420 ","End":"00:22.155","Text":"b is what goes in front of the x is 10,"},{"Start":"00:22.155 ","End":"00:24.390","Text":"and c is without x at all,"},{"Start":"00:24.390 ","End":"00:29.310","Text":"is the free co-efficient is minus 21."},{"Start":"00:29.310 ","End":"00:32.895","Text":"Next thing is to plug into the formula minus b."},{"Start":"00:32.895 ","End":"00:35.040","Text":"That\u0027s minus 10."},{"Start":"00:35.040 ","End":"00:43.230","Text":"Then plus or minus the square root of b^2 is 10^2 minus"},{"Start":"00:43.230 ","End":"00:51.650","Text":"4ac minus 4 times a is minus 1 and c is minus 21 and then there\u0027s a fraction,"},{"Start":"00:51.650 ","End":"00:57.650","Text":"it\u0027s all over 2a over 2 times a is minus 1."},{"Start":"00:57.650 ","End":"01:01.175","Text":"I\u0027m going to do the bit under the square root sign at the side here."},{"Start":"01:01.175 ","End":"01:04.925","Text":"Let\u0027s see. 10^2 is 100."},{"Start":"01:04.925 ","End":"01:07.175","Text":"Then I\u0027ll take this minus,"},{"Start":"01:07.175 ","End":"01:13.190","Text":"and then I\u0027ll do just this bit is 4 times minus 1 times minus 21 is 2 minuses."},{"Start":"01:13.190 ","End":"01:16.500","Text":"So it\u0027s 84."},{"Start":"01:16.820 ","End":"01:20.340","Text":"That is equal to 16."},{"Start":"01:20.340 ","End":"01:31.200","Text":"Back here, I have minus 10 plus or minus the square root of 16 over this is minus 2."},{"Start":"01:31.200 ","End":"01:34.920","Text":"The square root of 16 is 4."},{"Start":"01:34.920 ","End":"01:39.110","Text":"The two possibilities I get are choosing plus and minus."},{"Start":"01:39.110 ","End":"01:43.160","Text":"First the plus, minus 10."},{"Start":"01:43.160 ","End":"01:48.450","Text":"We said that this is 4 plus 4 over minus 2."},{"Start":"01:48.450 ","End":"01:51.350","Text":"Then the other one we\u0027ll get minus 10,"},{"Start":"01:51.350 ","End":"01:54.950","Text":"minus 4 over minus 2."},{"Start":"01:54.950 ","End":"02:00.360","Text":"In this case, it\u0027s minus 6 over minus 2 is plus 3,"},{"Start":"02:00.360 ","End":"02:07.100","Text":"minus over minus is plus and here we have minus 14 over minus 2 is 7."},{"Start":"02:07.100 ","End":"02:16.195","Text":"These are the two solutions and I\u0027ll highlight them there and there."},{"Start":"02:16.195 ","End":"02:17.690","Text":"Unlike in the previous example,"},{"Start":"02:17.690 ","End":"02:20.870","Text":"if you substitute 3 in this, you will get 0."},{"Start":"02:20.870 ","End":"02:23.720","Text":"If you substitute 7 in the left-hand side you\u0027ll also get 0."},{"Start":"02:23.720 ","End":"02:24.980","Text":"Both of these are solutions."},{"Start":"02:24.980 ","End":"02:26.735","Text":"But I\u0027m not going to check this time."},{"Start":"02:26.735 ","End":"02:28.370","Text":"You can do it yourself."},{"Start":"02:28.370 ","End":"02:31.050","Text":"Onto the next example."},{"Start":"02:31.520 ","End":"02:34.380","Text":"Here\u0027s the next example."},{"Start":"02:34.380 ","End":"02:36.105","Text":"Let\u0027s do it a bit quicker."},{"Start":"02:36.105 ","End":"02:37.610","Text":"We have to identify a,"},{"Start":"02:37.610 ","End":"02:39.625","Text":"b, and c in the formula."},{"Start":"02:39.625 ","End":"02:42.585","Text":"A is 4, that\u0027s this,"},{"Start":"02:42.585 ","End":"02:51.395","Text":"b is the coefficient of x minus 20 and c is the free co-efficient is 25."},{"Start":"02:51.395 ","End":"02:53.705","Text":"Plug it in the formula."},{"Start":"02:53.705 ","End":"02:55.985","Text":"Note that minus b,"},{"Start":"02:55.985 ","End":"03:02.220","Text":"and be careful here is plus 20 because b already is minus of minus of minus is plus,"},{"Start":"03:02.220 ","End":"03:03.635","Text":"so we have 20."},{"Start":"03:03.635 ","End":"03:12.350","Text":"Then plus or minus the square root of b^2,"},{"Start":"03:12.350 ","End":"03:16.619","Text":"is minus 20^2 minus 4ac minus 4 times a"},{"Start":"03:16.619 ","End":"03:22.680","Text":"times c. All this over,"},{"Start":"03:22.680 ","End":"03:26.250","Text":"2a, 2 times 4."},{"Start":"03:26.250 ","End":"03:30.175","Text":"I\u0027d like to compute what\u0027s under the square root sign at the side."},{"Start":"03:30.175 ","End":"03:35.095","Text":"Let\u0027s see, minus 20^2 is the same as 20^2,"},{"Start":"03:35.095 ","End":"03:39.530","Text":"which is 20 times 20 is 400."},{"Start":"03:39.530 ","End":"03:44.160","Text":"Then minus 4 times 4 times 25."},{"Start":"03:44.160 ","End":"03:50.805","Text":"Let\u0027s do 4 times 25 is 100 and then times 4 is 400."},{"Start":"03:50.805 ","End":"03:54.070","Text":"What I get is 0."},{"Start":"03:55.220 ","End":"04:06.270","Text":"We have 20 plus or minus the square root of 0 over 2 times 4 is 8."},{"Start":"04:06.270 ","End":"04:08.540","Text":"Now, what is the square root of 0?"},{"Start":"04:08.540 ","End":"04:11.850","Text":"The square root of 0 is 0."},{"Start":"04:12.770 ","End":"04:19.015","Text":"It\u0027s 20 plus or minus 0 over 8."},{"Start":"04:19.015 ","End":"04:22.190","Text":"Now this 0 is not going to do anything."},{"Start":"04:22.190 ","End":"04:23.900","Text":"If I take 20 plus 0,"},{"Start":"04:23.900 ","End":"04:27.670","Text":"it\u0027s 20, or 20 minus 0 it\u0027s still 20."},{"Start":"04:27.670 ","End":"04:29.870","Text":"This is one of those cases where we don\u0027t have"},{"Start":"04:29.870 ","End":"04:32.705","Text":"a branch because both branches are the same."},{"Start":"04:32.705 ","End":"04:34.585","Text":"It\u0027s 20 over 8."},{"Start":"04:34.585 ","End":"04:38.204","Text":"Adding or subtracting 0 is not going to make any difference."},{"Start":"04:38.204 ","End":"04:43.600","Text":"If I divide 8 into 20 goes twice,"},{"Start":"04:43.600 ","End":"04:47.410","Text":"and then 4/8, 4/8, so it\u0027s just 1/2."},{"Start":"04:47.410 ","End":"04:51.560","Text":"We only have one solution this time."},{"Start":"04:51.560 ","End":"04:56.030","Text":"Remember in the beginning I said that not always have 2 solutions."},{"Start":"04:56.030 ","End":"04:59.075","Text":"Well, here\u0027s one of those cases where although it\u0027s plus or minus,"},{"Start":"04:59.075 ","End":"05:01.715","Text":"we still only have 1 solution."},{"Start":"05:01.715 ","End":"05:03.830","Text":"Now in the next example,"},{"Start":"05:03.830 ","End":"05:06.215","Text":"we\u0027ll see something else that might happen besides"},{"Start":"05:06.215 ","End":"05:09.695","Text":"2 solutions or 1 solution and you\u0027ll see."},{"Start":"05:09.695 ","End":"05:13.055","Text":"Here\u0027s our next example."},{"Start":"05:13.055 ","End":"05:16.760","Text":"5x^2 plus 6x plus 2 equals 0."},{"Start":"05:16.760 ","End":"05:19.530","Text":"As usual, we start off by identifying a,"},{"Start":"05:19.530 ","End":"05:21.795","Text":"b and c. A is 5,"},{"Start":"05:21.795 ","End":"05:25.470","Text":"b is 6, c is 2."},{"Start":"05:25.470 ","End":"05:27.750","Text":"Use the formula."},{"Start":"05:27.750 ","End":"05:34.965","Text":"Minus b is minus 6 plus or minus the square root of b^2,"},{"Start":"05:34.965 ","End":"05:44.910","Text":"6^2 minus 4 times a times c. All this is over 2a,"},{"Start":"05:44.910 ","End":"05:47.835","Text":"which is 2 times 5."},{"Start":"05:47.835 ","End":"05:55.940","Text":"As usual, I like to do what\u0027s under the square root sign at the side. Let\u0027s see."},{"Start":"05:55.940 ","End":"06:03.425","Text":"6^2 is 36 minus 4 times 5 times 2 is 40."},{"Start":"06:03.425 ","End":"06:08.610","Text":"This equals minus 4."},{"Start":"06:08.610 ","End":"06:14.270","Text":"I could try to continue and I could write minus 6 plus or"},{"Start":"06:14.270 ","End":"06:20.725","Text":"minus the square root of minus 4 over 10."},{"Start":"06:20.725 ","End":"06:28.585","Text":"But I\u0027m stuck really because there is no square root of minus 4."},{"Start":"06:28.585 ","End":"06:34.010","Text":"How can a number squared be minus 4 everything squared is 0 or positive,"},{"Start":"06:34.010 ","End":"06:37.490","Text":"I can\u0027t have something times itself being negative."},{"Start":"06:37.490 ","End":"06:43.860","Text":"Because of this, the minus here, that\u0027s a problem."},{"Start":"06:43.860 ","End":"06:46.085","Text":"Then we just say there\u0027s no solution."},{"Start":"06:46.085 ","End":"06:51.500","Text":"There is nothing to solve."},{"Start":"06:51.500 ","End":"06:52.700","Text":"That\u0027s the third case."},{"Start":"06:52.700 ","End":"06:55.040","Text":"We either have 2 solutions."},{"Start":"06:55.040 ","End":"06:58.970","Text":"That\u0027s when this thing under the square root sign is positive."},{"Start":"06:58.970 ","End":"07:03.395","Text":"If it\u0027s 0, we get only one solution and if it\u0027s negative,"},{"Start":"07:03.395 ","End":"07:05.840","Text":"we have no solution has happened here."},{"Start":"07:05.840 ","End":"07:07.850","Text":"No value of x you can put here,"},{"Start":"07:07.850 ","End":"07:09.155","Text":"we\u0027ll make this thing equals 0."},{"Start":"07:09.155 ","End":"07:12.570","Text":"It can happen. We\u0027re done."}],"ID":4985},{"Watched":false,"Name":"Quadratic Equations, Missing C","Duration":"8m 3s","ChapterTopicVideoID":4995,"CourseChapterTopicPlaylistID":3868,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.080","Text":"I\u0027m continuing with quadratic equations and I forgot to clean the board from last time,"},{"Start":"00:05.080 ","End":"00:10.730","Text":"so mostly clean and let me just clean this bit also."},{"Start":"00:10.730 ","End":"00:13.390","Text":"This is going to be the topic for today."},{"Start":"00:13.390 ","End":"00:19.580","Text":"What happens when we have a missing term in this case where we have the missing c?"},{"Start":"00:19.940 ","End":"00:23.260","Text":"I added that to the title missing c,"},{"Start":"00:23.260 ","End":"00:25.810","Text":"that\u0027s the c from here."},{"Start":"00:25.810 ","End":"00:27.625","Text":"You could say it\u0027s missing,"},{"Start":"00:27.625 ","End":"00:29.980","Text":"or you could say that c is 0."},{"Start":"00:29.980 ","End":"00:35.660","Text":"I could write this as plus 0,"},{"Start":"00:35.660 ","End":"00:37.945","Text":"but that\u0027s not necessary."},{"Start":"00:37.945 ","End":"00:42.465","Text":"Now, there\u0027s a way of solving this which doesn\u0027t use the formula."},{"Start":"00:42.465 ","End":"00:44.945","Text":"You can still use the formula and, in fact, you know what?"},{"Start":"00:44.945 ","End":"00:48.545","Text":"I\u0027m going to start by using the formula and then I\u0027ll show you the shorter way."},{"Start":"00:48.545 ","End":"00:50.554","Text":"If I use the formula,"},{"Start":"00:50.554 ","End":"00:56.410","Text":"then I identify a=1, b=2."},{"Start":"00:56.410 ","End":"01:00.185","Text":"I deliberately erase that,"},{"Start":"01:00.185 ","End":"01:03.490","Text":"because you can imagine that c=0."},{"Start":"01:03.490 ","End":"01:05.000","Text":"If you were given the question,"},{"Start":"01:05.000 ","End":"01:06.500","Text":"they wouldn\u0027t write plus 0."},{"Start":"01:06.500 ","End":"01:08.210","Text":"But we would know that if it\u0027s missing,"},{"Start":"01:08.210 ","End":"01:10.610","Text":"it\u0027s like having c=0."},{"Start":"01:10.610 ","End":"01:12.895","Text":"Then we use the formula."},{"Start":"01:12.895 ","End":"01:15.900","Text":"We get x=minus b,"},{"Start":"01:15.900 ","End":"01:25.915","Text":"it\u0027s minus 2 plus or minus the square root of b^2 is 2^2 minus 4 times a times c,"},{"Start":"01:25.915 ","End":"01:30.775","Text":"all over 2a, 2 times 1."},{"Start":"01:30.775 ","End":"01:34.205","Text":"Now notice that anything multiplied by 0 is 0."},{"Start":"01:34.205 ","End":"01:36.710","Text":"It\u0027s as if this whole thing that I\u0027ve just"},{"Start":"01:36.710 ","End":"01:41.090","Text":"highlighted is not there because this is all 0."},{"Start":"01:41.090 ","End":"01:48.180","Text":"What we get is that x is equal to minus 2 plus or minus."},{"Start":"01:49.690 ","End":"01:52.745","Text":"I\u0027ll just write it over here."},{"Start":"01:52.745 ","End":"01:56.325","Text":"The square root of 2^2 squared is just 2."},{"Start":"01:56.325 ","End":"01:59.085","Text":"In fact, the square root of anything squared is just that anything,"},{"Start":"01:59.085 ","End":"02:00.945","Text":"I\u0027m assuming it\u0027s positive."},{"Start":"02:00.945 ","End":"02:08.745","Text":"What we get is minus 2 plus or minus 2 over 2 times 1 is 2."},{"Start":"02:08.745 ","End":"02:10.310","Text":"As usual, we take the 2 roots."},{"Start":"02:10.310 ","End":"02:13.265","Text":"We take the plus and we take the minus."},{"Start":"02:13.265 ","End":"02:16.730","Text":"If we take the plus, minus 2 plus 2 is 0."},{"Start":"02:16.730 ","End":"02:19.460","Text":"0 over 2 is 0."},{"Start":"02:19.460 ","End":"02:21.410","Text":"If we take the minus root,"},{"Start":"02:21.410 ","End":"02:27.105","Text":"we\u0027ve got minus 2 minus 2 over 2 we get minus 4 over 2,"},{"Start":"02:27.105 ","End":"02:29.305","Text":"which is minus 2."},{"Start":"02:29.305 ","End":"02:33.770","Text":"These are my 2 solutions using the formula."},{"Start":"02:33.770 ","End":"02:37.585","Text":"Now let\u0027s try it the other way, the new way."},{"Start":"02:37.585 ","End":"02:40.550","Text":"Now, if the c term is missing,"},{"Start":"02:40.550 ","End":"02:43.175","Text":"it means that every term contains x,"},{"Start":"02:43.175 ","End":"02:46.040","Text":"which means that I can take x outside the brackets."},{"Start":"02:46.040 ","End":"02:48.365","Text":"Let me continue over here."},{"Start":"02:48.365 ","End":"02:51.996","Text":"If I factorize it and take x out the brackets,"},{"Start":"02:51.996 ","End":"02:58.055","Text":"I\u0027ve got x times x plus 2 equals 0."},{"Start":"02:58.055 ","End":"03:02.210","Text":"Now there\u0027s an important thing about when a product is equal to 0."},{"Start":"03:02.210 ","End":"03:03.620","Text":"It doesn\u0027t happen for other numbers,"},{"Start":"03:03.620 ","End":"03:05.000","Text":"but for 0 it works."},{"Start":"03:05.000 ","End":"03:07.025","Text":"If I have 2 numbers,"},{"Start":"03:07.025 ","End":"03:10.090","Text":"I don\u0027t want to use a and b,"},{"Start":"03:10.090 ","End":"03:15.470","Text":"say u and v. If I have 2 numbers that the product is 0,"},{"Start":"03:15.470 ","End":"03:18.695","Text":"the only way this can happen is that one or the other is 0."},{"Start":"03:18.695 ","End":"03:21.350","Text":"Because if I have a non-zero number times a non-zero number,"},{"Start":"03:21.350 ","End":"03:23.090","Text":"it will also give me a non-zero number."},{"Start":"03:23.090 ","End":"03:28.100","Text":"The only way I can multiply and get 0 is if one of them is 0,"},{"Start":"03:28.100 ","End":"03:31.800","Text":"that means that u is 0 or v is 0."},{"Start":"03:31.800 ","End":"03:34.220","Text":"One or the other or both I suppose,"},{"Start":"03:34.220 ","End":"03:37.025","Text":"but at least one of them has to be 0."},{"Start":"03:37.025 ","End":"03:39.650","Text":"In this case, I have a product which is 0."},{"Start":"03:39.650 ","End":"03:43.775","Text":"Using this logic, either x is 0."},{"Start":"03:43.775 ","End":"03:46.160","Text":"That could happen if x was 0."},{"Start":"03:46.160 ","End":"03:49.384","Text":"Could also happen if x plus 2 was 0."},{"Start":"03:49.384 ","End":"03:51.815","Text":"What would make x plus 2 0?"},{"Start":"03:51.815 ","End":"03:55.015","Text":"X would equal minus 2."},{"Start":"03:55.015 ","End":"03:57.365","Text":"If I highlight it,"},{"Start":"03:57.365 ","End":"04:02.720","Text":"you\u0027ll see we got exactly the same solution as we did with the formula,"},{"Start":"04:02.720 ","End":"04:05.015","Text":"but a little bit less work."},{"Start":"04:05.015 ","End":"04:07.805","Text":"I\u0027m just showing you that this is possible."},{"Start":"04:07.805 ","End":"04:12.590","Text":"Let\u0027s do another example or 2 of this kind."},{"Start":"04:12.590 ","End":"04:15.260","Text":"Let\u0027s see let\u0027s change the previous example."},{"Start":"04:15.260 ","End":"04:21.390","Text":"I\u0027ll make it minus 5x^2 plus 15x equals 0."},{"Start":"04:21.390 ","End":"04:26.749","Text":"I deliberately left a bit of a blank here just to remind us that something is missing."},{"Start":"04:26.749 ","End":"04:28.475","Text":"Let\u0027s do it again."},{"Start":"04:28.475 ","End":"04:30.215","Text":"Not with the formula,"},{"Start":"04:30.215 ","End":"04:33.530","Text":"but with the shortcut way I just showed you."},{"Start":"04:33.530 ","End":"04:37.950","Text":"We take x out the brackets."},{"Start":"04:39.000 ","End":"04:46.000","Text":"But I\u0027d like to point out if we can do better than just taking x out, we should."},{"Start":"04:46.000 ","End":"04:50.505","Text":"For example, I noticed that 5 also goes into both of these."},{"Start":"04:50.505 ","End":"04:54.790","Text":"So it\u0027s quicker, it\u0027s better if you take the 5 out also."},{"Start":"04:54.790 ","End":"04:58.435","Text":"I\u0027m going to take 5 and x outside the brackets."},{"Start":"04:58.435 ","End":"05:00.250","Text":"What are we left with?"},{"Start":"05:00.250 ","End":"05:05.475","Text":"5 into minus 5 goes minus 1 time,"},{"Start":"05:05.475 ","End":"05:10.300","Text":"x into x^2 goes x times minus 1x,"},{"Start":"05:10.300 ","End":"05:13.770","Text":"and then 5x into 15x,"},{"Start":"05:13.770 ","End":"05:15.430","Text":"it goes 3 times."},{"Start":"05:15.430 ","End":"05:20.155","Text":"You can easily see that is equal to 0."},{"Start":"05:20.155 ","End":"05:26.590","Text":"Let me just drop the one part of the minus 1 anyway."},{"Start":"05:28.040 ","End":"05:31.970","Text":"Before we do this trick of product is 0,"},{"Start":"05:31.970 ","End":"05:34.250","Text":"we can actually cancel by 5."},{"Start":"05:34.250 ","End":"05:37.025","Text":"It\u0027s something worth doing because when you have a 0 here,"},{"Start":"05:37.025 ","End":"05:41.250","Text":"you can pretty much divide by anything that\u0027s not 0."},{"Start":"05:41.260 ","End":"05:45.260","Text":"In practice, I just put a line through it and say it\u0027s gone."},{"Start":"05:45.260 ","End":"05:50.030","Text":"But the justification is that I\u0027ve divided both sides by 5,"},{"Start":"05:50.030 ","End":"05:53.150","Text":"and I know that 0 divided by anything is 0."},{"Start":"05:53.150 ","End":"05:55.750","Text":"It\u0027s just like throwing the 5 away."},{"Start":"05:55.750 ","End":"05:58.610","Text":"I can\u0027t throw the x away because it could be 0."},{"Start":"05:58.610 ","End":"06:00.020","Text":"You can\u0027t divide by 0,"},{"Start":"06:00.020 ","End":"06:01.130","Text":"but if it\u0027s a number,"},{"Start":"06:01.130 ","End":"06:03.595","Text":"not 0, you can just throw it out."},{"Start":"06:03.595 ","End":"06:08.555","Text":"Now we have this same dichotomy or choice of 2,"},{"Start":"06:08.555 ","End":"06:12.935","Text":"product is 0, so one or the other has to be 0."},{"Start":"06:12.935 ","End":"06:22.299","Text":"We say that either x is 0 or that minus x plus 3 is 0."},{"Start":"06:23.230 ","End":"06:27.110","Text":"You can easily see that the solution to this is x equals 3."},{"Start":"06:27.110 ","End":"06:30.180","Text":"The only number that I can put here would be 3,"},{"Start":"06:30.180 ","End":"06:33.570","Text":"or you could solve it the long way."},{"Start":"06:33.570 ","End":"06:37.610","Text":"X is 3. These are the 2 solutions."},{"Start":"06:37.610 ","End":"06:42.970","Text":"X is 0 or x is 3 and that\u0027s all there is to it."},{"Start":"06:42.970 ","End":"06:49.315","Text":"Of course, you can check by substituting and 0 is always a solution in"},{"Start":"06:49.315 ","End":"06:56.850","Text":"this equation with missing c. Let me just do one more quick one."},{"Start":"06:56.850 ","End":"07:01.755","Text":"3x^2 minus 2x equals 0."},{"Start":"07:01.755 ","End":"07:06.735","Text":"Left a bit of space there just to remind us again that the c is missing."},{"Start":"07:06.735 ","End":"07:08.485","Text":"Not going to do it with the formula,"},{"Start":"07:08.485 ","End":"07:10.075","Text":"going to do it with the short way."},{"Start":"07:10.075 ","End":"07:12.550","Text":"We\u0027ll take x outside the brackets."},{"Start":"07:12.550 ","End":"07:14.230","Text":"But I can\u0027t do any better than that."},{"Start":"07:14.230 ","End":"07:15.915","Text":"Here, I could take a 5 out of both."},{"Start":"07:15.915 ","End":"07:18.115","Text":"Here, I don\u0027t see I can do any better."},{"Start":"07:18.115 ","End":"07:21.235","Text":"So we take out x and we\u0027re left with 3,"},{"Start":"07:21.235 ","End":"07:25.710","Text":"and then x minus 2 from here equals 0."},{"Start":"07:25.710 ","End":"07:29.480","Text":"Once again, we\u0027re in the situation that the product is 0,"},{"Start":"07:29.480 ","End":"07:31.910","Text":"so one or the other factors has to be 0."},{"Start":"07:31.910 ","End":"07:39.430","Text":"So x=0, or 3x minus 2 equals 0."},{"Start":"07:39.430 ","End":"07:41.810","Text":"For the other case, bring the 2 over to the other side,"},{"Start":"07:41.810 ","End":"07:45.080","Text":"divide by 3 x is 2/3,"},{"Start":"07:45.080 ","End":"07:49.365","Text":"and so our 2 solutions are 0 and 2/3."},{"Start":"07:49.365 ","End":"07:52.140","Text":"Like I said, in this situation with missing c,"},{"Start":"07:52.140 ","End":"07:56.885","Text":"you always get 0s in one of the solutions that\u0027s a check that something is going right."},{"Start":"07:56.885 ","End":"08:03.180","Text":"Done with missing c. Next chapter, missing b."}],"ID":4986},{"Watched":false,"Name":"Quadratic Equations, Missing B","Duration":"8m 2s","ChapterTopicVideoID":4994,"CourseChapterTopicPlaylistID":3868,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.670","Text":"Continuing with quadratic equations,"},{"Start":"00:02.670 ","End":"00:06.285","Text":"in the previous clip we had what we called missing c,"},{"Start":"00:06.285 ","End":"00:08.610","Text":"means that this thing was missing."},{"Start":"00:08.610 ","End":"00:12.585","Text":"This time we\u0027re going to take the case of a missing b."},{"Start":"00:12.585 ","End":"00:17.370","Text":"Notice that in the example I deliberately left a gap here to show"},{"Start":"00:17.370 ","End":"00:22.575","Text":"us that the bx part is missing."},{"Start":"00:22.575 ","End":"00:26.340","Text":"You could say it\u0027s missing or you could say b equals 0."},{"Start":"00:26.340 ","End":"00:28.360","Text":"It doesn\u0027t really matter."},{"Start":"00:28.360 ","End":"00:30.920","Text":"As in the case of missing c,"},{"Start":"00:30.920 ","End":"00:32.630","Text":"we could still use the formula,"},{"Start":"00:32.630 ","End":"00:34.160","Text":"but there was a shortcut way,"},{"Start":"00:34.160 ","End":"00:35.525","Text":"and that\u0027s the same here."},{"Start":"00:35.525 ","End":"00:38.495","Text":"First of all, I\u0027ll show you that the formula won\u0027t fail you."},{"Start":"00:38.495 ","End":"00:44.210","Text":"It might not be the easiest way to solve the problem but it will work."},{"Start":"00:44.210 ","End":"00:46.775","Text":"Let\u0027s do that first."},{"Start":"00:46.775 ","End":"00:53.040","Text":"As usual the first step is to identify the coefficients a,"},{"Start":"00:53.040 ","End":"00:57.295","Text":"b, and c. a equals 1 because this is 1x^2."},{"Start":"00:57.295 ","End":"01:00.440","Text":"Here\u0027s the thing, b equals 0,"},{"Start":"01:00.440 ","End":"01:03.605","Text":"because now I can imagine it\u0027s plus 0x,"},{"Start":"01:03.605 ","End":"01:05.760","Text":"but I don\u0027t have to write it,"},{"Start":"01:05.760 ","End":"01:09.740","Text":"and c is minus 25."},{"Start":"01:09.740 ","End":"01:13.721","Text":"Then I put it in the formula and we get minus b,"},{"Start":"01:13.721 ","End":"01:17.900","Text":"so that\u0027s minus 0 plus or minus"},{"Start":"01:17.900 ","End":"01:24.770","Text":"the square root of b^2 minus 4ac,"},{"Start":"01:24.770 ","End":"01:33.815","Text":"4 times 1 times minus 25 over 2a,"},{"Start":"01:33.815 ","End":"01:38.570","Text":"over 2 times 1 is a."},{"Start":"01:38.570 ","End":"01:41.930","Text":"Let me compute this bit at the side."},{"Start":"01:41.930 ","End":"01:47.360","Text":"What we have under the square root sign is 0^2 is 0 minus,"},{"Start":"01:47.360 ","End":"01:48.410","Text":"it\u0027s going to be a negative,"},{"Start":"01:48.410 ","End":"01:52.645","Text":"it\u0027s going to be plus 4 times 25 is 100."},{"Start":"01:52.645 ","End":"01:56.255","Text":"That\u0027s 100. Back here,"},{"Start":"01:56.255 ","End":"02:01.070","Text":"I don\u0027t have to write minus 0 or plus 0 or anything times 0 because that\u0027s just a 0."},{"Start":"02:01.070 ","End":"02:07.875","Text":"What we have is plus or minus the square root of 100 over,"},{"Start":"02:07.875 ","End":"02:10.325","Text":"2 times 1 is 2."},{"Start":"02:10.325 ","End":"02:12.650","Text":"If I take the plus,"},{"Start":"02:12.650 ","End":"02:14.210","Text":"square root of 100 is 10,"},{"Start":"02:14.210 ","End":"02:19.370","Text":"of course, so I get 10 over 2, which is 5."},{"Start":"02:19.370 ","End":"02:21.259","Text":"If I take the minus,"},{"Start":"02:21.259 ","End":"02:24.035","Text":"I\u0027ve got minus 10 over 2,"},{"Start":"02:24.035 ","End":"02:26.440","Text":"which is minus 5,"},{"Start":"02:26.440 ","End":"02:28.770","Text":"and these are my 2 solutions,"},{"Start":"02:28.770 ","End":"02:30.955","Text":"5 and minus 5."},{"Start":"02:30.955 ","End":"02:35.870","Text":"But like I said in a special case we don\u0027t have to use the formula,"},{"Start":"02:35.870 ","End":"02:38.020","Text":"let\u0027s do it another way."},{"Start":"02:38.020 ","End":"02:45.045","Text":"I\u0027m going to start again with x^2 minus 25 equals 0."},{"Start":"02:45.045 ","End":"02:53.600","Text":"Then I\u0027m going to bring the 25 to the other side and get x^2 equals 25."},{"Start":"02:53.600 ","End":"02:58.505","Text":"Now, I want to know what number squared is 25."},{"Start":"02:58.505 ","End":"03:00.440","Text":"You might say, well,"},{"Start":"03:00.440 ","End":"03:05.495","Text":"x equals 5 will do it because 5^2 is 25."},{"Start":"03:05.495 ","End":"03:14.650","Text":"But the thing is that that\u0027s not the only possibility because minus 5^2 is also 25."},{"Start":"03:14.650 ","End":"03:17.405","Text":"When we take the square root,"},{"Start":"03:17.405 ","End":"03:23.180","Text":"basically what we did was 5 is the square root of 25."},{"Start":"03:23.180 ","End":"03:25.303","Text":"That\u0027s what we did to get from here to here,"},{"Start":"03:25.303 ","End":"03:28.490","Text":"but at the end we really have to say this is not good enough,"},{"Start":"03:28.490 ","End":"03:32.330","Text":"x is plus or minus 5,"},{"Start":"03:32.330 ","End":"03:38.135","Text":"or plus or minus a square root of 25 and that gives us the answer straightaway."},{"Start":"03:38.135 ","End":"03:41.750","Text":"Of course, I can write it as 5 and minus 5,"},{"Start":"03:41.750 ","End":"03:46.400","Text":"but it\u0027s fine just to leave it as plus or minus 5."},{"Start":"03:46.400 ","End":"03:48.710","Text":"There is actually a third way,"},{"Start":"03:48.710 ","End":"03:50.030","Text":"this is entirely optional."},{"Start":"03:50.030 ","End":"03:54.440","Text":"I suggest you skip this bit if you don\u0027t care about the third way."},{"Start":"03:54.440 ","End":"04:00.830","Text":"But the third way is to write the x^2 minus 25 as x^2 minus 5 squared."},{"Start":"04:00.830 ","End":"04:02.945","Text":"After all, 25 is 5 squared,"},{"Start":"04:02.945 ","End":"04:05.105","Text":"and to say that this equals 0."},{"Start":"04:05.105 ","End":"04:12.200","Text":"Then remember the difference of squares formula that a^2 minus b^2 in general,"},{"Start":"04:12.200 ","End":"04:14.060","Text":"not this a and b here,"},{"Start":"04:14.060 ","End":"04:18.210","Text":"is a plus b, a minus b."},{"Start":"04:18.210 ","End":"04:22.290","Text":"Applying it in our case we get x plus 5,"},{"Start":"04:22.290 ","End":"04:25.430","Text":"x minus 5 equals 0."},{"Start":"04:25.430 ","End":"04:27.385","Text":"When we have a product as 0,"},{"Start":"04:27.385 ","End":"04:29.545","Text":"one or the other has to be 0."},{"Start":"04:29.545 ","End":"04:35.375","Text":"Either x plus 5 is 0 or x minus 5 is 0."},{"Start":"04:35.375 ","End":"04:38.480","Text":"This one gives us x equals minus 5,"},{"Start":"04:38.480 ","End":"04:42.425","Text":"and this one gives us x equals 5 or plus 5."},{"Start":"04:42.425 ","End":"04:44.390","Text":"Again, we get the same answer,"},{"Start":"04:44.390 ","End":"04:45.515","Text":"plus or minus 5."},{"Start":"04:45.515 ","End":"04:49.480","Text":"This is optional. Now I\u0027m going to do another example."},{"Start":"04:49.480 ","End":"04:52.345","Text":"Here is our next example,"},{"Start":"04:52.345 ","End":"04:55.695","Text":"18x^2 minus 8 equals 0."},{"Start":"04:55.695 ","End":"04:59.855","Text":"Again, I left a bit of a blank here to show that there\u0027s a missing b."},{"Start":"04:59.855 ","End":"05:07.085","Text":"But before I do this I\u0027d like to slightly revise the solution to the previous exercise."},{"Start":"05:07.085 ","End":"05:09.080","Text":"I did it for the first time and there was"},{"Start":"05:09.080 ","End":"05:11.930","Text":"too much explanation in it I wanted to get it more technical,"},{"Start":"05:11.930 ","End":"05:18.965","Text":"so I\u0027m going to erase part of it and show you how we would do it more technically."},{"Start":"05:18.965 ","End":"05:23.735","Text":"Here, we would bring the 25 over to the other side as before."},{"Start":"05:23.735 ","End":"05:28.145","Text":"Then, whenever I have x^2 equals something,"},{"Start":"05:28.145 ","End":"05:35.015","Text":"I immediately write x equals plus or minus the square root of what\u0027s written here."},{"Start":"05:35.015 ","End":"05:41.770","Text":"Then I\u0027d say, the square root of 25 is 5 so x equals plus or minus 5."},{"Start":"05:41.770 ","End":"05:44.535","Text":"That\u0027s the bare bones without the explanations,"},{"Start":"05:44.535 ","End":"05:48.035","Text":"and that\u0027s what we need really for solving this one."},{"Start":"05:48.035 ","End":"05:50.060","Text":"But in this one there is a slight difference."},{"Start":"05:50.060 ","End":"05:55.320","Text":"Here, it didn\u0027t have anything in front of the x^2 or more correctly I had 1x^2."},{"Start":"05:55.320 ","End":"05:56.760","Text":"Here I have 18x^2."},{"Start":"05:56.760 ","End":"05:59.170","Text":"Let\u0027s see how that affects things."},{"Start":"05:59.170 ","End":"06:03.380","Text":"Now, as before I bring the number to the other side."},{"Start":"06:03.380 ","End":"06:09.905","Text":"I\u0027ve got 18x^2 equals 0 plus 8, is 8."},{"Start":"06:09.905 ","End":"06:12.575","Text":"But because I don\u0027t have 1x^2,"},{"Start":"06:12.575 ","End":"06:14.900","Text":"no problem I\u0027ll just divide by 18."},{"Start":"06:14.900 ","End":"06:19.235","Text":"I\u0027m dividing both sides by the 18,"},{"Start":"06:19.235 ","End":"06:23.750","Text":"and I\u0027ll get 18x^2 over 18 is x^2,"},{"Start":"06:23.750 ","End":"06:27.860","Text":"and here I get 8 over 18."},{"Start":"06:27.860 ","End":"06:31.055","Text":"Let\u0027s reduce this fraction."},{"Start":"06:31.055 ","End":"06:34.970","Text":"I see that both numerator and denominator are divisible by 2,"},{"Start":"06:34.970 ","End":"06:36.635","Text":"so if I take out 2,"},{"Start":"06:36.635 ","End":"06:39.790","Text":"I\u0027ll get 4 here and 9 here."},{"Start":"06:39.790 ","End":"06:44.680","Text":"What we have is x^2 equals 4/9."},{"Start":"06:44.720 ","End":"06:47.270","Text":"Then like we did here,"},{"Start":"06:47.270 ","End":"06:53.250","Text":"we say x equals plus or minus the square root of 4/9."},{"Start":"06:53.590 ","End":"06:59.240","Text":"Now if you remember, when we take the square root or an exponent of a fraction,"},{"Start":"06:59.240 ","End":"07:02.915","Text":"we do it separately for the numerator and denominator."},{"Start":"07:02.915 ","End":"07:09.705","Text":"It\u0027s plus or minus the square root of 4 over the square root of 9."},{"Start":"07:09.705 ","End":"07:12.720","Text":"Square root of 4 is 2,"},{"Start":"07:12.720 ","End":"07:14.355","Text":"2 times 2 is 4."},{"Start":"07:14.355 ","End":"07:17.460","Text":"The square root of 9 is 3 because 3 times 3 is 9."},{"Start":"07:17.460 ","End":"07:21.885","Text":"This is plus or minus 2 over 3."},{"Start":"07:21.885 ","End":"07:23.610","Text":"These are 2 answers."},{"Start":"07:23.610 ","End":"07:25.365","Text":"I could write it as 2/3,"},{"Start":"07:25.365 ","End":"07:26.710","Text":"and minus 2/3,"},{"Start":"07:26.710 ","End":"07:30.050","Text":"but it\u0027s acceptable to leave it like this."},{"Start":"07:30.050 ","End":"07:34.910","Text":"There we are ready for maybe one more example."},{"Start":"07:34.910 ","End":"07:37.400","Text":"No, I think we\u0027re done."},{"Start":"07:37.400 ","End":"07:40.190","Text":"This is enough. I just want to point out one thing."},{"Start":"07:40.190 ","End":"07:41.585","Text":"You might wonder,"},{"Start":"07:41.585 ","End":"07:47.395","Text":"we had a section on missing c and then we had missing b,"},{"Start":"07:47.395 ","End":"07:49.265","Text":"what about missing a?"},{"Start":"07:49.265 ","End":"07:51.920","Text":"Well, remember, we can\u0027t have a missing"},{"Start":"07:51.920 ","End":"07:55.735","Text":"a because if a is missing it\u0027s not a quadratic equation."},{"Start":"07:55.735 ","End":"07:57.890","Text":"These are the 2 special cases,"},{"Start":"07:57.890 ","End":"07:59.480","Text":"missing c and missing b,"},{"Start":"07:59.480 ","End":"08:03.360","Text":"and we\u0027re done with this clip."}],"ID":4987},{"Watched":false,"Name":"Exercise 1","Duration":"2m 38s","ChapterTopicVideoID":4984,"CourseChapterTopicPlaylistID":3868,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.509","Text":"Here we have to solve this equation,"},{"Start":"00:03.509 ","End":"00:07.155","Text":"and this is a quadratic equation because of the x^2."},{"Start":"00:07.155 ","End":"00:11.145","Text":"Now, of course, we can always use the quadratic formula,"},{"Start":"00:11.145 ","End":"00:13.980","Text":"but in this case, it\u0027s not necessary."},{"Start":"00:13.980 ","End":"00:17.850","Text":"I\u0027ll show you a simple way of solving this."},{"Start":"00:17.850 ","End":"00:20.550","Text":"In fact, I might even show you two ways."},{"Start":"00:20.550 ","End":"00:28.590","Text":"One way is just to bring the 36 over to the other side. x^2 = 36."},{"Start":"00:28.590 ","End":"00:33.060","Text":"Then you might say, well, if something squared is 36,"},{"Start":"00:33.060 ","End":"00:40.170","Text":"this is going to be the square root of 36 and that\u0027s almost right."},{"Start":"00:40.170 ","End":"00:43.270","Text":"Then you would say that is equal to 6."},{"Start":"00:43.270 ","End":"00:45.590","Text":"The reason I say, almost right,"},{"Start":"00:45.590 ","End":"00:50.570","Text":"is because it could be plus or it could be minus, which is sometimes right."},{"Start":"00:50.570 ","End":"00:54.875","Text":"We usually write plus or minus in mathematics and as this symbol."},{"Start":"00:54.875 ","End":"00:57.445","Text":"So x is plus or minus 6."},{"Start":"00:57.445 ","End":"01:01.760","Text":"There is two possibilities because 6^2 is 36,"},{"Start":"01:01.760 ","End":"01:05.180","Text":"whereas minus -6 if you square that minus"},{"Start":"01:05.180 ","End":"01:08.615","Text":"times minus is plus so there\u0027s really two solutions,"},{"Start":"01:08.615 ","End":"01:16.480","Text":"so x could say is equal to either 6 or -6."},{"Start":"01:16.480 ","End":"01:19.790","Text":"Now, there\u0027s another way of doing this."},{"Start":"01:19.790 ","End":"01:22.175","Text":"One way, the other way of doing it,"},{"Start":"01:22.175 ","End":"01:23.690","Text":"I\u0027m just showing you,"},{"Start":"01:23.690 ","End":"01:29.600","Text":"whether I\u0027ll probably use this method is to say this is x^2 -"},{"Start":"01:29.600 ","End":"01:36.290","Text":"6^2= 0 because we straight away see that 36 is 6 squared."},{"Start":"01:36.290 ","End":"01:43.235","Text":"Then we use the formula of difference of squares and we get the sum times the difference,"},{"Start":"01:43.235 ","End":"01:49.565","Text":"(x + 6)( x - 6) = 0."},{"Start":"01:49.565 ","End":"01:51.470","Text":"Then you figure like this."},{"Start":"01:51.470 ","End":"01:55.129","Text":"How can you multiply two numbers together to get 0?"},{"Start":"01:55.129 ","End":"01:57.920","Text":"Has to be that one of them is 0,"},{"Start":"01:57.920 ","End":"02:06.795","Text":"so either this is 0 or this thing is equal to 0."},{"Start":"02:06.795 ","End":"02:15.550","Text":"In one case you get x + 6 is 0 so x is minus 6."},{"Start":"02:15.550 ","End":"02:17.195","Text":"In the other case,"},{"Start":"02:17.195 ","End":"02:20.120","Text":"you get that x minus 6 is 0,"},{"Start":"02:20.120 ","End":"02:22.760","Text":"in which case x = 6."},{"Start":"02:22.760 ","End":"02:26.100","Text":"Again, we get the two solutions."},{"Start":"02:26.330 ","End":"02:30.470","Text":"This is neat, but this is shorter and simpler."},{"Start":"02:30.470 ","End":"02:37.950","Text":"We\u0027ll go with this method in future exercises. That\u0027s it."}],"ID":4988},{"Watched":false,"Name":"Exercise 2","Duration":"1m 42s","ChapterTopicVideoID":4985,"CourseChapterTopicPlaylistID":3868,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.880","Text":"Here we have to solve this equation."},{"Start":"00:02.880 ","End":"00:06.570","Text":"It\u0027s a quadratic equation because of the x^2."},{"Start":"00:06.570 ","End":"00:11.955","Text":"Let\u0027s see, 32x^2-18=0."},{"Start":"00:11.955 ","End":"00:14.445","Text":"Because of the missing x term,"},{"Start":"00:14.445 ","End":"00:23.130","Text":"the easiest thing to do is bring the number over to the other side and say, 32x^2=18."},{"Start":"00:23.130 ","End":"00:24.660","Text":"Let\u0027s see if we can cancel."},{"Start":"00:24.660 ","End":"00:33.990","Text":"Yeah, 2 will go into both of them, so 16x^2=9."},{"Start":"00:33.990 ","End":"00:42.090","Text":"Now, I can say that x^2=9/16,"},{"Start":"00:42.090 ","End":"00:45.074","Text":"I\u0027ll bring the 16 to the other side."},{"Start":"00:45.074 ","End":"00:51.045","Text":"This works out nicely because 9 is 3^2 and 16 is 4^2."},{"Start":"00:51.045 ","End":"00:59.880","Text":"Notice that this is equal to (3/4)^2."},{"Start":"00:59.880 ","End":"01:03.290","Text":"You might be tempted to say that x is equal to"},{"Start":"01:03.290 ","End":"01:09.425","Text":"3/4 because (3/4)^2 is 9/16 and you\u0027d be partly right."},{"Start":"01:09.425 ","End":"01:14.570","Text":"In fact, we have to take the plus or the minus."},{"Start":"01:14.570 ","End":"01:16.790","Text":"Just make a small correction."},{"Start":"01:16.790 ","End":"01:21.065","Text":"It\u0027s plus or minus 3/4 because both of them,"},{"Start":"01:21.065 ","End":"01:23.790","Text":"when squared, will give you 9/16."},{"Start":"01:23.790 ","End":"01:27.335","Text":"The plus or the minus as minus times minus is plus."},{"Start":"01:27.335 ","End":"01:30.355","Text":"You could leave it like this or you could say"},{"Start":"01:30.355 ","End":"01:36.480","Text":"that x=3/4 or minus"},{"Start":"01:36.480 ","End":"01:42.570","Text":"3/4 in case you don\u0027t like this symbol for the plus or minus. That\u0027s it."}],"ID":4989},{"Watched":false,"Name":"Exercise 3","Duration":"1m 58s","ChapterTopicVideoID":4986,"CourseChapterTopicPlaylistID":3868,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"In this exercise, we have to solve this equation."},{"Start":"00:03.990 ","End":"00:08.220","Text":"It\u0027s a quadratic equation because of the x^2."},{"Start":"00:08.220 ","End":"00:12.660","Text":"But notice that there\u0027s no constant term, in general,"},{"Start":"00:12.660 ","End":"00:15.560","Text":"a quadratic equation is ax^2+bx+c,"},{"Start":"00:15.560 ","End":"00:17.115","Text":"but here, the c is missing,"},{"Start":"00:17.115 ","End":"00:19.785","Text":"so we don\u0027t use the formula."},{"Start":"00:19.785 ","End":"00:22.930","Text":"We use other techniques."},{"Start":"00:27.110 ","End":"00:29.415","Text":"Let\u0027s put it another way."},{"Start":"00:29.415 ","End":"00:34.859","Text":"I can cancel. I can reduce this by minus 7 in this case,"},{"Start":"00:34.859 ","End":"00:36.480","Text":"because it goes into both."},{"Start":"00:36.480 ","End":"00:40.425","Text":"So if I divide the whole thing by minus 7,"},{"Start":"00:40.425 ","End":"00:45.090","Text":"I get x^2,"},{"Start":"00:45.090 ","End":"00:48.450","Text":"and minus 14 over minus 7 is 2,"},{"Start":"00:48.450 ","End":"00:52.720","Text":"and 0 over minus 7 is still 0."},{"Start":"00:52.720 ","End":"00:55.775","Text":"Now you\u0027re allowed to divide both sides of an equation by 7,"},{"Start":"00:55.775 ","End":"00:57.935","Text":"so this is what we get."},{"Start":"00:57.935 ","End":"01:00.785","Text":"In this case, we can factorize."},{"Start":"01:00.785 ","End":"01:08.195","Text":"We can take x outside the brackets and what we\u0027re left with is x plus 2."},{"Start":"01:08.195 ","End":"01:11.690","Text":"If you\u0027re not sure, you can multiply it back out, expand,"},{"Start":"01:11.690 ","End":"01:13.295","Text":"x times x is x^2,"},{"Start":"01:13.295 ","End":"01:17.360","Text":"x times 2 is 2x. This is equal to 0."},{"Start":"01:17.360 ","End":"01:21.515","Text":"Now, if you multiply 2 numbers together and get 0,"},{"Start":"01:21.515 ","End":"01:24.155","Text":"then one or the other must be 0."},{"Start":"01:24.155 ","End":"01:30.830","Text":"Either this equals 0 or this thing"},{"Start":"01:30.830 ","End":"01:37.440","Text":"has to be equal to 0. x is 0 just means x is 0,"},{"Start":"01:37.440 ","End":"01:42.750","Text":"but x plus 2 is 0 means that x is equal to minus 2."},{"Start":"01:42.980 ","End":"01:46.535","Text":"There are 2 possible solutions,"},{"Start":"01:46.535 ","End":"01:50.555","Text":"either x is 0 or x is minus 2."},{"Start":"01:50.555 ","End":"01:54.395","Text":"This is typical quadratic equations typically have 2 solutions."},{"Start":"01:54.395 ","End":"01:58.530","Text":"Not always, but typically. We\u0027re done."}],"ID":4990},{"Watched":false,"Name":"Exercise 4","Duration":"1m 13s","ChapterTopicVideoID":4987,"CourseChapterTopicPlaylistID":3868,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.670","Text":"We have to solve this equation,"},{"Start":"00:02.670 ","End":"00:05.880","Text":"which is a quadratic equation,"},{"Start":"00:05.880 ","End":"00:08.340","Text":"but it\u0027s missing the constant term,"},{"Start":"00:08.340 ","End":"00:09.960","Text":"so we don\u0027t need the formula."},{"Start":"00:09.960 ","End":"00:12.105","Text":"Let\u0027s solve it over here."},{"Start":"00:12.105 ","End":"00:16.270","Text":"What we do in this case is factorize it."},{"Start":"00:16.340 ","End":"00:18.960","Text":"What\u0027s biggest we can take out,"},{"Start":"00:18.960 ","End":"00:26.820","Text":"we can take out x and we\u0027re left with x into x^2 goes x times and the 5 stays."},{"Start":"00:26.820 ","End":"00:31.260","Text":"You can check x times 5x is 5x^2 and from here we get"},{"Start":"00:31.260 ","End":"00:36.210","Text":"minus 1 because this is minus 1x equals 0."},{"Start":"00:36.210 ","End":"00:37.610","Text":"When a product is 0,"},{"Start":"00:37.610 ","End":"00:38.897","Text":"we said this before,"},{"Start":"00:38.897 ","End":"00:40.490","Text":"one of the other must be 0,"},{"Start":"00:40.490 ","End":"00:47.705","Text":"so we have to have x=0 or 5x-1=0."},{"Start":"00:47.705 ","End":"00:49.310","Text":"Now, x=0,"},{"Start":"00:49.310 ","End":"00:51.040","Text":"nothing further to say,"},{"Start":"00:51.040 ","End":"00:57.703","Text":"but 5x-1=0 means that 5x=1,"},{"Start":"00:57.703 ","End":"01:03.500","Text":"and then dividing by 5, x=1/5."},{"Start":"01:03.500 ","End":"01:06.880","Text":"We have 2 solutions,"},{"Start":"01:06.880 ","End":"01:13.740","Text":"either x=0 or x=1/5, and we\u0027re done."}],"ID":4991},{"Watched":false,"Name":"Exercise 5","Duration":"3m 27s","ChapterTopicVideoID":4988,"CourseChapterTopicPlaylistID":3868,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.885","Text":"Here we have to solve a quadratic equation."},{"Start":"00:03.885 ","End":"00:06.480","Text":"It\u0027s quadratic because it has the x^2,"},{"Start":"00:06.480 ","End":"00:11.610","Text":"and it generally fits the pattern of ax^2 plus bx plus c=0."},{"Start":"00:11.610 ","End":"00:16.350","Text":"Let\u0027s solve it using the formula that\u0027s given here that you should memorize."},{"Start":"00:16.350 ","End":"00:17.970","Text":"Let\u0027s see what we have."},{"Start":"00:17.970 ","End":"00:19.800","Text":"We have the coefficient of x^2,"},{"Start":"00:19.800 ","End":"00:21.855","Text":"it doesn\u0027t look like there is 1,"},{"Start":"00:21.855 ","End":"00:23.925","Text":"which means that it\u0027s x^2."},{"Start":"00:23.925 ","End":"00:26.670","Text":"I get that a =1,"},{"Start":"00:26.670 ","End":"00:29.700","Text":"b is the coefficient of x,"},{"Start":"00:29.700 ","End":"00:34.605","Text":"b=3, and c is the 3 co-efficient,"},{"Start":"00:34.605 ","End":"00:38.460","Text":"just the number that would make it minus 10."},{"Start":"00:38.460 ","End":"00:40.495","Text":"If you put all these in this formula,"},{"Start":"00:40.495 ","End":"00:42.810","Text":"I get that x =-b"},{"Start":"00:42.810 ","End":"00:47.640","Text":"minus 3 plus or minus"},{"Start":"00:47.640 ","End":"00:53.820","Text":"the square root of b^2 minus 4ac,"},{"Start":"00:53.820 ","End":"00:56.400","Text":"or what did I write as 9?"},{"Start":"00:56.400 ","End":"01:00.000","Text":"I meant to write a 2, 3^2 is 9,"},{"Start":"01:00.000 ","End":"01:03.630","Text":"so I was jumping the gun there so back here,"},{"Start":"01:03.630 ","End":"01:11.595","Text":"b^2 minus 4 times a times c 4 times 1 times minus 10."},{"Start":"01:11.595 ","End":"01:18.825","Text":"That\u0027s not all we still have a dividing line and on the bottom we need to put 2a,"},{"Start":"01:18.825 ","End":"01:22.020","Text":"which is 2 times 1."},{"Start":"01:22.020 ","End":"01:25.075","Text":"Now it\u0027s just straightforward substitution,"},{"Start":"01:25.075 ","End":"01:30.245","Text":"I start with the inside because under the square root is like a bracket,"},{"Start":"01:30.245 ","End":"01:34.070","Text":"so I have minus 3 plus or minus the square root,"},{"Start":"01:34.070 ","End":"01:36.110","Text":"and we\u0027ll see what it is in a moment."},{"Start":"01:36.110 ","End":"01:38.875","Text":"All over 2 times 1 is 2."},{"Start":"01:38.875 ","End":"01:41.420","Text":"Let\u0027s just figure out what\u0027s under the square root sign."},{"Start":"01:41.420 ","End":"01:42.890","Text":"We can do this mentally,"},{"Start":"01:42.890 ","End":"01:49.700","Text":"4 times 1 times minus 10 is negative 40 minus 40,"},{"Start":"01:49.700 ","End":"01:51.520","Text":"but there\u0027s a minus here."},{"Start":"01:51.520 ","End":"02:00.150","Text":"It\u0027s 3^2 is 9 plus 40 because it\u0027s minus a minus 4 times 1 times 10 is 40."},{"Start":"02:00.430 ","End":"02:10.010","Text":"What I get is minus 3 plus or minus the square root of 49 is 7."},{"Start":"02:10.010 ","End":"02:14.795","Text":"I\u0027m not going to write a whole extra line just to say that 9 plus 40 years 49."},{"Start":"02:14.795 ","End":"02:18.830","Text":"Think of this is 49,7^2 is 49,"},{"Start":"02:18.830 ","End":"02:21.175","Text":"So 7 is the square root."},{"Start":"02:21.175 ","End":"02:24.405","Text":"3 plus or minus 7/2,"},{"Start":"02:24.405 ","End":"02:30.305","Text":"and then I just take a plus 1 time and take minus the other time."},{"Start":"02:30.305 ","End":"02:31.700","Text":"If I take the plus,"},{"Start":"02:31.700 ","End":"02:37.190","Text":"I get minus 3 plus 7/2 to 7 minus 3 over 2,"},{"Start":"02:37.190 ","End":"02:39.830","Text":"4 over 2 is 2."},{"Start":"02:39.830 ","End":"02:41.525","Text":"Then if I take the minus,"},{"Start":"02:41.525 ","End":"02:46.510","Text":"minus 7, the 2 minus to strengthen each other, it\u0027s minus 10."},{"Start":"02:46.510 ","End":"02:48.635","Text":"But when I divide that into 2,"},{"Start":"02:48.635 ","End":"02:51.790","Text":"it\u0027s only minus 5."},{"Start":"02:51.790 ","End":"02:54.270","Text":"These are the 2 solutions,"},{"Start":"02:54.270 ","End":"03:02.620","Text":"and you could say that x = 2 or minus 5."},{"Start":"03:03.430 ","End":"03:06.830","Text":"When I say, or I mean both of them are a solution."},{"Start":"03:06.830 ","End":"03:11.900","Text":"If I put x = 2 here it\u0027ll work and x equals minus 51 here it\u0027ll work."},{"Start":"03:11.900 ","End":"03:14.030","Text":"Actually, it\u0027s not a bad idea at"},{"Start":"03:14.030 ","End":"03:17.660","Text":"the end of any equation to substitute and see if it does work."},{"Start":"03:17.660 ","End":"03:22.160","Text":"I\u0027ll leave that to you to put x= 2 and to check that the left-hand side does"},{"Start":"03:22.160 ","End":"03:27.690","Text":"indeed become 0. That\u0027s it?"}],"ID":4992},{"Watched":false,"Name":"Exercise 6","Duration":"2m 14s","ChapterTopicVideoID":4989,"CourseChapterTopicPlaylistID":3868,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:06.660","Text":"We have to solve this quadratic equation and it\u0027s a straight application of the formula."},{"Start":"00:06.660 ","End":"00:09.330","Text":"What you have to do is identify what a,"},{"Start":"00:09.330 ","End":"00:10.725","Text":"b, and c are."},{"Start":"00:10.725 ","End":"00:14.325","Text":"A is the coefficient of x^2 is minus 1."},{"Start":"00:14.325 ","End":"00:18.150","Text":"B is the coefficient of x, so that\u0027s 10."},{"Start":"00:18.150 ","End":"00:23.190","Text":"C is the free coefficient,"},{"Start":"00:23.190 ","End":"00:28.154","Text":"is minus 16 and then you just have to substitute in the formula."},{"Start":"00:28.154 ","End":"00:30.870","Text":"You should memorize this formula."},{"Start":"00:30.870 ","End":"00:40.260","Text":"But anyway, minus b plus or minus the square root of b^2 minus"},{"Start":"00:40.260 ","End":"00:49.334","Text":"4ac over"},{"Start":"00:49.334 ","End":"00:52.180","Text":"2a which is minus 2."},{"Start":"00:52.180 ","End":"00:53.720","Text":"Now we just evaluate,"},{"Start":"00:53.720 ","End":"00:55.580","Text":"I like to do what\u0027s under the square root first,"},{"Start":"00:55.580 ","End":"00:57.485","Text":"and that\u0027s what you\u0027re supposed to do anyway,"},{"Start":"00:57.485 ","End":"00:59.675","Text":"because square root is like a bracket."},{"Start":"00:59.675 ","End":"01:05.089","Text":"Let\u0027s see, we have 3 minuses; minus, minus,"},{"Start":"01:05.089 ","End":"01:07.760","Text":"minus, that makes it a minus,"},{"Start":"01:07.760 ","End":"01:12.705","Text":"and 4 times 16 is 64."},{"Start":"01:12.705 ","End":"01:15.315","Text":"I\u0027ll just do the bit under the square root."},{"Start":"01:15.315 ","End":"01:24.300","Text":"This is 10^2 is 100 minus 64, which is 36."},{"Start":"01:24.300 ","End":"01:30.070","Text":"We have minus 10 plus or minus the square root of 36 is 6,"},{"Start":"01:30.070 ","End":"01:35.255","Text":"because 6 times 6 is 36, over minus 2."},{"Start":"01:35.255 ","End":"01:37.854","Text":"If I take the plus,"},{"Start":"01:37.854 ","End":"01:42.640","Text":"I get -10+6/-2"},{"Start":"01:42.640 ","End":"01:49.980","Text":"which is -4/-2."},{"Start":"01:49.980 ","End":"01:51.210","Text":"The 2 minuses cancel,"},{"Start":"01:51.210 ","End":"01:53.520","Text":"4/2 is 2,"},{"Start":"01:53.520 ","End":"01:56.380","Text":"and if we do the other branch with the minus,"},{"Start":"01:56.380 ","End":"02:02.865","Text":"we get -16/-2,"},{"Start":"02:02.865 ","End":"02:07.395","Text":"the minuses also cancel, 16/2 is 8."},{"Start":"02:07.395 ","End":"02:09.770","Text":"These are the 2 solutions,"},{"Start":"02:09.770 ","End":"02:14.340","Text":"x=2 or x=8, and we\u0027re done."}],"ID":4993},{"Watched":false,"Name":"Exercise 7","Duration":"2m 18s","ChapterTopicVideoID":4990,"CourseChapterTopicPlaylistID":3868,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.790","Text":"Here we have a quadratic equation to solve quadratic because of the x^2."},{"Start":"00:05.790 ","End":"00:10.860","Text":"It fits the template ax^2 plus bx plus c = 0."},{"Start":"00:10.860 ","End":"00:14.400","Text":"Let\u0027s identify what is a,"},{"Start":"00:14.400 ","End":"00:16.275","Text":"that is 25,"},{"Start":"00:16.275 ","End":"00:19.320","Text":"b is minus 20,"},{"Start":"00:19.320 ","End":"00:21.525","Text":"c is 4,"},{"Start":"00:21.525 ","End":"00:26.145","Text":"substitute in this formula which you should memorize."},{"Start":"00:26.145 ","End":"00:29.775","Text":"We\u0027ve got that x = - b."},{"Start":"00:29.775 ","End":"00:31.320","Text":"If b is minus 20,"},{"Start":"00:31.320 ","End":"00:33.660","Text":"then minus b is plus 20."},{"Start":"00:33.660 ","End":"00:38.765","Text":"I need to put plus 20 just to emphasize that I didn\u0027t forget about the minus,"},{"Start":"00:38.765 ","End":"00:45.025","Text":"plus or minus the square root of b^2."},{"Start":"00:45.025 ","End":"00:48.840","Text":"Let\u0027s write it straightaway as 400."},{"Start":"00:48.840 ","End":"00:52.280","Text":"Of course, a number squared, the minus times minus is plus,"},{"Start":"00:52.280 ","End":"00:56.945","Text":"so it\u0027s 400, minus 4ac,"},{"Start":"00:56.945 ","End":"00:59.975","Text":"which is 4 times 20."},{"Start":"00:59.975 ","End":"01:04.160","Text":"I meant to say 25 times 4,"},{"Start":"01:04.160 ","End":"01:07.610","Text":"and all this over 2a."},{"Start":"01:07.610 ","End":"01:12.770","Text":"2a is twice 25,I\u0027ll write that already is 50."},{"Start":"01:12.770 ","End":"01:15.380","Text":"Let\u0027s see what\u0027s under the square root sign."},{"Start":"01:15.380 ","End":"01:17.540","Text":"That\u0027s usually where we start."},{"Start":"01:17.540 ","End":"01:21.089","Text":"We\u0027ve got 400 minus,"},{"Start":"01:21.089 ","End":"01:25.640","Text":"4 times 25 is 100 times 4 is 400."},{"Start":"01:25.640 ","End":"01:30.605","Text":"That gives me 0 and the square root of 0 is 0."},{"Start":"01:30.605 ","End":"01:41.055","Text":"What I get is that x=20 plus or minus 0 over 50."},{"Start":"01:41.055 ","End":"01:47.015","Text":"Clearly, plus 0 and minus 0 is the same and means not doing anything."},{"Start":"01:47.015 ","End":"01:49.130","Text":"So there\u0027s only one solution in fact,"},{"Start":"01:49.130 ","End":"01:52.590","Text":"which is 20 over 50, but it\u0027ll cancel it."},{"Start":"01:52.590 ","End":"01:54.780","Text":"We say x is 2/5."},{"Start":"01:54.780 ","End":"01:59.070","Text":"For those of you who just love decimals, that\u0027s 0.4."},{"Start":"01:59.070 ","End":"02:03.120","Text":"In any event, I\u0027m going to write the solution as 2/5,"},{"Start":"02:03.120 ","End":"02:05.365","Text":"so I like this the best."},{"Start":"02:05.365 ","End":"02:10.070","Text":"Sometimes it happens that a quadratic equation only has one solution."},{"Start":"02:10.070 ","End":"02:13.160","Text":"Clearly, that happens when what\u0027s under the square root is 0,"},{"Start":"02:13.160 ","End":"02:18.030","Text":"because then the plus or minus makes no difference. We\u0027re done."}],"ID":4994},{"Watched":false,"Name":"Exercise 8","Duration":"1m 27s","ChapterTopicVideoID":4991,"CourseChapterTopicPlaylistID":3868,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:06.375","Text":"We have another quadratic equation to solve and let\u0027s do it over here."},{"Start":"00:06.375 ","End":"00:11.925","Text":"Identify the a, b and c. a is 2,"},{"Start":"00:11.925 ","End":"00:14.295","Text":"b is minus 6,"},{"Start":"00:14.295 ","End":"00:15.720","Text":"c is 5,"},{"Start":"00:15.720 ","End":"00:19.635","Text":"and use the formula which you should have memorized."},{"Start":"00:19.635 ","End":"00:24.570","Text":"x=-b, so that\u0027s plus 6,"},{"Start":"00:24.570 ","End":"00:29.050","Text":"plus or minus the square root of b^2,"},{"Start":"00:29.050 ","End":"00:32.550","Text":"minus 6 times minus 6 is 36,"},{"Start":"00:32.550 ","End":"00:36.180","Text":"minus 4 times 2 times 5,"},{"Start":"00:36.180 ","End":"00:37.920","Text":"that\u0027s for 4ac,"},{"Start":"00:37.920 ","End":"00:41.020","Text":"over 2a is 4."},{"Start":"00:41.020 ","End":"00:44.180","Text":"The important thing is what\u0027s under the square root."},{"Start":"00:44.180 ","End":"00:45.845","Text":"Let\u0027s see what that is."},{"Start":"00:45.845 ","End":"00:51.240","Text":"We have 36 minus 4 times 2 times 5."},{"Start":"00:51.240 ","End":"00:57.660","Text":"8 times 5 is 40 and this is equal to negative 4."},{"Start":"00:57.660 ","End":"01:01.730","Text":"What we\u0027re left with is this part here,"},{"Start":"01:01.730 ","End":"01:05.490","Text":"which is the square root of minus 4."},{"Start":"01:05.990 ","End":"01:11.330","Text":"As you know, negative numbers do not have a square root."},{"Start":"01:11.330 ","End":"01:16.190","Text":"When this happens and we get into this impossible situation,"},{"Start":"01:16.190 ","End":"01:17.960","Text":"we just say that the equation,"},{"Start":"01:17.960 ","End":"01:27.090","Text":"this has no solution, and that\u0027s it."}],"ID":4995},{"Watched":false,"Name":"Exercise 9 - word problem","Duration":"6m 2s","ChapterTopicVideoID":8095,"CourseChapterTopicPlaylistID":3868,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.285","Text":"Here we have a word problem which is"},{"Start":"00:03.285 ","End":"00:08.055","Text":"geometrical and there\u0027s a rectangle and the diagonal."},{"Start":"00:08.055 ","End":"00:11.010","Text":"I\u0027m going to assume you\u0027ve studied Pythagoras\u0027 theorem."},{"Start":"00:11.010 ","End":"00:14.040","Text":"If not, there\u0027s no point in continuing."},{"Start":"00:14.040 ","End":"00:17.325","Text":"Most people will have studied Pythagoras\u0027 theorem."},{"Start":"00:17.325 ","End":"00:21.239","Text":"We\u0027re talking about a rectangle and we\u0027re given certain conditions"},{"Start":"00:21.239 ","End":"00:25.005","Text":"that it\u0027s length is 7 inches more than its width."},{"Start":"00:25.005 ","End":"00:29.600","Text":"We\u0027re told the diagonal is 17 inches and we have to find the dimensions."},{"Start":"00:29.600 ","End":"00:31.835","Text":"Dimensions mean the length and the width."},{"Start":"00:31.835 ","End":"00:35.345","Text":"A picture would certainly help."},{"Start":"00:35.345 ","End":"00:37.955","Text":"Here\u0027s the rectangle."},{"Start":"00:37.955 ","End":"00:40.580","Text":"I\u0027ll draw a diagonal too."},{"Start":"00:40.580 ","End":"00:44.420","Text":"Here\u0027s a diagonal and let\u0027s say we work in inches."},{"Start":"00:44.420 ","End":"00:47.165","Text":"We know that this is 17,"},{"Start":"00:47.165 ","End":"00:50.690","Text":"but it looks like we have 2 variables,"},{"Start":"00:50.690 ","End":"00:54.620","Text":"the width and the length."},{"Start":"00:54.620 ","End":"00:57.270","Text":"But we don\u0027t need"},{"Start":"00:57.270 ","End":"01:01.760","Text":"2 variables because we know that the length is 7 inches more than the width."},{"Start":"01:01.760 ","End":"01:07.310","Text":"Let\u0027s take the width as W and the length,"},{"Start":"01:07.310 ","End":"01:11.080","Text":"we won\u0027t give it a letter L we\u0027ll just say 7 more than"},{"Start":"01:11.080 ","End":"01:19.800","Text":"the width, so W+7."},{"Start":"01:19.800 ","End":"01:22.640","Text":"Now, look at say this triangle, by Pythagoras\u0027 theorem,"},{"Start":"01:22.640 ","End":"01:25.535","Text":"we can say that this squared plus this squared is this squared."},{"Start":"01:25.535 ","End":"01:30.083","Text":"In other words, (W^2) +"},{"Start":"01:30.083 ","End":"01:36.265","Text":"(W+7)^2 =17^2."},{"Start":"01:36.265 ","End":"01:38.880","Text":"Let\u0027s multiply out."},{"Start":"01:39.260 ","End":"01:46.040","Text":"Now there\u0027s a special expansion for this binomial."},{"Start":"01:46.040 ","End":"01:51.270","Text":"I\u0027m going to use the formula that (a+b)^2=a^2+2ab+b^2."},{"Start":"01:55.360 ","End":"02:01.505","Text":"Here we get W^2+2W times 7"},{"Start":"02:01.505 ","End":"02:02.900","Text":"is 14W"},{"Start":"02:02.900 ","End":"02:12.620","Text":"and 7^2=49,17^2=289."},{"Start":"02:12.620 ","End":"02:18.630","Text":"Bring everything to the left-hand side and collect these."},{"Start":"02:18.630 ","End":"02:23.850","Text":"We have 2W^2+14W,"},{"Start":"02:23.850 ","End":"02:29.430","Text":"then 49 minus 289 is"},{"Start":"02:29.430 ","End":"02:37.875","Text":"minus 240 equals 0."},{"Start":"02:37.875 ","End":"02:39.690","Text":"I like to reduce them."},{"Start":"02:39.690 ","End":"02:41.445","Text":"I see everything\u0027s divisible by 2."},{"Start":"02:41.445 ","End":"02:43.920","Text":"Let\u0027s divide by 2."},{"Start":"02:43.920 ","End":"02:52.855","Text":"We\u0027ve got W^2 + 7W -120 =0."},{"Start":"02:52.855 ","End":"02:55.430","Text":"This is a quadratic equation."},{"Start":"02:55.430 ","End":"02:58.865","Text":"I\u0027m assuming you\u0027ve solved quadratic equations,"},{"Start":"02:58.865 ","End":"03:00.950","Text":"I\u0027m going to use the formula method."},{"Start":"03:00.950 ","End":"03:09.263","Text":"There are several methods I\u0027ll use the formula where W equals,"},{"Start":"03:09.263 ","End":"03:11.549","Text":"you should have memorized this formula,"},{"Start":"03:11.549 ","End":"03:19.740","Text":"minus b plus or minus the square root of b^2."},{"Start":"03:19.820 ","End":"03:22.433","Text":"You might have just forgotten the formula,"},{"Start":"03:22.433 ","End":"03:28.280","Text":"so let me just say, a is the coefficient of the square term."},{"Start":"03:28.280 ","End":"03:29.635","Text":"a is 1,"},{"Start":"03:29.635 ","End":"03:31.860","Text":"b is 7,"},{"Start":"03:31.860 ","End":"03:39.860","Text":"c is minus 120 and the 2 x\u0027s are given by minus b plus or"},{"Start":"03:39.860 ","End":"03:49.605","Text":"minus the square root of b^2 minus 4ac over 2a."},{"Start":"03:49.605 ","End":"03:53.105","Text":"This in general, we have to figure it out in our case,"},{"Start":"03:53.105 ","End":"03:56.420","Text":"minus 7 plus or minus."},{"Start":"03:56.420 ","End":"04:06.150","Text":"Now, b^2=49 minus 4ac."},{"Start":"04:07.510 ","End":"04:17.250","Text":"4ac would be minus 4 times 1 times"},{"Start":"04:17.250 ","End":"04:28.215","Text":"120 so it\u0027s plus 480 all over,"},{"Start":"04:28.215 ","End":"04:33.579","Text":"2a, but a is 1 so this is 2."},{"Start":"04:33.579 ","End":"04:40.610","Text":"Notice that 49+480=529."},{"Start":"04:40.610 ","End":"04:44.525","Text":"If you look it up on your calculator or you can do it mentally,"},{"Start":"04:44.525 ","End":"04:48.540","Text":"this comes out to be 23."},{"Start":"04:48.560 ","End":"04:51.660","Text":"Move it out the way."},{"Start":"04:51.660 ","End":"04:55.760","Text":"We get 2 solutions 1 by taking the plus 1,"},{"Start":"04:55.760 ","End":"04:56.870","Text":"by taking the minus."},{"Start":"04:56.870 ","End":"04:58.160","Text":"If we have the plus,"},{"Start":"04:58.160 ","End":"05:00.080","Text":"it\u0027s minus 7,"},{"Start":"05:00.080 ","End":"05:03.870","Text":"plus this 23 over 2,"},{"Start":"05:03.870 ","End":"05:05.715","Text":"we take the minus,"},{"Start":"05:05.715 ","End":"05:10.995","Text":"it\u0027s minus 7 minus 23 over 2."},{"Start":"05:10.995 ","End":"05:16.380","Text":"Now here, 23-7=16 divided by 2 is 8."},{"Start":"05:16.380 ","End":"05:19.650","Text":"Here, I see it\u0027s going to come out negative."},{"Start":"05:19.650 ","End":"05:22.025","Text":"I know the width can\u0027t be negative,"},{"Start":"05:22.025 ","End":"05:24.080","Text":"so I don\u0027t care what it is."},{"Start":"05:24.080 ","End":"05:25.430","Text":"It comes out -15,"},{"Start":"05:25.430 ","End":"05:27.845","Text":"but I see it\u0027s negative so I don\u0027t bother."},{"Start":"05:27.845 ","End":"05:31.085","Text":"We know that the width is 8."},{"Start":"05:31.085 ","End":"05:33.875","Text":"We can determine all the dimensions."},{"Start":"05:33.875 ","End":"05:38.405","Text":"If the width is 8, then the length,"},{"Start":"05:38.405 ","End":"05:44.575","Text":"which is the width plus 7 is 15."},{"Start":"05:44.575 ","End":"05:47.540","Text":"We can write the answer, I\u0027ll even write it on the picture."},{"Start":"05:47.540 ","End":"05:57.840","Text":"The width is 8 inches and the length is 15 inches."},{"Start":"05:57.890 ","End":"06:02.530","Text":"That\u0027s the answer. We\u0027re done."}],"ID":8215}],"Thumbnail":null,"ID":3868},{"Name":"Disarranged Quadratic","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Disarranged Quadratic","Duration":"9m 38s","ChapterTopicVideoID":4996,"CourseChapterTopicPlaylistID":3869,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"Continuing with quadratic equations,"},{"Start":"00:03.270 ","End":"00:06.660","Text":"sometimes you get an equation that doesn\u0027t look like this."},{"Start":"00:06.660 ","End":"00:08.670","Text":"It doesn\u0027t look like a quadratic equation,"},{"Start":"00:08.670 ","End":"00:10.335","Text":"but after a bit of arranging,"},{"Start":"00:10.335 ","End":"00:12.150","Text":"it does become 1."},{"Start":"00:12.150 ","End":"00:14.640","Text":"I didn\u0027t know what name to give this kind of thing,"},{"Start":"00:14.640 ","End":"00:16.450","Text":"so I decided on the name,"},{"Start":"00:16.450 ","End":"00:20.610","Text":"disarranged quadratic equation is the name I made up."},{"Start":"00:20.610 ","End":"00:22.740","Text":"It just means that you need to do a bit of arranging"},{"Start":"00:22.740 ","End":"00:25.515","Text":"first and you\u0027ll see from the example what I mean."},{"Start":"00:25.515 ","End":"00:32.550","Text":"Let\u0027s take 5x^2 minus 6x"},{"Start":"00:32.550 ","End":"00:41.760","Text":"plus 6 = 2 plus 2x^2 plus 4."},{"Start":"00:41.760 ","End":"00:45.050","Text":"Here, it very much looks like a quadratic, as you can see,"},{"Start":"00:45.050 ","End":"00:49.010","Text":"the x^2, it\u0027s like a quadratic on the left and a quadratic on the right."},{"Start":"00:49.010 ","End":"00:51.920","Text":"But slowly we\u0027ll get them a bit more jumbled"},{"Start":"00:51.920 ","End":"00:54.935","Text":"up and it isn\u0027t always clear if something\u0027s quadratic or not."},{"Start":"00:54.935 ","End":"00:57.140","Text":"In this case, all we have to do is bring it into"},{"Start":"00:57.140 ","End":"01:00.020","Text":"the standard form so we can use the equation."},{"Start":"01:00.020 ","End":"01:04.280","Text":"I bring everything to the left and I think we can already do it in our heads."},{"Start":"01:04.280 ","End":"01:06.320","Text":"Let\u0027s take the x^2 first,"},{"Start":"01:06.320 ","End":"01:08.120","Text":"we got 5 from here,"},{"Start":"01:08.120 ","End":"01:10.385","Text":"and the 2 comes over as minus."},{"Start":"01:10.385 ","End":"01:13.640","Text":"We\u0027ve got 5 minus 2, that\u0027s going to be 3x^2."},{"Start":"01:13.640 ","End":"01:17.945","Text":"Then minus 6x,"},{"Start":"01:17.945 ","End":"01:23.810","Text":"there is no other x anywhere so it\u0027s just going to stay as minus 6x."},{"Start":"01:23.810 ","End":"01:28.890","Text":"If we take the plus 6 with minus 2, minus 4,"},{"Start":"01:28.890 ","End":"01:33.705","Text":"that will just give us zero because 2 and 4 is 6 on the other side it\u0027s minus 6,"},{"Start":"01:33.705 ","End":"01:38.890","Text":"so we get 1 of these quadratic equations with missing c, what we called."},{"Start":"01:38.890 ","End":"01:41.390","Text":"This is the main thing that we had to do for"},{"Start":"01:41.390 ","End":"01:43.670","Text":"this lesson is to bring something that\u0027s messed up,"},{"Start":"01:43.670 ","End":"01:45.110","Text":"to arrange it to be a quadratic."},{"Start":"01:45.110 ","End":"01:48.395","Text":"But I think we should also solve it to the end if we\u0027re here already."},{"Start":"01:48.395 ","End":"01:51.500","Text":"Let\u0027s solve it. Like I said,"},{"Start":"01:51.500 ","End":"01:55.190","Text":"if you can see a number that goes into everything, take that out."},{"Start":"01:55.190 ","End":"01:57.020","Text":"Let\u0027s take 3 out first."},{"Start":"01:57.020 ","End":"02:00.425","Text":"Then we have x^2 minus 2x=0,"},{"Start":"02:00.425 ","End":"02:02.225","Text":"and as always with a missing c,"},{"Start":"02:02.225 ","End":"02:03.725","Text":"we can take x out."},{"Start":"02:03.725 ","End":"02:07.760","Text":"That will become 3 and then an x and what I\u0027m left with now is"},{"Start":"02:07.760 ","End":"02:11.810","Text":"an x from here and a 2 from here =0."},{"Start":"02:11.810 ","End":"02:15.940","Text":"Remember I told you that if you just see a number with an equation that\u0027s =0,"},{"Start":"02:15.940 ","End":"02:18.470","Text":"you can just delete that number."},{"Start":"02:18.470 ","End":"02:20.740","Text":"What we\u0027re doing really is dividing by 3,"},{"Start":"02:20.740 ","End":"02:23.330","Text":"but 0 divided by 3 is still 0."},{"Start":"02:23.330 ","End":"02:27.785","Text":"Now we remember that when a product is 0,one or the other is 0,"},{"Start":"02:27.785 ","End":"02:35.640","Text":"so either x=0 or x-2=0 and this 1 gives us x=2,"},{"Start":"02:35.640 ","End":"02:36.950","Text":"and that\u0027s our answer,"},{"Start":"02:36.950 ","End":"02:40.160","Text":"is x=0 or x=2."},{"Start":"02:40.160 ","End":"02:42.050","Text":"Other thing is from time to time,"},{"Start":"02:42.050 ","End":"02:44.165","Text":"I suggest checking the answers,"},{"Start":"02:44.165 ","End":"02:45.410","Text":"not just in quadratics,"},{"Start":"02:45.410 ","End":"02:48.545","Text":"in anything to see that it satisfies the original equation,"},{"Start":"02:48.545 ","End":"02:50.900","Text":"It\u0027s good that you at least have a check if it\u0027s on"},{"Start":"02:50.900 ","End":"02:53.480","Text":"an exam or something as a way of checking yourself."},{"Start":"02:53.480 ","End":"02:56.350","Text":"Let\u0027s try plugging in x=0,"},{"Start":"02:56.350 ","End":"03:01.395","Text":"and see on the left we get zero and zero, we just get 6."},{"Start":"03:01.395 ","End":"03:04.605","Text":"On the right we get 2 and 0 and 4 is also 6."},{"Start":"03:04.605 ","End":"03:05.940","Text":"Let\u0027s try the 2,"},{"Start":"03:05.940 ","End":"03:12.285","Text":"5 times 4 =20,"},{"Start":"03:12.285 ","End":"03:14.415","Text":"20 minus 12= 8 plus 6 =14."},{"Start":"03:14.415 ","End":"03:17.415","Text":"On the left 14 and on the right,"},{"Start":"03:17.415 ","End":"03:19.920","Text":"2 plus 2 times 4, 4=2^2."},{"Start":"03:19.920 ","End":"03:26.025","Text":"2 plus 2 times 4 is 2 plus 8 =10 plus 4 =14."},{"Start":"03:26.025 ","End":"03:27.904","Text":"Both of these checkout,"},{"Start":"03:27.904 ","End":"03:30.110","Text":"you should really do it every time if you can,"},{"Start":"03:30.110 ","End":"03:32.315","Text":"but it\u0027s just something you should do."},{"Start":"03:32.315 ","End":"03:35.435","Text":"Check if the answer you get really satisfies."},{"Start":"03:35.435 ","End":"03:39.595","Text":"Let\u0027s go on for a slightly more mixed-up example."},{"Start":"03:39.595 ","End":"03:41.600","Text":"Here\u0027s our next example."},{"Start":"03:41.600 ","End":"03:45.230","Text":"We have to work a little bit more to get it into standard form."},{"Start":"03:45.230 ","End":"03:48.205","Text":"Let\u0027s just open brackets where we can."},{"Start":"03:48.205 ","End":"03:52.440","Text":"On the left of the equality minus x times x= minus x^2,"},{"Start":"03:52.440 ","End":"03:58.515","Text":"minus x times minus 9 = plus 9x."},{"Start":"03:58.515 ","End":"04:03.035","Text":"Then we have, remember we have to multiply each thing in here with each thing in here."},{"Start":"04:03.035 ","End":"04:05.660","Text":"I\u0027ll take the 1 first and do it with both of these 1 times"},{"Start":"04:05.660 ","End":"04:09.470","Text":"1 = 1,1 times minus x = minus x."},{"Start":"04:09.470 ","End":"04:12.230","Text":"Now I\u0027ll take the minus 4x with each."},{"Start":"04:12.230 ","End":"04:18.820","Text":"I get minus 4x plus 4x^2 and plus 7."},{"Start":"04:18.820 ","End":"04:23.644","Text":"Now I bring everything to the left-hand side."},{"Start":"04:23.644 ","End":"04:25.740","Text":"I\u0027ll take the x^2 first."},{"Start":"04:25.740 ","End":"04:28.620","Text":"Let\u0027s say I have 1 here and I have 1 here."},{"Start":"04:28.620 ","End":"04:30.480","Text":"On the left. It\u0027ll be minus 1,"},{"Start":"04:30.480 ","End":"04:33.810","Text":"minus 4, minus 5x^2."},{"Start":"04:33.810 ","End":"04:35.430","Text":"Let us see the Xs."},{"Start":"04:35.430 ","End":"04:40.010","Text":"I have 1 here and I have here,"},{"Start":"04:40.010 ","End":"04:41.450","Text":"and I have here."},{"Start":"04:41.450 ","End":"04:43.280","Text":"I\u0027ll take those together."},{"Start":"04:43.280 ","End":"04:47.855","Text":"I\u0027ve got plus 9 and then plus 1, plus 4 on the left."},{"Start":"04:47.855 ","End":"04:50.585","Text":"It\u0027ll be plus 14 x."},{"Start":"04:50.585 ","End":"04:53.150","Text":"Then what\u0027s leftover as numbers,"},{"Start":"04:53.150 ","End":"04:54.500","Text":"on the right-hand side,"},{"Start":"04:54.500 ","End":"04:57.859","Text":"I have 8 so on the left-hand side,"},{"Start":"04:57.859 ","End":"05:03.710","Text":"it becomes minus 8 =0."},{"Start":"05:03.710 ","End":"05:08.330","Text":"Of course, and then we have a formula and we"},{"Start":"05:08.330 ","End":"05:15.125","Text":"identify that a = minus 5,"},{"Start":"05:15.125 ","End":"05:22.035","Text":"b =14, c = minus 8 and just substitute."},{"Start":"05:22.035 ","End":"05:25.235","Text":"I\u0027m going to skip that. We\u0027ve done enough of this already."},{"Start":"05:25.235 ","End":"05:28.675","Text":"I\u0027ll just tell you the final answers."},{"Start":"05:28.675 ","End":"05:35.705","Text":"I make it x =2 or x = 4/5."},{"Start":"05:35.705 ","End":"05:39.770","Text":"The main thing I\u0027m trying to show here is how to arrange it to get it to a quadratic"},{"Start":"05:39.770 ","End":"05:43.925","Text":"in standard form and you already know how to solve it using the formula."},{"Start":"05:43.925 ","End":"05:48.245","Text":"I want to get to the third example and final 1 already."},{"Start":"05:48.245 ","End":"05:50.450","Text":"Let\u0027s go there."},{"Start":"05:50.450 ","End":"05:54.185","Text":"Here\u0027s our last exercise."},{"Start":"05:54.185 ","End":"05:59.330","Text":"Also, something that needs a bit of work before it becomes a standard quadratic,"},{"Start":"05:59.330 ","End":"06:00.980","Text":"we need to open brackets."},{"Start":"06:00.980 ","End":"06:04.140","Text":"I noticed that twice we have a difference of something squared,"},{"Start":"06:04.140 ","End":"06:06.905","Text":"that\u0027s going to mean we\u0027ll use a formula."},{"Start":"06:06.905 ","End":"06:08.750","Text":"Don\u0027t want to use the same letters a and b."},{"Start":"06:08.750 ","End":"06:10.760","Text":"You know what we\u0027ll make it A and B,"},{"Start":"06:10.760 ","End":"06:12.950","Text":"that A - B squared,"},{"Start":"06:12.950 ","End":"06:15.815","Text":"a special binomial expansion,"},{"Start":"06:15.815 ","End":"06:19.570","Text":"A^2 minus 2AB + B^2."},{"Start":"06:19.570 ","End":"06:23.060","Text":"I\u0027m going to use it twice here and here. I\u0027ve got twice."},{"Start":"06:23.060 ","End":"06:25.220","Text":"Now if I use the formula here on here,"},{"Start":"06:25.220 ","End":"06:30.125","Text":"get x^2 minus twice x times 3 =6x plus minus"},{"Start":"06:30.125 ","End":"06:35.450","Text":"3 plus 3^2 =9- this thing squared."},{"Start":"06:35.450 ","End":"06:41.945","Text":"I\u0027ll put it all in brackets 2x all squared is 4x^2 because it\u0027s 2^2 also,"},{"Start":"06:41.945 ","End":"06:45.675","Text":"minus and twice 2x times 5."},{"Start":"06:45.675 ","End":"06:49.305","Text":"That makes it minus 20x,"},{"Start":"06:49.305 ","End":"06:52.650","Text":"and then 5^2 =25."},{"Start":"06:52.650 ","End":"06:56.730","Text":"All this is still equal to 18x minus 29."},{"Start":"06:56.730 ","End":"07:00.990","Text":"Let\u0027s open the brackets 2x^2 minus 12x,"},{"Start":"07:00.990 ","End":"07:03.630","Text":"2 times 9 =18."},{"Start":"07:03.630 ","End":"07:07.850","Text":"Then minus 4x^2 with a minus means everything"},{"Start":"07:07.850 ","End":"07:13.085","Text":"reverses sign so it\u0027s plus 20x and minus 25."},{"Start":"07:13.085 ","End":"07:15.860","Text":"You know what? I\u0027ll just throw these on the left also."},{"Start":"07:15.860 ","End":"07:21.590","Text":"It\u0027ll be minus 18x plus 29 =0."},{"Start":"07:21.590 ","End":"07:24.410","Text":"Now let me collect like terms."},{"Start":"07:24.410 ","End":"07:26.780","Text":"Let\u0027s say I collect x^2 terms first."},{"Start":"07:26.780 ","End":"07:31.310","Text":"I could do it in color, but I just put some kind of mark underneath each 1,"},{"Start":"07:31.310 ","End":"07:32.760","Text":"say around it arc,"},{"Start":"07:32.760 ","End":"07:35.480","Text":"or something, and then there\u0027s no more of those."},{"Start":"07:35.480 ","End":"07:40.959","Text":"And maybe the Xs I\u0027ll put with a straight line and we have x here and we have x here,"},{"Start":"07:40.959 ","End":"07:43.010","Text":"and whatever is leftover is just the numbers."},{"Start":"07:43.010 ","End":"07:45.860","Text":"Maybe put a little some other side,"},{"Start":"07:45.860 ","End":"07:48.110","Text":"another color or whatever."},{"Start":"07:48.110 ","End":"07:55.880","Text":"x^2 together 2x^2 minus 4x^2 = minus 2x^2."},{"Start":"07:55.880 ","End":"08:00.350","Text":"Then Xs we have minus 12 plus 20 =8,"},{"Start":"08:00.350 ","End":"08:03.570","Text":"8 minus 18 = minus10x."},{"Start":"08:04.180 ","End":"08:06.620","Text":"Just numbers."},{"Start":"08:06.620 ","End":"08:10.940","Text":"18 minus 25 = minus 7 plus"},{"Start":"08:10.940 ","End":"08:17.160","Text":"29 = plus 22."},{"Start":"08:17.160 ","End":"08:18.420","Text":"I was supposed to get 12."},{"Start":"08:18.420 ","End":"08:20.150","Text":"I miss copied the exercise."},{"Start":"08:20.150 ","End":"08:22.700","Text":"This was supposed to be minus 19,"},{"Start":"08:22.700 ","End":"08:24.920","Text":"so this is also minus 19,"},{"Start":"08:24.920 ","End":"08:27.725","Text":"and then this will be plus 19,"},{"Start":"08:27.725 ","End":"08:30.125","Text":"and that will make this plus 12."},{"Start":"08:30.125 ","End":"08:34.100","Text":"I\u0027ll rewrite it. I fixed all those 2s to make them a 1,"},{"Start":"08:34.100 ","End":"08:37.130","Text":"sorry about that, =0."},{"Start":"08:37.130 ","End":"08:39.590","Text":"I noticed that everything is divisible by 2,"},{"Start":"08:39.590 ","End":"08:41.675","Text":"so we can divide everything by 2. But you know what?"},{"Start":"08:41.675 ","End":"08:43.715","Text":"I\u0027d like to get rid of that minus also,"},{"Start":"08:43.715 ","End":"08:47.750","Text":"let\u0027s divide everything by minus 2 and get smaller number."},{"Start":"08:47.750 ","End":"08:49.460","Text":"Dividing this by minus 2,"},{"Start":"08:49.460 ","End":"08:58.090","Text":"it\u0027s just x^2 by minus 2 makes it plus 5x and minus 6 =0."},{"Start":"08:58.090 ","End":"09:01.025","Text":"Before you know how to use the formula,"},{"Start":"09:01.025 ","End":"09:04.730","Text":"we know that a=1 and b=5,"},{"Start":"09:04.730 ","End":"09:10.050","Text":"and c= minus 6."},{"Start":"09:10.050 ","End":"09:14.444","Text":"I\u0027ll just cut to the chase and the answer is"},{"Start":"09:14.444 ","End":"09:20.565","Text":"that x= minus 6 or 1."},{"Start":"09:20.565 ","End":"09:22.790","Text":"Just check 1 of them, for example,"},{"Start":"09:22.790 ","End":"09:24.490","Text":"suppose we substitute x=1,"},{"Start":"09:24.490 ","End":"09:27.480","Text":"we get 1 plus 5 minus 6 = 0."},{"Start":"09:27.480 ","End":"09:29.630","Text":"I\u0027ll leave you to fill in the dot, dot,"},{"Start":"09:29.630 ","End":"09:36.350","Text":"dot because the idea was the first part how to arrange a mass into a quadratic equation."},{"Start":"09:36.350 ","End":"09:39.630","Text":"That\u0027s the third 1 and we\u0027re done."}],"ID":4996},{"Watched":false,"Name":"Exercise 1","Duration":"2m 48s","ChapterTopicVideoID":4997,"CourseChapterTopicPlaylistID":3869,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.865","Text":"We have here an equation to solve,"},{"Start":"00:02.865 ","End":"00:06.509","Text":"and I can see it\u0027s going to be a quadratic equation,"},{"Start":"00:06.509 ","End":"00:08.880","Text":"but it isn\u0027t quite yet."},{"Start":"00:08.880 ","End":"00:10.980","Text":"Well, at least it\u0027s not in standard form."},{"Start":"00:10.980 ","End":"00:12.150","Text":"It doesn\u0027t look like this."},{"Start":"00:12.150 ","End":"00:15.975","Text":"Let\u0027s just simplify and collect like terms."},{"Start":"00:15.975 ","End":"00:18.390","Text":"Let\u0027s move everything to the left."},{"Start":"00:18.390 ","End":"00:20.775","Text":"But in order x^2,"},{"Start":"00:20.775 ","End":"00:22.529","Text":"then x is the numbers."},{"Start":"00:22.529 ","End":"00:24.675","Text":"Let\u0027s see, x^2."},{"Start":"00:24.675 ","End":"00:27.805","Text":"Let those be like my apples."},{"Start":"00:27.805 ","End":"00:30.590","Text":"This will be like the oranges,"},{"Start":"00:30.590 ","End":"00:35.795","Text":"that\u0027s just the x\u0027s around no more and choose your third fruit."},{"Start":"00:35.795 ","End":"00:38.270","Text":"I don\u0027t know if it\u0027s a lemon or a banana."},{"Start":"00:38.270 ","End":"00:40.145","Text":"Anyway, we\u0027ll collect,"},{"Start":"00:40.145 ","End":"00:42.695","Text":"so how many x^2 do we have?"},{"Start":"00:42.695 ","End":"00:46.615","Text":"Let\u0027s see, yes, green that\u0027s 4 and minus 1."},{"Start":"00:46.615 ","End":"00:50.240","Text":"But the minus 1 cross over to the other side. Well, you know what?"},{"Start":"00:50.240 ","End":"00:52.145","Text":"Let\u0027s just first of all, I\u0027m jumping."},{"Start":"00:52.145 ","End":"00:53.630","Text":"Let\u0027s just take it slow."},{"Start":"00:53.630 ","End":"00:56.725","Text":"4x^2 plus x^2."},{"Start":"00:56.725 ","End":"00:58.325","Text":"That\u0027s the green ones."},{"Start":"00:58.325 ","End":"01:04.030","Text":"Then let\u0027s collect the orange ones minus 5x and then the numbers."},{"Start":"01:04.030 ","End":"01:06.090","Text":"So plus 7,"},{"Start":"01:06.090 ","End":"01:08.810","Text":"minus 4, minus 3."},{"Start":"01:08.810 ","End":"01:11.765","Text":"Notice that everything that I brought over a change sign,"},{"Start":"01:11.765 ","End":"01:13.460","Text":"and now I\u0027m left with just 0 here."},{"Start":"01:13.460 ","End":"01:15.005","Text":"Now let\u0027s gather them,"},{"Start":"01:15.005 ","End":"01:16.925","Text":"the x^2s together,"},{"Start":"01:16.925 ","End":"01:18.810","Text":"so that\u0027s 4 and 1."},{"Start":"01:18.810 ","End":"01:20.670","Text":"Notice there\u0027s a 1 here implicitly."},{"Start":"01:20.670 ","End":"01:24.150","Text":"So 4 and 1 is 5x^2,"},{"Start":"01:24.150 ","End":"01:29.460","Text":"4 apples and an apple is 5 apples minus 5x,"},{"Start":"01:29.460 ","End":"01:33.505","Text":"and then 7 minus 4 minus 3."},{"Start":"01:33.505 ","End":"01:36.754","Text":"That\u0027s nothing. I don\u0027t have to write the 0."},{"Start":"01:36.754 ","End":"01:39.335","Text":"I\u0027ll just write that this equals 0."},{"Start":"01:39.335 ","End":"01:42.814","Text":"Now, because we have a missing c,"},{"Start":"01:42.814 ","End":"01:45.650","Text":"we are not going to use the formula because"},{"Start":"01:45.650 ","End":"01:48.695","Text":"there are easier ways to do that and that\u0027s factorizing."},{"Start":"01:48.695 ","End":"01:57.440","Text":"For example, I could take 5x outside the brackets, or better still,"},{"Start":"01:57.440 ","End":"01:59.419","Text":"when we have something equals 0,"},{"Start":"01:59.419 ","End":"02:05.045","Text":"it\u0027s nice to divide both sides by 5 because 0 divided by anything is 0,"},{"Start":"02:05.045 ","End":"02:06.470","Text":"except 0 over 0,"},{"Start":"02:06.470 ","End":"02:08.120","Text":"of course, which is not the case here."},{"Start":"02:08.120 ","End":"02:14.285","Text":"Divide by 5 and we\u0027ve got x^2 minus x equals 0."},{"Start":"02:14.285 ","End":"02:17.240","Text":"Now let\u0027s factorize, take x out,"},{"Start":"02:17.240 ","End":"02:21.079","Text":"x times x minus 1 equals 0."},{"Start":"02:21.079 ","End":"02:25.160","Text":"Remember when you have the product of two or more things that gives"},{"Start":"02:25.160 ","End":"02:29.390","Text":"you 0 the only way a product and get 0 is if 1 of the factors is 0."},{"Start":"02:29.390 ","End":"02:35.840","Text":"We have a choice either this equals 0 or this equals 0,"},{"Start":"02:35.840 ","End":"02:38.060","Text":"x minus 1 equals 0,"},{"Start":"02:38.060 ","End":"02:40.790","Text":"which means that x equals 1."},{"Start":"02:40.790 ","End":"02:43.325","Text":"We found our two solutions,"},{"Start":"02:43.325 ","End":"02:49.140","Text":"either x equals 0 or x equals 1 and we\u0027re done."}],"ID":4997},{"Watched":false,"Name":"Exercise 2","Duration":"4m 11s","ChapterTopicVideoID":4998,"CourseChapterTopicPlaylistID":3869,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.800","Text":"Here we have an equation to solve and looks like it might be a quadratic because"},{"Start":"00:04.800 ","End":"00:09.465","Text":"there\u0027s an x times an x and just in case it is I have the formula on hand."},{"Start":"00:09.465 ","End":"00:12.810","Text":"Let\u0027s simplify, expand brackets,"},{"Start":"00:12.810 ","End":"00:15.720","Text":"and so on, collect like terms and see what we get."},{"Start":"00:15.720 ","End":"00:21.390","Text":"First 1, expanding minus x times x is minus x^2,"},{"Start":"00:21.390 ","End":"00:25.427","Text":"minus x times minus 5 plus 5x,"},{"Start":"00:25.427 ","End":"00:29.680","Text":"don\u0027t forget the minus times minus is plus equals."},{"Start":"00:29.680 ","End":"00:32.724","Text":"Now here we have 2 things times 2 things,"},{"Start":"00:32.724 ","End":"00:34.760","Text":"so it\u0027s time for the red arcs."},{"Start":"00:34.760 ","End":"00:36.530","Text":"Just like I had this and this here,"},{"Start":"00:36.530 ","End":"00:37.910","Text":"I have this with this,"},{"Start":"00:37.910 ","End":"00:41.733","Text":"this with this, this with this, this with this."},{"Start":"00:41.733 ","End":"00:44.385","Text":"1 times 1 is 1,"},{"Start":"00:44.385 ","End":"00:47.685","Text":"1 times minus x is minus x,"},{"Start":"00:47.685 ","End":"00:51.630","Text":"minus 3x times 1 minus 3x,"},{"Start":"00:51.630 ","End":"00:58.680","Text":"minus 3x times minus x is plus 3xx, which is x^2."},{"Start":"00:58.680 ","End":"01:02.330","Text":"There\u0027s a plus 4 at the end."},{"Start":"01:02.330 ","End":"01:05.360","Text":"Let\u0027s bring everything over to the left."},{"Start":"01:05.360 ","End":"01:07.900","Text":"You know what I\u0027ll just highlight."},{"Start":"01:07.900 ","End":"01:11.715","Text":"The x^2 will be like the apples,"},{"Start":"01:11.715 ","End":"01:19.910","Text":"x\u0027s are like oranges and numbers say lemons."},{"Start":"01:19.910 ","End":"01:28.790","Text":"Just reorder them minus x^2 plus 3x^2 that\u0027s the green ones."},{"Start":"01:28.790 ","End":"01:34.115","Text":"Then I switched sides and I forgot to change the sign."},{"Start":"01:34.115 ","End":"01:35.540","Text":"There we are minus,"},{"Start":"01:35.540 ","End":"01:36.880","Text":"minus of course,"},{"Start":"01:36.880 ","End":"01:40.190","Text":"and then the x\u0027s plus 5x there,"},{"Start":"01:40.190 ","End":"01:42.095","Text":"but everything else changes sign,"},{"Start":"01:42.095 ","End":"01:45.210","Text":"so it\u0027s plus x plus."}],"ID":4998},{"Watched":false,"Name":"Exercise 3","Duration":"5m 21s","ChapterTopicVideoID":4999,"CourseChapterTopicPlaylistID":3869,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"Here we have an equation to solve and it looks like it\u0027s going to be"},{"Start":"00:03.600 ","End":"00:08.522","Text":"a quadratic equation because of all the x^2\u0027s that we\u0027re going to get."},{"Start":"00:08.522 ","End":"00:10.890","Text":"But we have to simplify, expand brackets,"},{"Start":"00:10.890 ","End":"00:13.905","Text":"collect like terms, and then see where we are."},{"Start":"00:13.905 ","End":"00:17.340","Text":"I have the formula handy in case we need it."},{"Start":"00:17.340 ","End":"00:18.660","Text":"Let\u0027s begin."},{"Start":"00:18.660 ","End":"00:21.450","Text":"Let\u0027s use the formula,"},{"Start":"00:21.450 ","End":"00:23.400","Text":"and I\u0027ll write it at the side here."},{"Start":"00:23.400 ","End":"00:32.880","Text":"That a minus b^2 is a^2 minus 2ab plus b^2."},{"Start":"00:32.880 ","End":"00:37.073","Text":"That\u0027s one of those special binomial expansions,"},{"Start":"00:37.073 ","End":"00:40.135","Text":"and we can use it twice here and here."},{"Start":"00:40.135 ","End":"00:44.120","Text":"Now, I\u0027ll leave the 2 here and expand this according to this so we"},{"Start":"00:44.120 ","End":"00:49.000","Text":"get x^2 minus twice x times 5,"},{"Start":"00:49.000 ","End":"00:50.925","Text":"which is minus 10x,"},{"Start":"00:50.925 ","End":"00:55.080","Text":"and then plus 5^2, which is 25."},{"Start":"00:55.080 ","End":"00:57.750","Text":"Then the other one,"},{"Start":"00:57.750 ","End":"01:01.170","Text":"2x minus 3^2 so it\u0027s 2x all squared,"},{"Start":"01:01.170 ","End":"01:03.330","Text":"which is 2^2,"},{"Start":"01:03.330 ","End":"01:08.525","Text":"x^2, 4x^2 minus twice 2 times 3 times x."},{"Start":"01:08.525 ","End":"01:18.510","Text":"Let\u0027s see, that\u0027s makes it minus 12x and plus 3^2 which is 9,"},{"Start":"01:18.510 ","End":"01:22.920","Text":"and this equals 10x plus 21."},{"Start":"01:22.920 ","End":"01:27.890","Text":"Now I\u0027m going to expand this one, so 2x^2,"},{"Start":"01:27.890 ","End":"01:29.315","Text":"just multiply everything by 2,"},{"Start":"01:29.315 ","End":"01:32.633","Text":"minus 20x plus 50,"},{"Start":"01:32.633 ","End":"01:35.270","Text":"and I open this one because it\u0027s minus, it\u0027s minus 1."},{"Start":"01:35.270 ","End":"01:37.100","Text":"So just reverse all the signs."},{"Start":"01:37.100 ","End":"01:44.000","Text":"It\u0027s minus 4x^2 plus 12x minus 9, and you know what?"},{"Start":"01:44.000 ","End":"01:46.595","Text":"I\u0027ll bring this over to the left side."},{"Start":"01:46.595 ","End":"01:48.380","Text":"Then this leaves 0 on the right,"},{"Start":"01:48.380 ","End":"01:54.995","Text":"so it\u0027s minus 10x minus 2=0."},{"Start":"01:54.995 ","End":"02:02.045","Text":"To make it easier, I\u0027ll color the x^2\u0027s here and here and then the x\u0027s,"},{"Start":"02:02.045 ","End":"02:06.425","Text":"that\u0027s here, and here, and here."},{"Start":"02:06.425 ","End":"02:10.775","Text":"Then just the numbers here and here."},{"Start":"02:10.775 ","End":"02:13.925","Text":"Let\u0027s see the green ones, the x^2,"},{"Start":"02:13.925 ","End":"02:15.980","Text":"I have 2 minus 4,"},{"Start":"02:15.980 ","End":"02:18.500","Text":"which is minus 2x^2."},{"Start":"02:18.500 ","End":"02:26.205","Text":"Then the x\u0027s minus 20 plus 12 minus 10,"},{"Start":"02:26.205 ","End":"02:31.115","Text":"the minuses are minus 30 together and the pluses is 12."},{"Start":"02:31.115 ","End":"02:36.590","Text":"It\u0027s going to be a negative and the difference of 30 and 12 is 18,"},{"Start":"02:36.590 ","End":"02:39.480","Text":"so it\u0027s minus 18x."},{"Start":"02:40.040 ","End":"02:44.640","Text":"Let\u0027s see the yellow ones, the numbers,"},{"Start":"02:44.640 ","End":"02:55.065","Text":"50 minus 21 is plus 29 is equal to 0."},{"Start":"02:55.065 ","End":"02:59.030","Text":"I just noticed I forgot to mark this one,"},{"Start":"02:59.030 ","End":"03:01.280","Text":"so that will change things."},{"Start":"03:01.280 ","End":"03:02.750","Text":"I have to take off 9,"},{"Start":"03:02.750 ","End":"03:04.660","Text":"so that will be just 20."},{"Start":"03:04.660 ","End":"03:09.650","Text":"That\u0027s better, and remember I said to you that if you can see a common factor,"},{"Start":"03:09.650 ","End":"03:12.305","Text":"we should take it out like here."},{"Start":"03:12.305 ","End":"03:15.020","Text":"I could divide everything by 2."},{"Start":"03:15.020 ","End":"03:19.280","Text":"But why don\u0027t I just divide everything by minus 2 already?"},{"Start":"03:19.280 ","End":"03:22.440","Text":"Because I like quadratic equations to begin with a plus."},{"Start":"03:22.440 ","End":"03:24.600","Text":"Dividing everything by minus 2,"},{"Start":"03:24.600 ","End":"03:30.420","Text":"this leaves just x^2 minus 18 over minus 2 is plus 9,"},{"Start":"03:30.420 ","End":"03:36.395","Text":"and 20 over minus 2 is minus 10 is equal to 0."},{"Start":"03:36.395 ","End":"03:42.080","Text":"Finally, we\u0027re ready to use the formula and we just identify that a is 1."},{"Start":"03:42.080 ","End":"03:44.850","Text":"It doesn\u0027t say 1x^2 but it\u0027s implied,"},{"Start":"03:44.850 ","End":"03:47.320","Text":"so we take a=1,"},{"Start":"03:47.320 ","End":"03:53.750","Text":"b=9, c equals minus 10, and substitute here."},{"Start":"03:53.750 ","End":"03:57.500","Text":"What we get is minus b,"},{"Start":"03:57.500 ","End":"04:00.340","Text":"which is minus 9,"},{"Start":"04:00.340 ","End":"04:05.675","Text":"plus or minus the square root of b^2"},{"Start":"04:05.675 ","End":"04:14.160","Text":"minus 4ac and this is all over 2a,"},{"Start":"04:14.160 ","End":"04:16.015","Text":"2 times 1 is 2."},{"Start":"04:16.015 ","End":"04:18.800","Text":"The bit under the square root sign is most important,"},{"Start":"04:18.800 ","End":"04:22.070","Text":"in any event it should be done first because this is like a bracket."},{"Start":"04:22.070 ","End":"04:23.750","Text":"So what we get from here is,"},{"Start":"04:23.750 ","End":"04:25.775","Text":"let\u0027s see I\u0027ll just squeeze it here."},{"Start":"04:25.775 ","End":"04:28.708","Text":"9^2 is 81,"},{"Start":"04:28.708 ","End":"04:35.655","Text":"and a minus is a plus and 4 times 1 times 10 is 40."},{"Start":"04:35.655 ","End":"04:39.435","Text":"This is a 121,"},{"Start":"04:39.435 ","End":"04:43.129","Text":"the square root of a 121 is 11."},{"Start":"04:43.129 ","End":"04:50.685","Text":"What we get is minus 9 plus or minus 11 over 2,"},{"Start":"04:50.685 ","End":"04:53.585","Text":"so x can be one of two things."},{"Start":"04:53.585 ","End":"04:55.400","Text":"If we take the plus,"},{"Start":"04:55.400 ","End":"04:59.315","Text":"minus 9 plus 11 is 2."},{"Start":"04:59.315 ","End":"05:01.595","Text":"2 over 2 is 1."},{"Start":"05:01.595 ","End":"05:04.610","Text":"If I take the minus, minus 9,"},{"Start":"05:04.610 ","End":"05:12.345","Text":"minus 11 is minus 20 and over 2 is minus 10."},{"Start":"05:12.345 ","End":"05:14.450","Text":"There are two possible answers for x,"},{"Start":"05:14.450 ","End":"05:16.340","Text":"x could be equal to 1,"},{"Start":"05:16.340 ","End":"05:22.200","Text":"or x could be equal to minus 10. We\u0027re done."}],"ID":4999}],"Thumbnail":null,"ID":3869},{"Name":"Quadratics in Disguise","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Disguised Quadratic","Duration":"9m 27s","ChapterTopicVideoID":5000,"CourseChapterTopicPlaylistID":3870,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"Previously on quadratic equations,"},{"Start":"00:03.240 ","End":"00:07.380","Text":"we had a clip which I called disarranged quadratic equations."},{"Start":"00:07.380 ","End":"00:09.360","Text":"This is an informal word,"},{"Start":"00:09.360 ","End":"00:12.765","Text":"which just means that we have to rearrange a bit,"},{"Start":"00:12.765 ","End":"00:15.735","Text":"tidy up, and then get to this form."},{"Start":"00:15.735 ","End":"00:19.410","Text":"But you can still see it\u0027s going to be quadratic because you know when you square this,"},{"Start":"00:19.410 ","End":"00:22.875","Text":"you\u0027re going to get x squared times something and so on."},{"Start":"00:22.875 ","End":"00:28.320","Text":"I just want to get a bit more into the disarranged to the point of disguised."},{"Start":"00:28.320 ","End":"00:31.740","Text":"Disguised is also an informal word, but I didn\u0027t make it up."},{"Start":"00:31.740 ","End":"00:33.270","Text":"I\u0027ve seen others use it,"},{"Start":"00:33.270 ","End":"00:35.700","Text":"other professors, but it\u0027s still,"},{"Start":"00:35.700 ","End":"00:40.170","Text":"I just mean something gets even harder to recognize as being quadratic,"},{"Start":"00:40.170 ","End":"00:42.025","Text":"but which ends up being quadratic,"},{"Start":"00:42.025 ","End":"00:44.030","Text":"especially when there\u0027s a denominator."},{"Start":"00:44.030 ","End":"00:45.619","Text":"Let me give you an example."},{"Start":"00:45.619 ","End":"00:52.605","Text":"Let\u0027s see, 5x minus 11 over 4"},{"Start":"00:52.605 ","End":"00:58.095","Text":"minus 2x minus 5"},{"Start":"00:58.095 ","End":"01:05.275","Text":"over 3 is equal to 2 over x."},{"Start":"01:05.275 ","End":"01:10.130","Text":"You probably agree with me that it doesn\u0027t look like a quadratic equation."},{"Start":"01:10.130 ","End":"01:13.775","Text":"Just a lot of work with denominators maybe."},{"Start":"01:13.775 ","End":"01:16.130","Text":"Let\u0027s get to it,"},{"Start":"01:16.130 ","End":"01:18.545","Text":"and you\u0027ll see why it\u0027s quadratic."},{"Start":"01:18.545 ","End":"01:23.045","Text":"Let\u0027s look for a common denominator for all 3 denominators."},{"Start":"01:23.045 ","End":"01:27.870","Text":"I don\u0027t see anything less than multiplying them all together,"},{"Start":"01:27.870 ","End":"01:29.250","Text":"4 times 3 times x,"},{"Start":"01:29.250 ","End":"01:30.594","Text":"which is 12x,"},{"Start":"01:30.594 ","End":"01:34.880","Text":"but I prefer to leave it for the moment as 4 times 3"},{"Start":"01:34.880 ","End":"01:39.775","Text":"times x and to do some coloring here,"},{"Start":"01:39.775 ","End":"01:42.515","Text":"here, and here."},{"Start":"01:42.515 ","End":"01:46.280","Text":"The reason I\u0027m doing this is because when I see these,"},{"Start":"01:46.280 ","End":"01:48.020","Text":"like the x is in yellow,"},{"Start":"01:48.020 ","End":"01:50.525","Text":"I\u0027ll be able to see which colors are missing."},{"Start":"01:50.525 ","End":"01:54.810","Text":"This one I\u0027ll put in this green,"},{"Start":"01:54.810 ","End":"01:57.485","Text":"this light turquoise here."},{"Start":"01:57.485 ","End":"02:01.160","Text":"Now what I say is this my common denominator,"},{"Start":"02:01.160 ","End":"02:07.520","Text":"4 into 4 times 3 times x leaves me 3 times x. I write the 3x here."},{"Start":"02:07.520 ","End":"02:09.470","Text":"This is just like with fractions."},{"Start":"02:09.470 ","End":"02:18.635","Text":"Then I see that 3 goes into 4 times 3 times x 4x times the 2 missing colors,"},{"Start":"02:18.635 ","End":"02:20.930","Text":"and x goes into 4 times 3 times x,"},{"Start":"02:20.930 ","End":"02:23.225","Text":"4 times 3 times."},{"Start":"02:23.225 ","End":"02:28.650","Text":"I\u0027ll write this as 12, 4 times 3."},{"Start":"02:30.400 ","End":"02:33.170","Text":"Now what we do is we multiply out,"},{"Start":"02:33.170 ","End":"02:34.550","Text":"we forget the denominators,"},{"Start":"02:34.550 ","End":"02:40.535","Text":"and we multiply the numerators by this extra factor respectively and see what we get."},{"Start":"02:40.535 ","End":"02:43.235","Text":"This is good practice in fractions."},{"Start":"02:43.235 ","End":"02:47.000","Text":"3x times this, I\u0027ll do it bit by bit."},{"Start":"02:47.000 ","End":"02:48.620","Text":"I\u0027ll just put it in brackets first,"},{"Start":"02:48.620 ","End":"02:50.000","Text":"I won\u0027t actually multiply out,"},{"Start":"02:50.000 ","End":"02:51.920","Text":"but indicate that I\u0027m multiplying."},{"Start":"02:51.920 ","End":"02:57.040","Text":"Then minus 4x(2x - 5)."},{"Start":"02:57.040 ","End":"03:00.110","Text":"Remember dividing line is effectively a bracket and"},{"Start":"03:00.110 ","End":"03:03.845","Text":"the rules of operations, their order apply."},{"Start":"03:03.845 ","End":"03:08.170","Text":"Then equals 12 times 2."},{"Start":"03:08.170 ","End":"03:10.670","Text":"Don\u0027t forget to put the little dot."},{"Start":"03:10.670 ","End":"03:14.830","Text":"In algebra, you can\u0027t just leave it and say 1,"},{"Start":"03:14.830 ","End":"03:17.210","Text":"2, 2, it\u0027d be 122."},{"Start":"03:17.210 ","End":"03:20.900","Text":"We don\u0027t usually put the cross because then it gets confused with the x,"},{"Start":"03:20.900 ","End":"03:25.265","Text":"so dot is the preferred multiplication in algebra."},{"Start":"03:25.265 ","End":"03:26.795","Text":"Here it\u0027s optional."},{"Start":"03:26.795 ","End":"03:28.340","Text":"Like I put a dot here or not,"},{"Start":"03:28.340 ","End":"03:30.380","Text":"won\u0027t make any difference because the brackets and"},{"Start":"03:30.380 ","End":"03:33.635","Text":"the variable name indicate that it\u0027s a multiplication."},{"Start":"03:33.635 ","End":"03:36.650","Text":"That\u0027s besides the point, but good to know."},{"Start":"03:36.650 ","End":"03:38.705","Text":"Now multiplying out,"},{"Start":"03:38.705 ","End":"03:40.610","Text":"each one of these with each one of these,"},{"Start":"03:40.610 ","End":"03:46.550","Text":"so 3x times 5x is 3 times 5 times x times x is 15x squared,"},{"Start":"03:46.550 ","End":"03:56.320","Text":"and 3x times minus 11 minus 33x minus similarly 8x squared."},{"Start":"03:56.320 ","End":"03:58.310","Text":"Now watch out on the last one."},{"Start":"03:58.310 ","End":"04:02.430","Text":"Minus with minus is plus and 5 times 4 is 20,"},{"Start":"04:02.430 ","End":"04:04.665","Text":"so it\u0027s plus 20x."},{"Start":"04:04.665 ","End":"04:07.272","Text":"Tell you what I\u0027m going to do,"},{"Start":"04:07.272 ","End":"04:09.510","Text":"the 12 times 2,"},{"Start":"04:09.510 ","End":"04:11.595","Text":"we all know that\u0027s 24."},{"Start":"04:11.595 ","End":"04:16.265","Text":"I\u0027ll just straight away put it into the other side as minus 24 and equals 0."},{"Start":"04:16.265 ","End":"04:18.305","Text":"You see I\u0027m aiming to get it to this 4,"},{"Start":"04:18.305 ","End":"04:21.330","Text":"I already see that I\u0027ve got x squared in it."},{"Start":"04:22.750 ","End":"04:25.115","Text":"I don\u0027t think I\u0027ll use coloring here,"},{"Start":"04:25.115 ","End":"04:26.240","Text":"not that many terms."},{"Start":"04:26.240 ","End":"04:27.620","Text":"The x squared terms first."},{"Start":"04:27.620 ","End":"04:33.600","Text":"We have 15 minus 8 is 7x squared."},{"Start":"04:33.770 ","End":"04:41.890","Text":"Let\u0027s see, x is, I have an x here, here,"},{"Start":"04:41.890 ","End":"04:45.395","Text":"so minus 33 plus 20,"},{"Start":"04:45.395 ","End":"04:50.345","Text":"that gives me minus 13x."},{"Start":"04:50.345 ","End":"04:56.925","Text":"Then numbers just minus 24, and that\u0027s equal to 0."},{"Start":"04:56.925 ","End":"05:01.835","Text":"Now I can use the quadratic formula that is,"},{"Start":"05:01.835 ","End":"05:04.490","Text":"but I like to identify a, b, and c first,"},{"Start":"05:04.490 ","End":"05:06.610","Text":"so a is 7,"},{"Start":"05:06.610 ","End":"05:09.175","Text":"b is minus 13,"},{"Start":"05:09.175 ","End":"05:11.510","Text":"c is minus 24,"},{"Start":"05:11.510 ","End":"05:14.060","Text":"and then I plug it in this formula."},{"Start":"05:14.060 ","End":"05:15.680","Text":"Why don\u0027t I continue over here,"},{"Start":"05:15.680 ","End":"05:17.030","Text":"it\u0027ll be close at hand."},{"Start":"05:17.030 ","End":"05:20.810","Text":"Looking minus b, b is minus 13."},{"Start":"05:20.810 ","End":"05:23.540","Text":"Notice the plus 13,"},{"Start":"05:23.540 ","End":"05:25.535","Text":"I\u0027m almost tempted to write a plus,"},{"Start":"05:25.535 ","End":"05:31.400","Text":"but I will, just to show you that I didn\u0027t forget the minus, it\u0027s minus minus."},{"Start":"05:31.400 ","End":"05:39.530","Text":"Then plus or minus the square root of b squared."},{"Start":"05:39.530 ","End":"05:41.340","Text":"It\u0027s minus 13 squared,"},{"Start":"05:41.340 ","End":"05:47.015","Text":"but I\u0027ll write it as 13 squared because we know that negative times negative is positive."},{"Start":"05:47.015 ","End":"05:51.850","Text":"Then minus 4 times a,"},{"Start":"05:51.850 ","End":"05:53.988","Text":"which is 7,"},{"Start":"05:53.988 ","End":"05:59.625","Text":"and c which is minus 24."},{"Start":"05:59.625 ","End":"06:02.625","Text":"All this going to be over 2a,"},{"Start":"06:02.625 ","End":"06:08.060","Text":"so over 2a is 2 times 7."},{"Start":"06:08.060 ","End":"06:12.355","Text":"I just like to compute the bit under the square root at the side, I often do this."},{"Start":"06:12.355 ","End":"06:17.270","Text":"Let\u0027s see, 13 squared is 169."},{"Start":"06:17.270 ","End":"06:22.965","Text":"Then 4 times 7 times 24,"},{"Start":"06:22.965 ","End":"06:25.800","Text":"have to do the calculator,"},{"Start":"06:25.800 ","End":"06:28.260","Text":"I make it 672."},{"Start":"06:28.260 ","End":"06:29.620","Text":"Let\u0027s just see what sign it is,"},{"Start":"06:29.620 ","End":"06:30.910","Text":"is a minus and there\u0027s minus,"},{"Start":"06:30.910 ","End":"06:34.540","Text":"is a plus is 672."},{"Start":"06:34.540 ","End":"06:37.345","Text":"If I add these two together,"},{"Start":"06:37.345 ","End":"06:41.590","Text":"I make that 841."},{"Start":"06:41.590 ","End":"06:45.814","Text":"I\u0027m now back here,"},{"Start":"06:45.814 ","End":"06:47.930","Text":"put that under the square root sign."},{"Start":"06:47.930 ","End":"06:57.170","Text":"13 plus or minus the square root of 841 over 14."},{"Start":"06:57.170 ","End":"06:59.405","Text":"Now if I do the square root of this,"},{"Start":"06:59.405 ","End":"07:01.640","Text":"the square root of 841,"},{"Start":"07:01.640 ","End":"07:06.470","Text":"I also checked on the calculator, is 29."},{"Start":"07:06.470 ","End":"07:11.090","Text":"What I\u0027ll get if I take the plus above and the minus below,"},{"Start":"07:11.090 ","End":"07:19.040","Text":"I get 13 plus 29 over 14."},{"Start":"07:19.040 ","End":"07:27.710","Text":"The other way I get 13 minus 29 over 14."},{"Start":"07:27.710 ","End":"07:31.265","Text":"Let\u0027s see what I get each way."},{"Start":"07:31.265 ","End":"07:34.346","Text":"13 plus 29 is 42,"},{"Start":"07:34.346 ","End":"07:38.505","Text":"42 over 14 is an even 3."},{"Start":"07:38.505 ","End":"07:45.465","Text":"The other way, 13 minus 29 is,"},{"Start":"07:45.465 ","End":"07:52.990","Text":"let\u0027s see, it\u0027s minus 16 over 14."},{"Start":"07:53.690 ","End":"08:00.735","Text":"I can cancel by 2,"},{"Start":"08:00.735 ","End":"08:06.635","Text":"and I can get that this is equal to minus 8 over 7,"},{"Start":"08:06.635 ","End":"08:08.310","Text":"which is the answer,"},{"Start":"08:08.310 ","End":"08:10.510","Text":"but I prefer to write it as a mixed number."},{"Start":"08:10.510 ","End":"08:16.030","Text":"It\u0027s more commonly accepted to leave the answer as a mixed number,"},{"Start":"08:16.030 ","End":"08:20.670","Text":"that\u0027s minus 1 and 1/7."},{"Start":"08:20.670 ","End":"08:25.385","Text":"These are my two answers for x, this and this."},{"Start":"08:25.385 ","End":"08:30.350","Text":"There\u0027s something else I have to mention in these cases where you have denominators or"},{"Start":"08:30.350 ","End":"08:35.360","Text":"variables in the denominator such as x. I wanted to get more technical,"},{"Start":"08:35.360 ","End":"08:37.310","Text":"but then you may not have studied functions,"},{"Start":"08:37.310 ","End":"08:39.230","Text":"so let me just say it this way."},{"Start":"08:39.230 ","End":"08:42.320","Text":"You have to make sure when you have a fraction,"},{"Start":"08:42.320 ","End":"08:44.330","Text":"that the denominator is not 0."},{"Start":"08:44.330 ","End":"08:48.200","Text":"Now 4 I know is not 0 and 3, I know is not 0."},{"Start":"08:48.200 ","End":"08:50.720","Text":"But x of hand I didn\u0027t know,"},{"Start":"08:50.720 ","End":"08:55.088","Text":"so I have to make sure at the end that I didn\u0027t get 0 as one of my answers,"},{"Start":"08:55.088 ","End":"08:56.300","Text":"and indeed I didn\u0027t,"},{"Start":"08:56.300 ","End":"08:57.610","Text":"x could be 3,"},{"Start":"08:57.610 ","End":"08:59.805","Text":"x could be minus 1, and a 1/7."},{"Start":"08:59.805 ","End":"09:03.170","Text":"But sometimes it\u0027s not x in some expression involving x."},{"Start":"09:03.170 ","End":"09:05.570","Text":"Let me just say generally that when you get to the end,"},{"Start":"09:05.570 ","End":"09:07.910","Text":"you have to check your answers to make sure they don\u0027t"},{"Start":"09:07.910 ","End":"09:11.050","Text":"cause any other denominators to be 0."},{"Start":"09:11.050 ","End":"09:13.670","Text":"Having checked that, now we can safely say"},{"Start":"09:13.670 ","End":"09:16.235","Text":"that we\u0027re done and these are the two solutions."},{"Start":"09:16.235 ","End":"09:18.505","Text":"I\u0027ll leave it to you to substitute."},{"Start":"09:18.505 ","End":"09:20.390","Text":"You should once in a while substitute"},{"Start":"09:20.390 ","End":"09:23.495","Text":"your answers to check that you\u0027d indeed get an inequality,"},{"Start":"09:23.495 ","End":"09:27.960","Text":"but I won\u0027t do it this time. We\u0027re done."}],"ID":5000},{"Watched":false,"Name":"Exercise 1","Duration":"4m 11s","ChapterTopicVideoID":5001,"CourseChapterTopicPlaylistID":3870,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:03.705","Text":"Here we have an equation to solve,"},{"Start":"00:03.705 ","End":"00:07.020","Text":"and it\u0027s not exactly clear what kind it is."},{"Start":"00:07.020 ","End":"00:09.600","Text":"It looks quadratic in actual fact,"},{"Start":"00:09.600 ","End":"00:14.160","Text":"this is one of those quadratics in disguise with denominators."},{"Start":"00:14.160 ","End":"00:18.465","Text":"First thing to do is to get rid of denominators."},{"Start":"00:18.465 ","End":"00:24.284","Text":"Let\u0027s look over here and we see these two and we could multiply the two of them together."},{"Start":"00:24.284 ","End":"00:26.774","Text":"But that won\u0027t be the most efficient."},{"Start":"00:26.774 ","End":"00:32.448","Text":"What is good to do is to factorize wherever it\u0027s possible."},{"Start":"00:32.448 ","End":"00:36.966","Text":"In this case, I think it\u0027s obvious that you have x here and here,"},{"Start":"00:36.966 ","End":"00:39.510","Text":"and both these divide by 3."},{"Start":"00:39.510 ","End":"00:42.135","Text":"Let\u0027s, first of all,"},{"Start":"00:42.135 ","End":"00:46.390","Text":"rewrite it as x^2 + 2."},{"Start":"00:46.390 ","End":"00:48.110","Text":"Here I\u0027ll take the x out,"},{"Start":"00:48.110 ","End":"00:53.070","Text":"so I have x(3x+5)."},{"Start":"00:53.240 ","End":"00:57.995","Text":"Here also the numerator is the same, 3x-1."},{"Start":"00:57.995 ","End":"01:00.410","Text":"But the denominator, I\u0027m taking 3 out,"},{"Start":"01:00.410 ","End":"01:05.540","Text":"so I have 3(3x+5)."},{"Start":"01:05.540 ","End":"01:10.340","Text":"Notice that the 3x+5 is common to both denominators and this will"},{"Start":"01:10.340 ","End":"01:16.130","Text":"save us from getting too big a common denominator."},{"Start":"01:16.130 ","End":"01:20.750","Text":"In fact, we can make do with just taking one of each of what we have."},{"Start":"01:20.750 ","End":"01:25.520","Text":"We have a 3, we have x,"},{"Start":"01:25.520 ","End":"01:29.785","Text":"and we have 3x+5."},{"Start":"01:29.785 ","End":"01:32.620","Text":"This should cover everything."},{"Start":"01:32.720 ","End":"01:38.750","Text":"If I look at how many times this denominator goes into this common denominator,"},{"Start":"01:38.750 ","End":"01:40.250","Text":"I look at what\u0027s missing."},{"Start":"01:40.250 ","End":"01:42.140","Text":"What\u0027s missing is the 3,"},{"Start":"01:42.140 ","End":"01:43.850","Text":"so it goes in three times."},{"Start":"01:43.850 ","End":"01:45.440","Text":"I put a 3 here."},{"Start":"01:45.440 ","End":"01:48.260","Text":"If I look at this as opposed to this,"},{"Start":"01:48.260 ","End":"01:51.230","Text":"what\u0027s missing, what\u0027s added."},{"Start":"01:51.230 ","End":"01:58.760","Text":"The other way is the x. I put x over here."},{"Start":"01:58.760 ","End":"02:01.460","Text":"Now we proceed as usual."},{"Start":"02:01.460 ","End":"02:04.910","Text":"We get multiplying this, in fact,"},{"Start":"02:04.910 ","End":"02:10.500","Text":"I will multiply it out already, we get 3x^2."},{"Start":"02:12.260 ","End":"02:22.140","Text":"Better write an extra step, 3(X^2+2)= x(3x-1)."},{"Start":"02:22.140 ","End":"02:26.230","Text":"Now I\u0027ll open it up and get"},{"Start":"02:35.030 ","End":"02:45.720","Text":"3x^2+6=3x^2-x. That certainly"},{"Start":"02:45.720 ","End":"02:47.535","Text":"works out nicely."},{"Start":"02:47.535 ","End":"02:51.930","Text":"The 3x^2 and the 3x^2 cancel."},{"Start":"02:51.930 ","End":"02:57.270","Text":"We\u0027re left with 6=-x or"},{"Start":"02:57.270 ","End":"03:03.405","Text":"in other words, x=-6."},{"Start":"03:03.405 ","End":"03:07.520","Text":"You can think of it as I bring the x over to the left and the 6 over to the right."},{"Start":"03:07.520 ","End":"03:09.890","Text":"This is what I get."},{"Start":"03:09.890 ","End":"03:14.360","Text":"Now, when you have denominators,"},{"Start":"03:14.360 ","End":"03:20.120","Text":"it\u0027s important to go back and check that the value you got makes sense."},{"Start":"03:20.120 ","End":"03:23.164","Text":"In other words, it doesn\u0027t make any other denominator is 0."},{"Start":"03:23.164 ","End":"03:29.350","Text":"If I put x=-6 in either one of them, in fact,"},{"Start":"03:29.350 ","End":"03:34.300","Text":"it may be easier to put them here and here because these denominators are the same,"},{"Start":"03:34.300 ","End":"03:36.435","Text":"if I put in -6,"},{"Start":"03:36.435 ","End":"03:37.950","Text":"this factor is not 0,"},{"Start":"03:37.950 ","End":"03:41.310","Text":"and -18+5 is also not 0,"},{"Start":"03:41.310 ","End":"03:42.690","Text":"this is not 0."},{"Start":"03:42.690 ","End":"03:48.770","Text":"3x+5 is not 0, so we\u0027re okay."},{"Start":"03:48.770 ","End":"03:51.370","Text":"As long as the denominators are not 0,"},{"Start":"03:51.370 ","End":"03:53.060","Text":"which can sometimes happen."},{"Start":"03:53.060 ","End":"03:54.865","Text":"We have to check against that."},{"Start":"03:54.865 ","End":"03:57.025","Text":"Of course, we could have gone all the way and"},{"Start":"03:57.025 ","End":"04:01.510","Text":"checked that -6 actually satisfies the equation."},{"Start":"04:01.510 ","End":"04:03.505","Text":"Sometimes I do that, sometimes not."},{"Start":"04:03.505 ","End":"04:04.750","Text":"Today I will pass."},{"Start":"04:04.750 ","End":"04:10.550","Text":"Let\u0027s go on to the next exercise in the next clip."}],"ID":5001},{"Watched":false,"Name":"Exercise 2","Duration":"7m 7s","ChapterTopicVideoID":5002,"CourseChapterTopicPlaylistID":3870,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.844","Text":"Here we have this equation to solve."},{"Start":"00:02.844 ","End":"00:09.060","Text":"It\u0027s going to turn out to be one of those quadratics,"},{"Start":"00:09.060 ","End":"00:12.345","Text":"barely disguised, but there\u0027s a denominator."},{"Start":"00:12.345 ","End":"00:17.100","Text":"We must remember at the end to check for possible 0 in the denominator,"},{"Start":"00:17.100 ","End":"00:18.600","Text":"which we can\u0027t allow."},{"Start":"00:18.600 ","End":"00:23.580","Text":"Actually, this one is so clear that if x turns out to be different from 3,"},{"Start":"00:23.580 ","End":"00:27.675","Text":"then we\u0027re all right, because only 3 will make the denominator 0."},{"Start":"00:27.675 ","End":"00:31.525","Text":"Anyway, let\u0027s get started and find that common denominator."},{"Start":"00:31.525 ","End":"00:33.800","Text":"Would think that\u0027s fairly obvious here."},{"Start":"00:33.800 ","End":"00:38.020","Text":"We just have to take x-3 as the denominator."},{"Start":"00:38.020 ","End":"00:42.670","Text":"If I multiply it on the left,"},{"Start":"00:42.670 ","End":"00:46.900","Text":"(x /3)/( x/3) is just going to leave me with 1."},{"Start":"00:46.900 ","End":"00:50.120","Text":"This is 12, doesn\u0027t have a denominator,"},{"Start":"00:50.120 ","End":"00:52.775","Text":"so I need the whole x-3."},{"Start":"00:52.775 ","End":"00:55.430","Text":"Some like to think of it as 12/1."},{"Start":"00:55.430 ","End":"01:00.590","Text":"In either case, 1 goes into (x-3) (x-3) times either way."},{"Start":"01:00.590 ","End":"01:10.900","Text":"What we get is 1 times this whole thing is just going to copy it 4x^2-24x+36."},{"Start":"01:10.900 ","End":"01:15.090","Text":"Here, I get this times this."},{"Start":"01:15.090 ","End":"01:16.730","Text":"Doesn\u0027t matter which order I multiply,"},{"Start":"01:16.730 ","End":"01:18.725","Text":"I prefer the number first,"},{"Start":"01:18.725 ","End":"01:22.020","Text":"12 times x minus 3."},{"Start":"01:22.550 ","End":"01:25.220","Text":"I\u0027d like to point out here, that I wouldn\u0027t have used"},{"Start":"01:25.220 ","End":"01:27.560","Text":"the common denominator method myself here,"},{"Start":"01:27.560 ","End":"01:29.330","Text":"there\u0027s some simple cases,"},{"Start":"01:29.330 ","End":"01:34.280","Text":"I\u0027ll show you a numerical example."},{"Start":"01:34.280 ","End":"01:41.725","Text":"If I say that 15 over 5=3,"},{"Start":"01:41.725 ","End":"01:48.870","Text":"then I know right away that 15 is 3x5,"},{"Start":"01:48.870 ","End":"01:53.515","Text":"and I also know that 15/3=5."},{"Start":"01:53.515 ","End":"01:56.450","Text":"In fact, in some schools, these three truths,"},{"Start":"01:56.450 ","End":"01:59.510","Text":"the total at the same time because they\u0027re really all the same thing."},{"Start":"01:59.510 ","End":"02:06.320","Text":"When I say something over something equals something like if I saw a/ b=c,"},{"Start":"02:06.320 ","End":"02:12.585","Text":"I will straight away write that from here that a=b×c,"},{"Start":"02:12.585 ","End":"02:15.275","Text":"the same a,b, and c from the quadratic."},{"Start":"02:15.275 ","End":"02:17.360","Text":"I would tell you if this over this is this,"},{"Start":"02:17.360 ","End":"02:19.055","Text":"then this is this times this."},{"Start":"02:19.055 ","End":"02:24.400","Text":"That\u0027s what I did here. But we did it formerly with the common denominator."},{"Start":"02:24.400 ","End":"02:29.630","Text":"Let\u0027s continue. Let\u0027s open the brackets on the right."},{"Start":"02:29.630 ","End":"02:36.815","Text":"I have here 12x-36,"},{"Start":"02:36.815 ","End":"02:39.400","Text":"and now let\u0027s bring everything to the left,"},{"Start":"02:39.400 ","End":"02:41.410","Text":"so it will look something like this,"},{"Start":"02:41.410 ","End":"02:44.725","Text":"and we get 4x^2."},{"Start":"02:44.725 ","End":"02:47.680","Text":"We can pick the ones we want as we need them."},{"Start":"02:47.680 ","End":"02:48.700","Text":"That\u0027s the x^2 now."},{"Start":"02:48.700 ","End":"02:51.850","Text":"X is I have minus 24 from here."},{"Start":"02:51.850 ","End":"02:55.370","Text":"This goes over to the other side,"},{"Start":"02:55.370 ","End":"02:57.940","Text":"when it does that, it becomes minus."},{"Start":"02:57.940 ","End":"03:00.290","Text":"So, I\u0027ve got minus 36x."},{"Start":"03:01.250 ","End":"03:08.230","Text":"Then here I have a plus 36 and this goes over and becomes a plus 36."},{"Start":"03:08.230 ","End":"03:09.535","Text":"It strengthens."},{"Start":"03:09.535 ","End":"03:14.165","Text":"So it\u0027s plus 72 equals 0."},{"Start":"03:14.165 ","End":"03:17.150","Text":"Remember my advice, if you see something,"},{"Start":"03:17.150 ","End":"03:19.670","Text":"some common factor that stands out,"},{"Start":"03:19.670 ","End":"03:21.455","Text":"just take it out."},{"Start":"03:21.455 ","End":"03:25.565","Text":"4, obviously everything divides by 4 here."},{"Start":"03:25.565 ","End":"03:29.690","Text":"I\u0027m going to divide both sides by 4."},{"Start":"03:29.690 ","End":"03:33.600","Text":"That means I divide each term by 4,"},{"Start":"03:33.600 ","End":"03:35.630","Text":"I\u0027ve got x^2,"},{"Start":"03:35.630 ","End":"03:41.463","Text":"36/4 is 9x,"},{"Start":"03:41.463 ","End":"03:48.230","Text":"72/4 is 18, and 0/4 is just 0."},{"Start":"03:48.230 ","End":"03:51.260","Text":"It\u0027s not that I forgot to divide the right-hand side also,"},{"Start":"03:51.260 ","End":"03:52.910","Text":"I did divide this 0 by 4,"},{"Start":"03:52.910 ","End":"03:56.825","Text":"it just stayed 0. Quadratic equation."},{"Start":"03:56.825 ","End":"03:58.840","Text":"Whip out the formula."},{"Start":"03:58.840 ","End":"04:02.650","Text":"I\u0027ll just get myself a bit more space here."},{"Start":"04:03.290 ","End":"04:06.705","Text":"We identify that."},{"Start":"04:06.705 ","End":"04:14.670","Text":"I\u0027ll do it over here, a=1 because x^2 is 1x^2,"},{"Start":"04:14.670 ","End":"04:18.990","Text":"b=-9 because it\u0027s +b,"},{"Start":"04:18.990 ","End":"04:26.730","Text":"so it\u0027s plus minus 9, and c=18."},{"Start":"04:26.730 ","End":"04:29.400","Text":"Let\u0027s see what that gives us."},{"Start":"04:29.400 ","End":"04:32.010","Text":"X=-b and it\u0027s minus,"},{"Start":"04:32.010 ","End":"04:33.195","Text":"so it\u0027s plus 9,"},{"Start":"04:33.195 ","End":"04:35.030","Text":"I\u0027ll write plus 9, just to emphasize,"},{"Start":"04:35.030 ","End":"04:37.860","Text":"I didn\u0027t forget the minus, plus,"},{"Start":"04:37.860 ","End":"04:40.875","Text":"or minus the square root of b^2."},{"Start":"04:40.875 ","End":"04:44.180","Text":"That\u0027s supposed to put (-9)^2 But you\u0027ll forgive me if"},{"Start":"04:44.180 ","End":"04:47.840","Text":"I just put 9^2 because negative times negative is positive,"},{"Start":"04:47.840 ","End":"04:52.340","Text":"-4×a, which is 1,"},{"Start":"04:52.340 ","End":"04:55.210","Text":"times c, which is 18."},{"Start":"04:55.210 ","End":"05:02.060","Text":"Instead of writing dot, I can put brackets for multiplication over 2a."},{"Start":"05:02.060 ","End":"05:07.310","Text":"2 is this, a is 1, not 21."},{"Start":"05:07.310 ","End":"05:12.410","Text":"2×1. I\u0027d like to compute"},{"Start":"05:12.410 ","End":"05:17.615","Text":"what\u0027s under the square root sign at the side. Let\u0027s see what we get."},{"Start":"05:17.615 ","End":"05:29.180","Text":"9^2 is 81, 4×18 is 72."},{"Start":"05:29.180 ","End":"05:32.150","Text":"Also, know this because here I got 72 divided by 4."},{"Start":"05:32.150 ","End":"05:40.560","Text":"Yeah, so it\u0027s 81-72, which equals 9."},{"Start":"05:40.560 ","End":"05:43.460","Text":"I even like to go one step further and say this is"},{"Start":"05:43.460 ","End":"05:46.760","Text":"3^2 because that\u0027ll be useful when I do the square root."},{"Start":"05:46.760 ","End":"05:49.150","Text":"Okay, back here."},{"Start":"05:49.150 ","End":"05:58.410","Text":"x=9 plus or minus the square root of 9 is 3 already."},{"Start":"05:58.410 ","End":"06:02.895","Text":"2×1 is 2, take two paths."},{"Start":"06:02.895 ","End":"06:05.055","Text":"Take the plus,"},{"Start":"06:05.055 ","End":"06:10.110","Text":"9+3 is 12/2 is 6."},{"Start":"06:10.110 ","End":"06:11.610","Text":"Take the other path,"},{"Start":"06:11.610 ","End":"06:17.140","Text":"9-3 is 6, 6/2 is 3."},{"Start":"06:18.170 ","End":"06:23.165","Text":"It would be tempting to say that there are two solutions,"},{"Start":"06:23.165 ","End":"06:26.645","Text":"x=6, and x=3."},{"Start":"06:26.645 ","End":"06:28.790","Text":"Wait a minute."},{"Start":"06:28.790 ","End":"06:32.750","Text":"I remembered that if we put x=3,"},{"Start":"06:32.750 ","End":"06:35.330","Text":"this will make the denominator be 0."},{"Start":"06:35.330 ","End":"06:36.350","Text":"We have to check."},{"Start":"06:36.350 ","End":"06:39.545","Text":"6 has no problem with the denominator,"},{"Start":"06:39.545 ","End":"06:43.290","Text":"but 3 makes the denominator 0."},{"Start":"06:43.910 ","End":"06:47.075","Text":"I will highlight this with a different color."},{"Start":"06:47.075 ","End":"06:50.300","Text":"This meaning bad and to make it really clear,"},{"Start":"06:50.300 ","End":"06:51.965","Text":"I\u0027ll say yes,"},{"Start":"06:51.965 ","End":"06:53.240","Text":"hooray for 6,"},{"Start":"06:53.240 ","End":"06:56.120","Text":"and boo 3 does not work."},{"Start":"06:56.120 ","End":"07:03.610","Text":"Our answer is x=6 only."},{"Start":"07:03.700 ","End":"07:06.720","Text":"That\u0027s it for this one."}],"ID":5002},{"Watched":false,"Name":"Exercise 3","Duration":"7m 47s","ChapterTopicVideoID":5003,"CourseChapterTopicPlaylistID":3870,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"Here we have an equation and we see a lot of denominators,"},{"Start":"00:03.930 ","End":"00:06.510","Text":"but it doesn\u0027t look like a quadratic."},{"Start":"00:06.510 ","End":"00:08.070","Text":"No x squared here."},{"Start":"00:08.070 ","End":"00:12.315","Text":"Nevertheless, this is one of those disguised quadratics and you\u0027ll soon see."},{"Start":"00:12.315 ","End":"00:16.710","Text":"Let\u0027s work over here. First thing to do is find a common denominator."},{"Start":"00:16.710 ","End":"00:19.260","Text":"We have 3 and 2 and x here."},{"Start":"00:19.260 ","End":"00:24.060","Text":"I can\u0027t see anything less than taking 3 times 2 times"},{"Start":"00:24.060 ","End":"00:29.970","Text":"x. I deliberately didn\u0027t write it as 6x because I would like to see all 3 factors."},{"Start":"00:29.970 ","End":"00:32.730","Text":"It helps, for example, here,"},{"Start":"00:32.730 ","End":"00:35.715","Text":"when I want to know how many times 3 goes into it,"},{"Start":"00:35.715 ","End":"00:39.120","Text":"I can see that it\u0027s 2x times."},{"Start":"00:39.120 ","End":"00:41.010","Text":"Just ignore the 3."},{"Start":"00:41.010 ","End":"00:43.620","Text":"When I want to know what goes with the 2,"},{"Start":"00:43.620 ","End":"00:47.205","Text":"I just ignore the 2 and I get 3x."},{"Start":"00:47.205 ","End":"00:50.190","Text":"For the x, I just get 3 times 2."},{"Start":"00:50.190 ","End":"00:52.990","Text":"This time I\u0027ll write it as 6."},{"Start":"00:55.610 ","End":"00:58.470","Text":"Following the usual procedure,"},{"Start":"00:58.470 ","End":"01:00.590","Text":"but let me also remind you at this point,"},{"Start":"01:00.590 ","End":"01:01.835","Text":"I should have said at the beginning,"},{"Start":"01:01.835 ","End":"01:03.785","Text":"not to forget at the end,"},{"Start":"01:03.785 ","End":"01:09.950","Text":"to check that we can put x in the denominator,"},{"Start":"01:09.950 ","End":"01:12.950","Text":"that we don\u0027t get a 0 there, basically."},{"Start":"01:12.950 ","End":"01:15.155","Text":"In this case, you can easily say reject"},{"Start":"01:15.155 ","End":"01:18.710","Text":"x=0 if you get it and everything else will be okay."},{"Start":"01:18.710 ","End":"01:21.925","Text":"Back here, yeah, multiplying out."},{"Start":"01:21.925 ","End":"01:26.505","Text":"2x times 4x plus 1,"},{"Start":"01:26.505 ","End":"01:28.545","Text":"I need parentheses around the 4x"},{"Start":"01:28.545 ","End":"01:32.655","Text":"plus 1 or brackets because the dividing lines like a bracket,"},{"Start":"01:32.655 ","End":"01:36.630","Text":"also minus and then 3x,"},{"Start":"01:36.630 ","End":"01:40.275","Text":"again brackets x plus 2."},{"Start":"01:40.275 ","End":"01:44.710","Text":"Here 6 times 2."},{"Start":"01:45.020 ","End":"01:51.840","Text":"Expand, 2x times 4x is 8x^2 plus"},{"Start":"01:51.840 ","End":"02:00.400","Text":"2x minus 3x^2 minus 3 times 2,"},{"Start":"02:03.140 ","End":"02:06.835","Text":"6x and equals 12."},{"Start":"02:06.835 ","End":"02:10.730","Text":"Let\u0027s collect the terms in order of first x^2,"},{"Start":"02:10.730 ","End":"02:12.260","Text":"then x, and just numbers."},{"Start":"02:12.260 ","End":"02:13.610","Text":"For x^2, let\u0027s see,"},{"Start":"02:13.610 ","End":"02:15.460","Text":"you have here and here,"},{"Start":"02:15.460 ","End":"02:19.265","Text":"8 minus 3, 5x^2."},{"Start":"02:19.265 ","End":"02:23.735","Text":"Let\u0027s pick the x\u0027s we have from here and from here."},{"Start":"02:23.735 ","End":"02:25.535","Text":"That\u0027s 2 minus 6,"},{"Start":"02:25.535 ","End":"02:30.875","Text":"so it\u0027s minus 4x and numbers just on the right,"},{"Start":"02:30.875 ","End":"02:34.580","Text":"12, but bring it over to the left and it becomes"},{"Start":"02:34.580 ","End":"02:39.790","Text":"minus 12 and what we\u0027re left with on the right is nothing, 0."},{"Start":"02:39.790 ","End":"02:42.605","Text":"Now, it really looks like a quadratic equation."},{"Start":"02:42.605 ","End":"02:47.945","Text":"Let\u0027s go over here and say that a equals 5."},{"Start":"02:47.945 ","End":"02:49.880","Text":"I\u0027m looking over here,"},{"Start":"02:49.880 ","End":"02:52.135","Text":"b equals minus 4,"},{"Start":"02:52.135 ","End":"02:55.230","Text":"and c equals minus 12."},{"Start":"02:55.230 ","End":"03:00.920","Text":"Now, I substitute and get x equals minus b,"},{"Start":"03:00.920 ","End":"03:02.359","Text":"b is already negative,"},{"Start":"03:02.359 ","End":"03:04.250","Text":"so it\u0027s plus 4,"},{"Start":"03:04.250 ","End":"03:10.570","Text":"plus or minus the square root of b^2."},{"Start":"03:10.570 ","End":"03:15.310","Text":"If it\u0027s negative, I can ignore the fact that it\u0027s negative because when I square it,"},{"Start":"03:15.310 ","End":"03:17.050","Text":"it\u0027ll come up positive."},{"Start":"03:17.050 ","End":"03:21.510","Text":"Minus 4, a is 5."},{"Start":"03:21.510 ","End":"03:24.265","Text":"I need to indicate it\u0027s multiplication,"},{"Start":"03:24.265 ","End":"03:28.330","Text":"and c is minus 12."},{"Start":"03:28.330 ","End":"03:32.185","Text":"I can use brackets for multiplication."},{"Start":"03:32.185 ","End":"03:39.275","Text":"All this is over 2a, 2 times 5."},{"Start":"03:39.275 ","End":"03:42.475","Text":"Let\u0027s see what we have under the square root sign, we have this."},{"Start":"03:42.475 ","End":"03:45.124","Text":"I\u0027d like to compute this at the side."},{"Start":"03:45.124 ","End":"03:53.970","Text":"4^2 is 16, 4 times 5 times 12 is 20 times 12 is 240."},{"Start":"03:53.970 ","End":"03:56.880","Text":"We have minus and minus, so it\u0027s plus,"},{"Start":"03:56.880 ","End":"04:01.455","Text":"this comes out to be 256."},{"Start":"04:01.455 ","End":"04:07.935","Text":"I know that 256 is 16 squared,"},{"Start":"04:07.935 ","End":"04:12.310","Text":"especially if you\u0027re into computers and powers of 2, you would know this."},{"Start":"04:12.310 ","End":"04:14.545","Text":"Many people know this anyway."},{"Start":"04:14.545 ","End":"04:18.505","Text":"Let\u0027s say you\u0027re using a calculator later for the square root."},{"Start":"04:18.505 ","End":"04:23.255","Text":"Back here, what do we get?"},{"Start":"04:23.255 ","End":"04:25.760","Text":"We get 4,"},{"Start":"04:25.760 ","End":"04:28.645","Text":"I\u0027m running out of space here,"},{"Start":"04:28.645 ","End":"04:30.436","Text":"let me just scroll a bit,"},{"Start":"04:30.436 ","End":"04:38.950","Text":"plus or minus the square root"},{"Start":"04:38.950 ","End":"04:41.725","Text":"of this is just 16."},{"Start":"04:41.725 ","End":"04:46.270","Text":"That\u0027s the reason when I saw that I know that it\u0027s something squared, I wrote it."},{"Start":"04:46.270 ","End":"04:50.965","Text":"Otherwise, I do on the calculator square root of 256, I\u0027ll get 16."},{"Start":"04:50.965 ","End":"04:54.025","Text":"Over 2 times 5 is 10."},{"Start":"04:54.025 ","End":"04:57.370","Text":"Now, we take the fork in the road or"},{"Start":"04:57.370 ","End":"05:01.539","Text":"the left side is the plus and the right side is the minus."},{"Start":"05:01.539 ","End":"05:06.080","Text":"4 plus 16 is 20/10 is 2,"},{"Start":"05:06.080 ","End":"05:12.115","Text":"4 minus 16 is minus 12/10."},{"Start":"05:12.115 ","End":"05:16.175","Text":"A decimal that comes out nicely,"},{"Start":"05:16.175 ","End":"05:19.340","Text":"minus 1.2, but I think we want it as a fraction,"},{"Start":"05:19.340 ","End":"05:24.800","Text":"so let me write it as minus 1 and 2/10,"},{"Start":"05:24.800 ","End":"05:29.030","Text":"which is minus 1 and 1/5."},{"Start":"05:29.030 ","End":"05:31.550","Text":"I have 2 answers,"},{"Start":"05:31.550 ","End":"05:35.670","Text":"2 and minus 1 and 1/5."},{"Start":"05:35.670 ","End":"05:38.330","Text":"Notice that neither of them is 0,"},{"Start":"05:38.330 ","End":"05:42.210","Text":"so we have no problems with denominators."},{"Start":"05:43.400 ","End":"05:46.690","Text":"Let\u0027s just do a check."},{"Start":"05:47.450 ","End":"05:50.670","Text":"I\u0027ll check on 1 of them. We\u0027ll check on the whole number 1."},{"Start":"05:50.670 ","End":"05:53.165","Text":"It\u0027s easier, so it\u0027s better than nothing."},{"Start":"05:53.165 ","End":"05:57.150","Text":"Let\u0027s see if we got the right answer. Let\u0027s see."},{"Start":"05:57.150 ","End":"06:06.515","Text":"I\u0027ll take it from here and I\u0027ll substitute x equals 2 and see what we get."},{"Start":"06:06.515 ","End":"06:12.685","Text":"The first term is 4 times 2 plus 1 is 9/3."},{"Start":"06:12.685 ","End":"06:15.705","Text":"Then we have a minus,"},{"Start":"06:15.705 ","End":"06:21.360","Text":"then (2+2)/2 is 4/2"},{"Start":"06:21.360 ","End":"06:27.395","Text":"but I put equals with a question mark because we don\u0027t know this is what we\u0027re checking."},{"Start":"06:27.395 ","End":"06:28.130","Text":"Is it"},{"Start":"06:28.130 ","End":"06:38.120","Text":"equal to 2/2?"},{"Start":"06:38.120 ","End":"06:41.520","Text":"A dot from the question mark came on the line, Oh, never mind."},{"Start":"06:41.520 ","End":"06:45.810","Text":"9/3 is 3,"},{"Start":"06:45.810 ","End":"06:52.590","Text":"4/2 is 2, 2/2 is 1."},{"Start":"06:52.590 ","End":"06:59.205","Text":"What we\u0027re asking is 3 minus 2 equal by any chance to 1?"},{"Start":"06:59.205 ","End":"07:01.650","Text":"The answer is yes."},{"Start":"07:01.650 ","End":"07:06.250","Text":"At least the 2 is the right answer."},{"Start":"07:06.970 ","End":"07:09.650","Text":"Let\u0027s do levels of checking."},{"Start":"07:09.650 ","End":"07:16.790","Text":"What I\u0027m saying is you can just check that nothing is 0 in the denominator. That\u0027s fine."},{"Start":"07:16.790 ","End":"07:21.085","Text":"You can check that they are also solutions."},{"Start":"07:21.085 ","End":"07:24.275","Text":"I with the middle approach just checked on one of them,"},{"Start":"07:24.275 ","End":"07:26.600","Text":"just so there\u0027s no gross mistake,"},{"Start":"07:26.600 ","End":"07:27.680","Text":"but if I had time,"},{"Start":"07:27.680 ","End":"07:29.975","Text":"I\u0027d do the other one also."},{"Start":"07:29.975 ","End":"07:33.845","Text":"That\u0027s what I suggest you do on exams if you have any time leftover,"},{"Start":"07:33.845 ","End":"07:37.235","Text":"is for all the equations to substitute"},{"Start":"07:37.235 ","End":"07:42.110","Text":"the answer in the original equation and see if you get a true statement."},{"Start":"07:42.110 ","End":"07:44.495","Text":"That\u0027s it for this one."},{"Start":"07:44.495 ","End":"07:47.520","Text":"Onto the next and the next clip."}],"ID":5003},{"Watched":false,"Name":"Exercise 4","Duration":"6m 51s","ChapterTopicVideoID":5004,"CourseChapterTopicPlaylistID":3870,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.250","Text":"Here we have an exercise,"},{"Start":"00:02.250 ","End":"00:04.140","Text":"it has a denominator."},{"Start":"00:04.140 ","End":"00:07.785","Text":"It\u0027s in the section on disguised quadratics."},{"Start":"00:07.785 ","End":"00:09.555","Text":"That\u0027s what was supposed to get,"},{"Start":"00:09.555 ","End":"00:11.430","Text":"but it\u0027s a bit tricky."},{"Start":"00:11.430 ","End":"00:13.620","Text":"When I first tried to solve it by myself,"},{"Start":"00:13.620 ","End":"00:15.570","Text":"I thought it\u0027s straightforward."},{"Start":"00:15.570 ","End":"00:18.270","Text":"It\u0027s x plus 3 is a common denominator,"},{"Start":"00:18.270 ","End":"00:21.240","Text":"I multiply both sides by x plus 3."},{"Start":"00:21.240 ","End":"00:25.260","Text":"But notice that if you multiply the right-hand side by x plus 3,"},{"Start":"00:25.260 ","End":"00:29.820","Text":"you\u0027ll get x cubed and there won\u0027t be anything to cancel with that x cubed."},{"Start":"00:29.820 ","End":"00:32.805","Text":"We haven\u0027t learned to solve cubic equations."},{"Start":"00:32.805 ","End":"00:37.290","Text":"I got stuck. I had to look again and I looked a bit harder."},{"Start":"00:37.290 ","End":"00:43.070","Text":"Then I noticed, if you look at the numerator and denominator here,"},{"Start":"00:43.070 ","End":"00:45.035","Text":"something rang a bell,"},{"Start":"00:45.035 ","End":"00:47.075","Text":"difference of squares formula."},{"Start":"00:47.075 ","End":"00:49.700","Text":"Let me show you what I mean."},{"Start":"00:49.700 ","End":"00:53.900","Text":"There was a formula that a squared minus b"},{"Start":"00:53.900 ","End":"01:00.930","Text":"squared equals a minus b, a plus b."},{"Start":"01:00.930 ","End":"01:04.290","Text":"Not the same a and b as here totally different."},{"Start":"01:04.290 ","End":"01:07.890","Text":"The difference of squares formula."},{"Start":"01:07.890 ","End":"01:11.906","Text":"Then if you look"},{"Start":"01:11.906 ","End":"01:16.410","Text":"at this and see it is x squared minus 3 squared looks like something could cancel."},{"Start":"01:16.410 ","End":"01:22.490","Text":"I\u0027m going to go with this approach because just multiplying out leads to a dead end."},{"Start":"01:22.490 ","End":"01:25.565","Text":"Like I said, I don\u0027t want to go through it with you."},{"Start":"01:25.565 ","End":"01:34.735","Text":"Let\u0027s do that, x minus 3, x plus 3."},{"Start":"01:34.735 ","End":"01:38.930","Text":"That\u0027s what happens when I interpret this in here."},{"Start":"01:38.930 ","End":"01:42.050","Text":"Of course, I am looking at 9 as 3 squared in"},{"Start":"01:42.050 ","End":"01:45.680","Text":"order to be able to apply the formula with b as 3,"},{"Start":"01:45.680 ","End":"01:49.100","Text":"then over x plus"},{"Start":"01:49.100 ","End":"01:58.030","Text":"3 plus x equals x squared minus 18."},{"Start":"01:58.930 ","End":"02:01.160","Text":"Now if we look at this,"},{"Start":"02:01.160 ","End":"02:04.955","Text":"we have a factor that\u0027s in the numerator and in the denominator,"},{"Start":"02:04.955 ","End":"02:07.160","Text":"that\u0027s the x plus 3."},{"Start":"02:07.160 ","End":"02:09.640","Text":"I can cancel it out,"},{"Start":"02:09.640 ","End":"02:12.260","Text":"but I\u0027ll have to give a little warning also."},{"Start":"02:12.260 ","End":"02:13.715","Text":"Let\u0027s first of all,"},{"Start":"02:13.715 ","End":"02:18.335","Text":"just put a line through here and a line through here."},{"Start":"02:18.335 ","End":"02:25.610","Text":"The reason it\u0027s not so simple is that if x plus 3 was 0,"},{"Start":"02:25.610 ","End":"02:28.352","Text":"then this thing wouldn\u0027t be defined,"},{"Start":"02:28.352 ","End":"02:31.415","Text":"you couldn\u0027t divide by 0."},{"Start":"02:31.415 ","End":"02:33.605","Text":"But after I\u0027ve canceled,"},{"Start":"02:33.605 ","End":"02:35.060","Text":"there\u0027s no restriction anymore."},{"Start":"02:35.060 ","End":"02:39.560","Text":"Let me just make a note here that we have to check at"},{"Start":"02:39.560 ","End":"02:44.420","Text":"the end that x plus 3 is not equal to 0 with whatever x we get."},{"Start":"02:44.420 ","End":"02:48.650","Text":"In other words, x is not equal to minus 3."},{"Start":"02:48.650 ","End":"02:51.290","Text":"We get minus 3 will have to rule it out."},{"Start":"02:51.290 ","End":"02:54.155","Text":"Other than that, we proceed as usual."},{"Start":"02:54.155 ","End":"02:57.590","Text":"What I have now is just x minus 3,"},{"Start":"02:57.590 ","End":"02:59.000","Text":"that\u0027s all it\u0027s left from here,"},{"Start":"02:59.000 ","End":"03:03.935","Text":"plus x equals x squared minus 18."},{"Start":"03:03.935 ","End":"03:06.110","Text":"Now I want to make it look like this."},{"Start":"03:06.110 ","End":"03:10.835","Text":"Just for everything onto the left-hand side and x squared first,"},{"Start":"03:10.835 ","End":"03:12.650","Text":"let\u0027s pick the x squared."},{"Start":"03:12.650 ","End":"03:19.680","Text":"This one comes over as minus x squared and then what do I have?"},{"Start":"03:19.680 ","End":"03:21.150","Text":"I need x\u0027s,"},{"Start":"03:21.150 ","End":"03:27.535","Text":"the x\u0027s I have here and here 2x they\u0027re on the correct side."},{"Start":"03:27.535 ","End":"03:30.835","Text":"Then numbers, I\u0027m going to have to throw this over,"},{"Start":"03:30.835 ","End":"03:40.625","Text":"there\u0027ll be plus 18 together with the minus 3 will be plus 15 equals 0."},{"Start":"03:40.625 ","End":"03:44.020","Text":"Let\u0027s continue over here where the formula is."},{"Start":"03:44.020 ","End":"03:47.410","Text":"In our case we have that a equals minus 1,"},{"Start":"03:47.410 ","End":"03:49.030","Text":"that\u0027s minus 1x squared,"},{"Start":"03:49.030 ","End":"03:51.910","Text":"and b is 2,"},{"Start":"03:51.910 ","End":"03:56.955","Text":"and c is 15."},{"Start":"03:56.955 ","End":"04:02.790","Text":"We get x equals minus b,"},{"Start":"04:02.790 ","End":"04:12.510","Text":"that\u0027s minus 2 plus or minus the square root of b squared minus 4 times a is"},{"Start":"04:12.510 ","End":"04:16.980","Text":"minus 1 times c is"},{"Start":"04:16.980 ","End":"04:23.040","Text":"15 over 2a,"},{"Start":"04:23.040 ","End":"04:27.025","Text":"which is twice minus 1."},{"Start":"04:27.025 ","End":"04:34.260","Text":"Let\u0027s compute what\u0027s under the square root sign at the side. Keep things tidier."},{"Start":"04:34.490 ","End":"04:38.760","Text":"Now, 2^2 is 4,"},{"Start":"04:38.760 ","End":"04:45.805","Text":"so I have 4, and then 4 times 1 times 15 is 60."},{"Start":"04:45.805 ","End":"04:48.560","Text":"Do I want it plus or minus?"},{"Start":"04:48.560 ","End":"04:51.170","Text":"Well, there\u0027s a minus here and a minus 1 here,"},{"Start":"04:51.170 ","End":"04:54.890","Text":"so even number of minuses is a plus,"},{"Start":"04:54.890 ","End":"05:03.670","Text":"that gives me 64 and I already see that 64 is 8 squared."},{"Start":"05:03.670 ","End":"05:08.510","Text":"I\u0027m writing this because I\u0027ll need the square root of 64,"},{"Start":"05:08.510 ","End":"05:12.370","Text":"so I won\u0027t need to do it on the calculator because I can see it\u0027s going to be 8."},{"Start":"05:12.370 ","End":"05:21.475","Text":"Back here, x equals minus 2 plus or minus 8,"},{"Start":"05:21.475 ","End":"05:24.905","Text":"because the square root of 64 is 8 from here,"},{"Start":"05:24.905 ","End":"05:28.910","Text":"over 2 times minus 1 is minus 2."},{"Start":"05:28.910 ","End":"05:32.420","Text":"So let\u0027s take the plus and the minus cases."},{"Start":"05:32.420 ","End":"05:34.505","Text":"If I take the plus,"},{"Start":"05:34.505 ","End":"05:38.330","Text":"I have minus 2 plus 8 is 6,"},{"Start":"05:38.330 ","End":"05:43.500","Text":"6 over minus 2 is minus 3."},{"Start":"05:43.500 ","End":"05:47.450","Text":"If I take the other case of the minus,"},{"Start":"05:47.450 ","End":"05:49.250","Text":"I\u0027ve got minus 2 minus 8,"},{"Start":"05:49.250 ","End":"05:50.390","Text":"which is minus 10."},{"Start":"05:50.390 ","End":"05:56.440","Text":"Minus 10 over minus 2 is the same as 10/2 because it\u0027s plus,"},{"Start":"05:56.440 ","End":"05:59.290","Text":"so that would be 5."},{"Start":"05:59.290 ","End":"06:03.245","Text":"It looks like we\u0027ve solved this question and I would"},{"Start":"06:03.245 ","End":"06:07.700","Text":"like to say x could be 5 or minus 3,"},{"Start":"06:07.700 ","End":"06:11.930","Text":"but I\u0027m holding back on this minus 3 because I"},{"Start":"06:11.930 ","End":"06:17.565","Text":"remember that at the beginning we said here it is."},{"Start":"06:17.565 ","End":"06:22.275","Text":"That we can\u0027t have x being minus 3."},{"Start":"06:22.275 ","End":"06:24.685","Text":"Because if x is minus 3,"},{"Start":"06:24.685 ","End":"06:28.669","Text":"the original question is not defined."},{"Start":"06:28.669 ","End":"06:35.464","Text":"You cannot substitute minus 3 in here because an x plus 3 is 0 and we\u0027re dividing by 0."},{"Start":"06:35.464 ","End":"06:38.195","Text":"This is off limits,"},{"Start":"06:38.195 ","End":"06:40.460","Text":"I\u0027ll put a cross through it,"},{"Start":"06:40.460 ","End":"06:42.349","Text":"that is not allowed."},{"Start":"06:42.349 ","End":"06:50.280","Text":"Our answer is just x equals 5, and we\u0027re done."}],"ID":5004},{"Watched":false,"Name":"Exercise 5","Duration":"12m 44s","ChapterTopicVideoID":5005,"CourseChapterTopicPlaylistID":3870,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.085","Text":"Here I have an equation to solve."},{"Start":"00:02.085 ","End":"00:05.670","Text":"The thing that\u0027s most prominent is all these denominators."},{"Start":"00:05.670 ","End":"00:07.935","Text":"Anyway, copy the exercise over here."},{"Start":"00:07.935 ","End":"00:13.410","Text":"The first task will be to find a common denominator for all these."},{"Start":"00:13.410 ","End":"00:16.140","Text":"There\u0027s a method."},{"Start":"00:16.140 ","End":"00:18.210","Text":"I have already used it in the past."},{"Start":"00:18.210 ","End":"00:21.600","Text":"You\u0027ve seen it. Where we factorize each one and that way we can"},{"Start":"00:21.600 ","End":"00:26.115","Text":"save just multiplying it all out and getting a huge denominator."},{"Start":"00:26.115 ","End":"00:28.740","Text":"Let\u0027s take the 3 denominators separately."},{"Start":"00:28.740 ","End":"00:32.310","Text":"Let\u0027s see. We have a 2x plus 2."},{"Start":"00:32.310 ","End":"00:34.515","Text":"I\u0027ll just write them first,"},{"Start":"00:34.515 ","End":"00:43.050","Text":"2(x minus 1)^2 and the other denominator is 1 minus x^2."},{"Start":"00:43.050 ","End":"00:45.930","Text":"I want to factorize each of the denominators."},{"Start":"00:45.930 ","End":"00:53.780","Text":"Here easily you can see that I take the 2 out and get left with x plus 1."},{"Start":"00:53.780 ","End":"00:56.525","Text":"Here we already have a 2,"},{"Start":"00:56.525 ","End":"00:58.300","Text":"and x minus 1^2,"},{"Start":"00:58.300 ","End":"01:00.620","Text":"I\u0027ll just write it as x minus 1,"},{"Start":"01:00.620 ","End":"01:03.320","Text":"x minus 1,"},{"Start":"01:03.320 ","End":"01:06.570","Text":"and 1 minus x^2,"},{"Start":"01:06.800 ","End":"01:11.250","Text":"it\u0027s like 1^2 minus x^2."},{"Start":"01:11.250 ","End":"01:12.915","Text":"I\u0027ll do that at the side."},{"Start":"01:12.915 ","End":"01:16.257","Text":"1^2 minus x^2."},{"Start":"01:16.257 ","End":"01:22.453","Text":"I\u0027ll just move this a bit to the left, there we go."},{"Start":"01:22.453 ","End":"01:27.500","Text":"Is equal to 1 plus x times 1 minus x."},{"Start":"01:27.500 ","End":"01:29.000","Text":"If you\u0027re wondering how I got this,"},{"Start":"01:29.000 ","End":"01:32.510","Text":"you may have forgotten the a^2 minus b^2,"},{"Start":"01:32.510 ","End":"01:34.705","Text":"the difference of squares formula,"},{"Start":"01:34.705 ","End":"01:37.332","Text":"a plus b, a minus b,"},{"Start":"01:37.332 ","End":"01:39.065","Text":"here it is just as a reminder."},{"Start":"01:39.065 ","End":"01:43.063","Text":"Anyway, getting back here I\u0027ll just copy this here,"},{"Start":"01:43.063 ","End":"01:47.900","Text":"but I\u0027d also like to use a trick I\u0027ve used before."},{"Start":"01:47.900 ","End":"01:51.725","Text":"The trick is this. When you have a difference,"},{"Start":"01:51.725 ","End":"01:56.375","Text":"you can reverse the order provided you also throw in a minus."},{"Start":"01:56.375 ","End":"02:02.065","Text":"This is actually equal to minus of x minus 1."},{"Start":"02:02.065 ","End":"02:06.410","Text":"You can always check by multiplying by minus,"},{"Start":"02:06.410 ","End":"02:10.390","Text":"we get minus x plus 1 which is the same as 1 minus x."},{"Start":"02:10.390 ","End":"02:15.800","Text":"The reason I do this is because I see that I already have an x minus 1,"},{"Start":"02:15.800 ","End":"02:17.930","Text":"so I don\u0027t need also a 1 minus x,"},{"Start":"02:17.930 ","End":"02:20.360","Text":"the only difference is a minus."},{"Start":"02:20.360 ","End":"02:22.250","Text":"Also, of course,"},{"Start":"02:22.250 ","End":"02:25.415","Text":"if it\u0027s a plus, I can just change the order freely."},{"Start":"02:25.415 ","End":"02:29.645","Text":"This will just be x plus 1 and I\u0027ll leave it in brackets."},{"Start":"02:29.645 ","End":"02:33.210","Text":"Here I just change the order, nothing else."},{"Start":"02:33.210 ","End":"02:35.280","Text":"Here I change the order and throw in a minus."},{"Start":"02:35.280 ","End":"02:36.795","Text":"That\u0027s what you\u0027re doing when there\u0027s a difference."},{"Start":"02:36.795 ","End":"02:38.750","Text":"Now going back to here,"},{"Start":"02:38.750 ","End":"02:44.020","Text":"what I have is I\u0027ll throw the minus right upfront."},{"Start":"02:44.020 ","End":"02:50.395","Text":"I have a minus x plus 1 times x minus 1."},{"Start":"02:50.395 ","End":"02:54.780","Text":"The minus that was in front of the x minus 1 I moved it up to the front."},{"Start":"02:55.090 ","End":"02:58.550","Text":"Now I look at these 3 things and I try to see"},{"Start":"02:58.550 ","End":"03:02.390","Text":"what do I have here altogether that I need to take?"},{"Start":"03:02.390 ","End":"03:06.800","Text":"Well, I have x minus 1 twice,"},{"Start":"03:06.800 ","End":"03:09.785","Text":"here once and twice."},{"Start":"03:09.785 ","End":"03:16.575","Text":"I have an x plus 1 here and here,"},{"Start":"03:16.575 ","End":"03:19.755","Text":"and oops, I forgot to mark the minus 1 here."},{"Start":"03:19.755 ","End":"03:22.770","Text":"This is also an x minus 1."},{"Start":"03:22.770 ","End":"03:26.090","Text":"Minus is not that important,"},{"Start":"03:26.090 ","End":"03:29.075","Text":"but I also need the 2,"},{"Start":"03:29.075 ","End":"03:31.895","Text":"I use this color."},{"Start":"03:31.895 ","End":"03:34.265","Text":"Hard to see but there we are."},{"Start":"03:34.265 ","End":"03:40.860","Text":"What I need is to make sure for the common denominator,"},{"Start":"03:42.830 ","End":"03:46.250","Text":"is that I have all of these represented,"},{"Start":"03:46.250 ","End":"03:49.250","Text":"but I don\u0027t need unnecessary duplication."},{"Start":"03:49.250 ","End":"03:51.110","Text":"I need a 2,"},{"Start":"03:51.110 ","End":"03:57.035","Text":"I need an x plus 1,"},{"Start":"03:57.035 ","End":"04:04.670","Text":"and I need x minus 1 twice in order to cover this case."},{"Start":"04:04.670 ","End":"04:08.705","Text":"The minus we just put in as needed."},{"Start":"04:08.705 ","End":"04:12.770","Text":"I\u0027ll just highlight this also, and there we are."},{"Start":"04:12.770 ","End":"04:15.080","Text":"This is a common denominator."},{"Start":"04:15.080 ","End":"04:20.700","Text":"Now, when we multiply this common denominator by all of the terms here,"},{"Start":"04:20.880 ","End":"04:24.845","Text":"part of the common denominator will be used to canceling"},{"Start":"04:24.845 ","End":"04:28.483","Text":"the denominator in each of the 3 cases,"},{"Start":"04:28.483 ","End":"04:30.545","Text":"and we\u0027ll have to see what\u0027s left."},{"Start":"04:30.545 ","End":"04:34.175","Text":"Let\u0027s look at the first one, the 2x minus 2."},{"Start":"04:34.175 ","End":"04:38.510","Text":"This is it. What is it still missing to become this?"},{"Start":"04:38.510 ","End":"04:42.830","Text":"We see it\u0027s missing x minus 1, x minus 1."},{"Start":"04:42.830 ","End":"04:47.390","Text":"Over here I\u0027ll write the bit that\u0027s missing which is the x minus 1,"},{"Start":"04:47.390 ","End":"04:49.045","Text":"x minus 1,"},{"Start":"04:49.045 ","End":"04:52.770","Text":"also known as (x minus 1)^2."},{"Start":"04:52.770 ","End":"04:59.600","Text":"Now here, I have 2(x minus 1)^2."},{"Start":"04:59.600 ","End":"05:03.200","Text":"Already I have the (x minus 1)^2,"},{"Start":"05:03.200 ","End":"05:06.035","Text":"all I\u0027m missing is the x plus 1."},{"Start":"05:06.035 ","End":"05:10.550","Text":"This bit, the top needs to be multiplied by x plus 1."},{"Start":"05:10.550 ","End":"05:12.395","Text":"Let me leave it in brackets."},{"Start":"05:12.395 ","End":"05:15.160","Text":"Here I have the (1 minus x)^2,"},{"Start":"05:15.160 ","End":"05:18.960","Text":"so I have this and this,"},{"Start":"05:18.960 ","End":"05:29.720","Text":"and what I\u0027m missing is the 2 and I\u0027m missing one of the x minus 1s."},{"Start":"05:29.720 ","End":"05:32.930","Text":"But I also have a minus,"},{"Start":"05:32.930 ","End":"05:34.870","Text":"so I have to take care of the sign."},{"Start":"05:34.870 ","End":"05:40.683","Text":"What I have to throw in here is minus the 2,"},{"Start":"05:40.683 ","End":"05:43.528","Text":"did we say, and another x minus 1."},{"Start":"05:43.528 ","End":"05:51.715","Text":"The minus when I multiply by or divide by will just change its sign to what this was."},{"Start":"05:51.715 ","End":"05:52.960","Text":"If I multiply this by minus,"},{"Start":"05:52.960 ","End":"05:55.636","Text":"the minus goes away."},{"Start":"05:55.636 ","End":"06:01.255","Text":"The right-hand side, I have to multiply by the whole common denominator,"},{"Start":"06:01.255 ","End":"06:04.635","Text":"which is this, but it doesn\u0027t matter."},{"Start":"06:04.635 ","End":"06:07.865","Text":"It doesn\u0027t matter whatever I multiply by,"},{"Start":"06:07.865 ","End":"06:09.455","Text":"which would be this,"},{"Start":"06:09.455 ","End":"06:12.530","Text":"except that it\u0027s 0, so don\u0027t need to,"},{"Start":"06:12.530 ","End":"06:14.885","Text":"because everything times 0 is 0."},{"Start":"06:14.885 ","End":"06:17.585","Text":"Hope you followed that bit with the 0."},{"Start":"06:17.585 ","End":"06:23.355","Text":"I\u0027m going to expand,"},{"Start":"06:23.355 ","End":"06:24.960","Text":"and let\u0027s do this carefully."},{"Start":"06:24.960 ","End":"06:26.220","Text":"We have this times this,"},{"Start":"06:26.220 ","End":"06:27.840","Text":"I\u0027ll put the 3 in front."},{"Start":"06:27.840 ","End":"06:34.230","Text":"For the first thing I have 3 times (x minus 1)^2."},{"Start":"06:34.230 ","End":"06:39.060","Text":"Next, I have minus,"},{"Start":"06:39.060 ","End":"06:44.870","Text":"and then the x plus 1 from here and the 2x minus 5,"},{"Start":"06:44.870 ","End":"06:49.228","Text":"which has to be put in brackets from here,"},{"Start":"06:49.228 ","End":"06:50.431","Text":"and then minus,"},{"Start":"06:50.431 ","End":"06:57.198","Text":"straightaway I\u0027m going to combine this minus with this minus and make it a plus,"},{"Start":"06:57.198 ","End":"07:01.790","Text":"and I\u0027ll emphasize the plus so I haven\u0027t forgotten the minuses,"},{"Start":"07:01.790 ","End":"07:11.325","Text":"and I get 2 and I\u0027ll put the 4 in front,"},{"Start":"07:11.325 ","End":"07:17.240","Text":"times 4 times x minus 1,"},{"Start":"07:17.240 ","End":"07:23.040","Text":"and equals, we said that 0 times the common denominator is still 0."},{"Start":"07:23.260 ","End":"07:27.060","Text":"Now let\u0027s open brackets."},{"Start":"07:29.750 ","End":"07:35.580","Text":"The first thing I have to do is multiply this out and I\u0027m going to use the formula,"},{"Start":"07:35.620 ","End":"07:41.300","Text":"which I\u0027ll write over here that (a minus b)^2 is a^2"},{"Start":"07:41.300 ","End":"07:47.000","Text":"minus 2ab plus b^2 in case you forgot."},{"Start":"07:47.000 ","End":"07:54.051","Text":"Back here, this is x^2 minus 2 times x times 1 plus 1^2,"},{"Start":"07:54.051 ","End":"07:56.170","Text":"this is what we get."},{"Start":"07:56.170 ","End":"08:00.230","Text":"Then here we have minus,"},{"Start":"08:00.230 ","End":"08:02.495","Text":"and I\u0027ll multiply this out, I\u0027ll tell you why."},{"Start":"08:02.495 ","End":"08:05.795","Text":"I\u0027ll leave the minus but I\u0027ll multiply it all out in brackets,"},{"Start":"08:05.795 ","End":"08:09.580","Text":"so x times 2x is 2x^2,"},{"Start":"08:09.580 ","End":"08:13.290","Text":"x times minus 5 is minus 5x,"},{"Start":"08:13.290 ","End":"08:19.494","Text":"plus 1 times 2x and plus 1 times minus 5,"},{"Start":"08:19.494 ","End":"08:24.285","Text":"and then the 2 times the 4 I\u0027ll think of as 8,"},{"Start":"08:24.285 ","End":"08:26.985","Text":"so I get 8 times x,"},{"Start":"08:26.985 ","End":"08:28.900","Text":"8 times minus 1,"},{"Start":"08:28.900 ","End":"08:30.965","Text":"all this equals 0."},{"Start":"08:30.965 ","End":"08:35.485","Text":"I still have to do another level of expanding."},{"Start":"08:35.485 ","End":"08:42.840","Text":"3x^2 minus 3 times 2 is 6x plus 3,"},{"Start":"08:42.840 ","End":"08:44.855","Text":"the minus is a minus 1,"},{"Start":"08:44.855 ","End":"08:46.890","Text":"means I reverse all the signs,"},{"Start":"08:46.890 ","End":"08:54.240","Text":"so minus 2x^2 plus 5x minus 2x plus 5,"},{"Start":"08:54.240 ","End":"09:01.280","Text":"and then plus 8x minus 8 equals 0."},{"Start":"09:01.280 ","End":"09:10.490","Text":"I think I\u0027ll just continue with scrolling a bit. Let\u0027s see now."},{"Start":"09:10.490 ","End":"09:12.620","Text":"Let\u0027s collect all the x^2s together."},{"Start":"09:12.620 ","End":"09:14.450","Text":"Where do I have x^2?"},{"Start":"09:14.450 ","End":"09:22.320","Text":"I have x^2 here and here,"},{"Start":"09:22.320 ","End":"09:26.848","Text":"and x\u0027s I have here,"},{"Start":"09:26.848 ","End":"09:30.925","Text":"and here, and here, and here."},{"Start":"09:30.925 ","End":"09:33.260","Text":"The rest of it is numbers,"},{"Start":"09:33.260 ","End":"09:35.570","Text":"here, here, and here."},{"Start":"09:35.570 ","End":"09:38.750","Text":"Let\u0027s collect the like terms,"},{"Start":"09:38.750 ","End":"09:40.910","Text":"the green ones first."},{"Start":"09:40.910 ","End":"09:46.960","Text":"Let\u0027s see, 3x^2 minus 2x^2, x^2."},{"Start":"09:46.960 ","End":"09:50.790","Text":"Just counting the coefficients of the x\u0027s,"},{"Start":"09:50.790 ","End":"09:52.170","Text":"minus 6 plus 5,"},{"Start":"09:52.170 ","End":"09:54.015","Text":"minus 2 plus 8."},{"Start":"09:54.015 ","End":"09:55.860","Text":"Let\u0027s do the positives,"},{"Start":"09:55.860 ","End":"09:57.870","Text":"5 and 8 is 13,"},{"Start":"09:57.870 ","End":"10:02.550","Text":"the negative 6 and 2 is 8,"},{"Start":"10:02.550 ","End":"10:06.060","Text":"13 minus 8 is 5."},{"Start":"10:06.060 ","End":"10:08.295","Text":"Let me just recheck that."},{"Start":"10:08.295 ","End":"10:12.860","Text":"8 and 5 is 13."},{"Start":"10:12.860 ","End":"10:15.070","Text":"6 plus 2 is 8,13 minus 8."},{"Start":"10:15.070 ","End":"10:17.390","Text":"Numbers, that\u0027s the yellow ones,"},{"Start":"10:17.390 ","End":"10:22.260","Text":"plus 3 plus 5 is 8,"},{"Start":"10:22.260 ","End":"10:25.305","Text":"minus 8, it\u0027s 0."},{"Start":"10:25.305 ","End":"10:29.795","Text":"We get nothing more and is equal to 0."},{"Start":"10:29.795 ","End":"10:32.630","Text":"We\u0027ve worked hard so far."},{"Start":"10:32.630 ","End":"10:36.785","Text":"At least we get a break here because this doesn\u0027t need the formula."},{"Start":"10:36.785 ","End":"10:38.870","Text":"When there\u0027s a c missing,"},{"Start":"10:38.870 ","End":"10:42.005","Text":"I can just take x outside the brackets."},{"Start":"10:42.005 ","End":"10:51.915","Text":"I\u0027ll continue over here. I get that x times x plus 5 equals 0."},{"Start":"10:51.915 ","End":"10:55.490","Text":"Now, remember if the product of 2 things is 0,"},{"Start":"10:55.490 ","End":"10:59.240","Text":"the only way you can multiply and get 0 is if one of them is 0,"},{"Start":"10:59.240 ","End":"11:00.913","Text":"so we have 2 choices,"},{"Start":"11:00.913 ","End":"11:07.515","Text":"either x is 0 or x plus 5 is 0."},{"Start":"11:07.515 ","End":"11:17.044","Text":"To say that x plus 5 is 0 means that x is minus 5 and so this gives us a solution."},{"Start":"11:17.044 ","End":"11:19.670","Text":"We expect 2 possible values of x,"},{"Start":"11:19.670 ","End":"11:23.075","Text":"x is 0 or x is minus 5."},{"Start":"11:23.075 ","End":"11:25.940","Text":"There\u0027s one more thing I almost forgot,"},{"Start":"11:25.940 ","End":"11:29.155","Text":"because we\u0027re working with denominators,"},{"Start":"11:29.155 ","End":"11:34.190","Text":"we have to make sure that the solution that we have,"},{"Start":"11:34.190 ","End":"11:36.350","Text":"or both of them in this case,"},{"Start":"11:36.350 ","End":"11:38.495","Text":"don\u0027t cause a problem,"},{"Start":"11:38.495 ","End":"11:41.000","Text":"meaning they don\u0027t make"},{"Start":"11:41.000 ","End":"11:46.174","Text":"any other denominator 0 that the equation is defined for these values."},{"Start":"11:46.174 ","End":"11:48.995","Text":"Let\u0027s take them one at a time."},{"Start":"11:48.995 ","End":"11:51.020","Text":"Let\u0027s take x=0,"},{"Start":"11:51.020 ","End":"11:53.240","Text":"and there\u0027s 3 denominators,"},{"Start":"11:53.240 ","End":"11:57.215","Text":"this one, this one, and this one."},{"Start":"11:57.215 ","End":"11:59.600","Text":"Now if I put x=0 here,"},{"Start":"11:59.600 ","End":"12:01.880","Text":"I don\u0027t get 0, I get 2."},{"Start":"12:01.880 ","End":"12:03.650","Text":"If I put x=0 here,"},{"Start":"12:03.650 ","End":"12:06.110","Text":"I get 2 times 1^2, it\u0027s also 2."},{"Start":"12:06.110 ","End":"12:07.820","Text":"If I put 0 here I get 1."},{"Start":"12:07.820 ","End":"12:10.490","Text":"I don\u0027t get 0 with any case."},{"Start":"12:10.490 ","End":"12:12.800","Text":"If I put minus 5 in each of these,"},{"Start":"12:12.800 ","End":"12:16.390","Text":"you can see that twice minus 5 plus 2 is not 0,"},{"Start":"12:16.390 ","End":"12:18.570","Text":"minus 5 minus 1,"},{"Start":"12:18.570 ","End":"12:20.027","Text":"not 0, square it,"},{"Start":"12:20.027 ","End":"12:22.050","Text":"double it, still not 0."},{"Start":"12:22.050 ","End":"12:25.710","Text":"1 plus 25 is not 0."},{"Start":"12:25.710 ","End":"12:30.000","Text":"In other words, these really are okay."},{"Start":"12:30.000 ","End":"12:33.710","Text":"They do not make the denominator 0."},{"Start":"12:33.710 ","End":"12:35.300","Text":"The equation is defined for them."},{"Start":"12:35.300 ","End":"12:38.990","Text":"That\u0027s just that little extra check you have to do whenever you have"},{"Start":"12:38.990 ","End":"12:45.660","Text":"denominators in your equation. We\u0027re done."}],"ID":5005}],"Thumbnail":null,"ID":3870},{"Name":"Nonlinear Equations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Not-Quite-Linear in Two Unknowns","Duration":"13m 20s","ChapterTopicVideoID":5027,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:03.705","Text":"I didn\u0027t know what to call this clip."},{"Start":"00:03.705 ","End":"00:07.425","Text":"I finally chose the expression not quite linear."},{"Start":"00:07.425 ","End":"00:11.025","Text":"It could have been special equations in 2 unknowns."},{"Start":"00:11.025 ","End":"00:15.780","Text":"Perhaps I\u0027ll just start with an example then you\u0027ll see what I mean."},{"Start":"00:15.780 ","End":"00:18.300","Text":"Let\u0027s take the first equation."},{"Start":"00:18.300 ","End":"00:24.360","Text":"Let\u0027s put the curly braces and say 3 over x"},{"Start":"00:24.360 ","End":"00:32.970","Text":"plus 2 over y is equal to 5."},{"Start":"00:32.970 ","End":"00:42.640","Text":"Let\u0027s take 7 over x minus 2 over y, also equals 5."},{"Start":"00:42.650 ","End":"00:49.540","Text":"Now this system of equations is not linear."},{"Start":"00:49.540 ","End":"00:52.690","Text":"In linear we have to have just x, we can\u0027t have x^2,"},{"Start":"00:52.690 ","End":"00:54.370","Text":"or the square root of x,"},{"Start":"00:54.370 ","End":"00:57.385","Text":"or 1 over x. X cannot be in the denominator,"},{"Start":"00:57.385 ","End":"00:58.915","Text":"so it\u0027s not linear."},{"Start":"00:58.915 ","End":"01:02.110","Text":"Nevertheless, the way we go about solving"},{"Start":"01:02.110 ","End":"01:06.100","Text":"this system is almost exactly the same as a linear equation."},{"Start":"01:06.100 ","End":"01:11.050","Text":"Let me show you how we get it to look very much like the usual situation."},{"Start":"01:11.050 ","End":"01:12.610","Text":"Suppose I rewrite this,"},{"Start":"01:12.610 ","End":"01:14.425","Text":"instead of 3 over x,"},{"Start":"01:14.425 ","End":"01:18.520","Text":"that\u0027s the same as 3 times 1 over x."},{"Start":"01:18.520 ","End":"01:21.040","Text":"Because this times this,"},{"Start":"01:21.040 ","End":"01:22.450","Text":"you multiply in the numerator,"},{"Start":"01:22.450 ","End":"01:25.260","Text":"3 times 1 is 3 over x, and so on."},{"Start":"01:25.260 ","End":"01:31.310","Text":"Plus twice 1 over y equals 5,"},{"Start":"01:31.310 ","End":"01:40.720","Text":"and 7 times 1 over x minus 2 times 1 over y equals 5."},{"Start":"01:40.720 ","End":"01:43.550","Text":"Now let\u0027s use a bit of coloring."},{"Start":"01:43.550 ","End":"01:46.595","Text":"Let\u0027s put a bit of color here,"},{"Start":"01:46.595 ","End":"01:49.160","Text":"and a bit of color here,"},{"Start":"01:49.160 ","End":"01:54.315","Text":"and here, and here."},{"Start":"01:54.315 ","End":"01:59.555","Text":"Notice that we only have 2 kind of things."},{"Start":"01:59.555 ","End":"02:01.760","Text":"We have this color that\u0027s 1 over x,"},{"Start":"02:01.760 ","End":"02:03.875","Text":"and this color, the 1 over y."},{"Start":"02:03.875 ","End":"02:07.610","Text":"This is very similar to,"},{"Start":"02:07.610 ","End":"02:10.370","Text":"I\u0027ll show you what it\u0027s similar to."},{"Start":"02:10.370 ","End":"02:15.800","Text":"I copy this over here and just erase the 1 over x and 1 over y."},{"Start":"02:15.800 ","End":"02:17.810","Text":"Suppose it was x here,"},{"Start":"02:17.810 ","End":"02:22.265","Text":"and x here, and y here, and y here."},{"Start":"02:22.265 ","End":"02:25.700","Text":"Don\u0027t these 2 systems look very similar?"},{"Start":"02:25.700 ","End":"02:29.165","Text":"Now this is a linear system and we know how to solve it."},{"Start":"02:29.165 ","End":"02:31.910","Text":"But here it\u0027s very much like it,"},{"Start":"02:31.910 ","End":"02:34.130","Text":"except that instead of x we have 1 over x,"},{"Start":"02:34.130 ","End":"02:36.725","Text":"and instead of y we have 1 over y."},{"Start":"02:36.725 ","End":"02:39.560","Text":"Let\u0027s continue and solve it and maybe we\u0027ll end up with"},{"Start":"02:39.560 ","End":"02:42.535","Text":"what 1 over x equals and 1 over y equals,"},{"Start":"02:42.535 ","End":"02:49.160","Text":"and then we\u0027ll worry about how to get x and y. I\u0027m going to erase this."},{"Start":"02:49.160 ","End":"02:51.830","Text":"This was just an analogy."},{"Start":"02:51.830 ","End":"02:56.435","Text":"Now I notice that the coefficient here is plus 2,"},{"Start":"02:56.435 ","End":"02:58.100","Text":"and here it\u0027s minus 2,"},{"Start":"02:58.100 ","End":"03:00.680","Text":"and I\u0027m thinking elimination."},{"Start":"03:00.680 ","End":"03:07.480","Text":"What I\u0027m going to do is add these 2 equations and see what we get."},{"Start":"03:07.480 ","End":"03:11.550","Text":"We have 3 of these and 7 of these,"},{"Start":"03:11.550 ","End":"03:14.070","Text":"doesn\u0027t matter what, 3 apples and 7 apples,"},{"Start":"03:14.070 ","End":"03:15.855","Text":"so I\u0027ve got 10 apples."},{"Start":"03:15.855 ","End":"03:22.275","Text":"I\u0027ve got 10 times 1 over x."},{"Start":"03:22.275 ","End":"03:25.470","Text":"Here plus 2 pairs,"},{"Start":"03:25.470 ","End":"03:28.590","Text":"minus 2 pairs, or whatever they are."},{"Start":"03:28.590 ","End":"03:31.985","Text":"Cancel, just leaves nothing,"},{"Start":"03:31.985 ","End":"03:34.790","Text":"I\u0027m not even going to write plus 0, it\u0027s just eliminated."},{"Start":"03:34.790 ","End":"03:37.100","Text":"That\u0027s the method of elimination."},{"Start":"03:37.100 ","End":"03:40.440","Text":"I\u0027ll put a dot here."},{"Start":"03:40.440 ","End":"03:43.825","Text":"5 plus 5 is equal to 10."},{"Start":"03:43.825 ","End":"03:47.630","Text":"Now I can divide both sides by 10 equations,"},{"Start":"03:47.630 ","End":"03:48.980","Text":"or bring this to the other side."},{"Start":"03:48.980 ","End":"03:54.570","Text":"In short, I get 1 over x is equal to 1."},{"Start":"03:54.580 ","End":"03:59.150","Text":"I\u0027m just going to highlight this just to indicate this is the same 1 over"},{"Start":"03:59.150 ","End":"04:03.200","Text":"x. I know I could easily get x from this,"},{"Start":"04:03.200 ","End":"04:05.135","Text":"but I\u0027ll hold off on that just for now."},{"Start":"04:05.135 ","End":"04:07.130","Text":"I\u0027ll leave it as 1 over x."},{"Start":"04:07.130 ","End":"04:10.430","Text":"Now just like in the method of elimination,"},{"Start":"04:10.430 ","End":"04:12.529","Text":"once I\u0027ve found one of the things,"},{"Start":"04:12.529 ","End":"04:16.715","Text":"I just substitute it in either one. Let\u0027s see."},{"Start":"04:16.715 ","End":"04:19.009","Text":"This has slightly smaller numbers,"},{"Start":"04:19.009 ","End":"04:20.900","Text":"I\u0027m going to go for the first one."},{"Start":"04:20.900 ","End":"04:24.525","Text":"Substitute this in this one,"},{"Start":"04:24.525 ","End":"04:29.025","Text":"and then what we get is, let\u0027s see."},{"Start":"04:29.025 ","End":"04:35.160","Text":"Now this thing is 1 from here,"},{"Start":"04:35.160 ","End":"04:38.238","Text":"so 3 times 1."},{"Start":"04:38.238 ","End":"04:41.180","Text":"Just underline that to show it was substituted."},{"Start":"04:41.180 ","End":"04:47.930","Text":"Plus 2 times 1 over y equals 5."},{"Start":"04:47.930 ","End":"04:51.830","Text":"3 times 1 is 3."},{"Start":"04:51.830 ","End":"04:54.200","Text":"If I bring it to the other side,"},{"Start":"04:54.200 ","End":"04:58.110","Text":"5 minus 3 is 2."},{"Start":"04:58.110 ","End":"05:04.640","Text":"I\u0027m going to do that without indicating the steps in between, equals 2."},{"Start":"05:04.640 ","End":"05:07.790","Text":"But maybe I\u0027ll just write that 2 is 5 minus 3,"},{"Start":"05:07.790 ","End":"05:10.805","Text":"3 times 1 is 3."},{"Start":"05:10.805 ","End":"05:13.655","Text":"Divide both sides by 2,"},{"Start":"05:13.655 ","End":"05:17.770","Text":"and then I\u0027ve got 1 over y equals 1."},{"Start":"05:17.770 ","End":"05:23.250","Text":"I\u0027ll highlight this also so we\u0027ve got both colored things."},{"Start":"05:23.250 ","End":"05:24.995","Text":"That\u0027s very good."},{"Start":"05:24.995 ","End":"05:27.935","Text":"But actually what we need is x and y,"},{"Start":"05:27.935 ","End":"05:30.436","Text":"so no problem,"},{"Start":"05:30.436 ","End":"05:31.905","Text":"just a little bit more work."},{"Start":"05:31.905 ","End":"05:33.835","Text":"If 1 over x equals 1,"},{"Start":"05:33.835 ","End":"05:39.160","Text":"let\u0027s multiply both sides by x, a common denominator."},{"Start":"05:39.160 ","End":"05:42.330","Text":"In short, we get that x equals 1."},{"Start":"05:42.330 ","End":"05:44.570","Text":"I trust you to solve that."},{"Start":"05:44.570 ","End":"05:45.800","Text":"You can always check anyway,"},{"Start":"05:45.800 ","End":"05:46.940","Text":"1 over 1 is 1."},{"Start":"05:46.940 ","End":"05:50.630","Text":"If I ask what 1 over what is 1, you\u0027ll know it\u0027s 1."},{"Start":"05:50.630 ","End":"05:53.834","Text":"That\u0027s x. The same thing here,"},{"Start":"05:53.834 ","End":"05:55.275","Text":"1 over y is 1."},{"Start":"05:55.275 ","End":"05:56.850","Text":"Multiply both sides by y,"},{"Start":"05:56.850 ","End":"05:59.800","Text":"I get 1 equals y, y equals 1."},{"Start":"05:59.930 ","End":"06:04.620","Text":"To be precise or formal,"},{"Start":"06:04.620 ","End":"06:11.430","Text":"I just say the solution is x equals 1 and y equals 1."},{"Start":"06:11.430 ","End":"06:14.355","Text":"Yes, that\u0027s the solution."},{"Start":"06:14.355 ","End":"06:16.200","Text":"That wasn\u0027t so bad."},{"Start":"06:16.200 ","End":"06:18.260","Text":"This is, as I say,"},{"Start":"06:18.260 ","End":"06:22.175","Text":"not linear but very close to or similar to linear."},{"Start":"06:22.175 ","End":"06:24.950","Text":"Let\u0027s do another one of these."},{"Start":"06:24.950 ","End":"06:28.190","Text":"To save time I copied it out."},{"Start":"06:28.190 ","End":"06:29.885","Text":"This is what it is."},{"Start":"06:29.885 ","End":"06:35.140","Text":"Let\u0027s do the same thing we did last time and write this as 2 times 1 over x."},{"Start":"06:35.140 ","End":"06:39.670","Text":"Here we are. I\u0027m not going to bother with the coloring this time."},{"Start":"06:39.670 ","End":"06:45.024","Text":"If I look at the coefficients and I want to use the method of elimination,"},{"Start":"06:45.024 ","End":"06:47.710","Text":"first of all look to see if I have a 1 somewhere,"},{"Start":"06:47.710 ","End":"06:50.410","Text":"and I don\u0027t, but then I look to see if 1 goes into the other."},{"Start":"06:50.410 ","End":"06:52.315","Text":"I see that 2 goes into 4."},{"Start":"06:52.315 ","End":"06:55.090","Text":"I want to save multiplying both of them by a constant."},{"Start":"06:55.090 ","End":"06:58.500","Text":"I\u0027ll just multiply the top by minus 2,"},{"Start":"06:58.500 ","End":"06:59.940","Text":"and then I\u0027ll get a minus 4,"},{"Start":"06:59.940 ","End":"07:01.860","Text":"and then I\u0027ll get an elimination."},{"Start":"07:01.860 ","End":"07:06.925","Text":"If I multiply the minus 2 by the whole first equation I will get,"},{"Start":"07:06.925 ","End":"07:11.150","Text":"minus 2 times 2 is minus 4."},{"Start":"07:11.780 ","End":"07:16.335","Text":"Minus 2 times minus 2 is plus 4."},{"Start":"07:16.335 ","End":"07:18.210","Text":"1 over y stays."},{"Start":"07:18.210 ","End":"07:20.925","Text":"Minus 2 times 3 is minus 6."},{"Start":"07:20.925 ","End":"07:24.800","Text":"Now what I want to do is take these 2 and add them,"},{"Start":"07:24.800 ","End":"07:27.185","Text":"because then this and this will cancel."},{"Start":"07:27.185 ","End":"07:29.060","Text":"What I get is this plus this,"},{"Start":"07:29.060 ","End":"07:31.740","Text":"nothing, this is eliminated."},{"Start":"07:31.740 ","End":"07:39.855","Text":"I get 3 plus 4 is 7 times 1 over y equals minus 1 and minus 6 minus 7,"},{"Start":"07:39.855 ","End":"07:45.300","Text":"so 1 over y equals minus 1."},{"Start":"07:45.300 ","End":"07:48.215","Text":"Let me leave it as 1 over y for the moment."},{"Start":"07:48.215 ","End":"07:55.420","Text":"Now let\u0027s put this into one of these first 2 equations."},{"Start":"07:55.420 ","End":"07:58.325","Text":"Let\u0027s go for the top one."},{"Start":"07:58.325 ","End":"08:02.940","Text":"I\u0027m going to put this one into here."},{"Start":"08:03.140 ","End":"08:09.690","Text":"If I copy it out I get 2 times 1 over x minus 2 times,"},{"Start":"08:09.690 ","End":"08:12.800","Text":"now 1 over y I have already, I\u0027ve just found it,"},{"Start":"08:12.800 ","End":"08:18.210","Text":"it\u0027s minus 1, equals 3."},{"Start":"08:19.340 ","End":"08:26.445","Text":"Then 2 times 1 over x plus 2 equals 3,"},{"Start":"08:26.445 ","End":"08:32.440","Text":"2 times 1 over x equals 3 minus 2 is 1."},{"Start":"08:33.280 ","End":"08:39.628","Text":"Let\u0027s see now, multiply both sides by x,"},{"Start":"08:39.628 ","End":"08:40.775","Text":"the x will cancel,"},{"Start":"08:40.775 ","End":"08:43.025","Text":"I will get 2 equals x."},{"Start":"08:43.025 ","End":"08:52.515","Text":"In other words, x equals"},{"Start":"08:52.515 ","End":"08:54.010","Text":"2."},{"Start":"08:54.010 ","End":"09:01.580","Text":"Let us go back here and also multiply both sides by y. I get 1 equals minus y."},{"Start":"09:01.580 ","End":"09:09.780","Text":"In short you\u0027ll get that y equals minus 1."},{"Start":"09:09.780 ","End":"09:11.480","Text":"I\u0027ll leave you to check that."},{"Start":"09:11.480 ","End":"09:13.520","Text":"In fact you can just substitute and see that it works,"},{"Start":"09:13.520 ","End":"09:14.990","Text":"1 over minus 1 is minus 1,"},{"Start":"09:14.990 ","End":"09:16.610","Text":"so it must be right."},{"Start":"09:16.610 ","End":"09:27.020","Text":"At the end just write it properly as x equals y equals 2 minus 1,"},{"Start":"09:27.020 ","End":"09:29.150","Text":"a bit of highlighting,"},{"Start":"09:29.150 ","End":"09:31.505","Text":"and that\u0027s the solution."},{"Start":"09:31.505 ","End":"09:34.415","Text":"Now let\u0027s go and do a third one."},{"Start":"09:34.415 ","End":"09:37.295","Text":"Here\u0027s a third example."},{"Start":"09:37.295 ","End":"09:39.170","Text":"It\u0027s a bit of a hybrid."},{"Start":"09:39.170 ","End":"09:44.120","Text":"The x\u0027s are linear, just x\u0027s themselves."},{"Start":"09:44.120 ","End":"09:46.400","Text":"The y is like the 1 over y."},{"Start":"09:46.400 ","End":"09:48.950","Text":"You know what, I don\u0027t feel like copying the whole thing out."},{"Start":"09:48.950 ","End":"09:52.070","Text":"Don\u0027t make me do that. I\u0027ll just write this as 3 times 1 over y,"},{"Start":"09:52.070 ","End":"09:56.440","Text":"and this is 2 times 1 over y. There we are."},{"Start":"09:56.440 ","End":"10:01.565","Text":"Now again we want to use the method of elimination."},{"Start":"10:01.565 ","End":"10:04.250","Text":"We\u0027re not worried that it\u0027s not quite linear that we have 1"},{"Start":"10:04.250 ","End":"10:07.265","Text":"over y instead of y, that\u0027s not a problem."},{"Start":"10:07.265 ","End":"10:10.100","Text":"There is a more technical difficulty here that we"},{"Start":"10:10.100 ","End":"10:13.190","Text":"can\u0027t get away with just multiplying one of them."},{"Start":"10:13.190 ","End":"10:15.890","Text":"In this case, nothing goes into anything,"},{"Start":"10:15.890 ","End":"10:17.060","Text":"so we\u0027ve got no choice,"},{"Start":"10:17.060 ","End":"10:19.655","Text":"you must multiply both of them by some number."},{"Start":"10:19.655 ","End":"10:23.200","Text":"Tell you what though, here we got a minus and here I\u0027ve got a plus,"},{"Start":"10:23.200 ","End":"10:29.380","Text":"so I can multiply by positive numbers and multiply this one by 2,"},{"Start":"10:29.380 ","End":"10:32.570","Text":"and this equation by 3."},{"Start":"10:32.570 ","End":"10:35.540","Text":"The idea is, then I\u0027ll have minus 6 here,"},{"Start":"10:35.540 ","End":"10:37.160","Text":"then I\u0027ll have plus 6 here,"},{"Start":"10:37.160 ","End":"10:40.310","Text":"and I can eliminate the 1 over y and go straight for x."},{"Start":"10:40.310 ","End":"10:41.810","Text":"That\u0027s the plan."},{"Start":"10:41.810 ","End":"10:44.795","Text":"Let me rewrite this."},{"Start":"10:44.795 ","End":"10:48.200","Text":"I\u0027ll start with the curly braces, it\u0027s the easy bit."},{"Start":"10:48.200 ","End":"10:50.273","Text":"Now this multiplied by 2,"},{"Start":"10:50.273 ","End":"10:52.220","Text":"everything is multiplied by 2,"},{"Start":"10:52.220 ","End":"10:55.685","Text":"so 8x, just doubling everything,"},{"Start":"10:55.685 ","End":"11:00.255","Text":"minus 6 times 1 over y, equals minus 4."},{"Start":"11:00.255 ","End":"11:02.250","Text":"Here I\u0027m tripling everything,"},{"Start":"11:02.250 ","End":"11:09.540","Text":"so it\u0027s 9 x plus 6 times 1 over y equals,"},{"Start":"11:09.540 ","End":"11:11.690","Text":"7 times 3 is 21."},{"Start":"11:11.690 ","End":"11:16.768","Text":"Now I\u0027m going to add these 2 together at the equations,"},{"Start":"11:16.768 ","End":"11:19.655","Text":"8 and 9 is 17x."},{"Start":"11:19.655 ","End":"11:24.150","Text":"These 2 cancel, that\u0027s the elimination."},{"Start":"11:24.150 ","End":"11:29.540","Text":"This is equal to,21 minus 4 is 17, what nice numbers."},{"Start":"11:29.540 ","End":"11:32.030","Text":"All we have to do now is divide by 17,"},{"Start":"11:32.030 ","End":"11:33.890","Text":"and we\u0027ve got x equals 1."},{"Start":"11:33.890 ","End":"11:36.560","Text":"That\u0027s one of them found. What do we do?"},{"Start":"11:36.560 ","End":"11:41.045","Text":"We substitute in either one of them, whichever looks simpler."},{"Start":"11:41.045 ","End":"11:44.645","Text":"I like to go with this one because it\u0027s all positive and,"},{"Start":"11:44.645 ","End":"11:47.660","Text":"I don\u0027t know, just fill this it\u0027ll be luckier."},{"Start":"11:47.660 ","End":"11:51.080","Text":"Let\u0027s go and substitute into this one,"},{"Start":"11:51.080 ","End":"11:54.605","Text":"whichever, but we have to decide."},{"Start":"11:54.605 ","End":"11:57.905","Text":"What we get now is,"},{"Start":"11:57.905 ","End":"12:00.510","Text":"just putting x equals 1 here,"},{"Start":"12:00.510 ","End":"12:04.440","Text":"so 3 times, I\u0027ll just put the 1 in brackets parenthesis,"},{"Start":"12:04.440 ","End":"12:05.690","Text":"that\u0027s what I substituted."},{"Start":"12:05.690 ","End":"12:11.525","Text":"Everything else copied exactly as is: 2 times 1 over y equals 7."},{"Start":"12:11.525 ","End":"12:18.500","Text":"Now here I get 2 times 1 over y. I\u0027ll take the 3 times 1,"},{"Start":"12:18.500 ","End":"12:20.149","Text":"which is 3 over to the other side,"},{"Start":"12:20.149 ","End":"12:22.955","Text":"7 minus 3 is 4."},{"Start":"12:22.955 ","End":"12:28.260","Text":"Divide both sides by 2,"},{"Start":"12:28.260 ","End":"12:34.170","Text":"and I\u0027ve got 1 over y equals 2."},{"Start":"12:34.170 ","End":"12:36.930","Text":"If 1 over y is 2,"},{"Start":"12:36.930 ","End":"12:40.045","Text":"then y equals a half."},{"Start":"12:40.045 ","End":"12:42.295","Text":"Let\u0027s see how would I explain that."},{"Start":"12:42.295 ","End":"12:45.595","Text":"I\u0027ll write it first, y equals one-half."},{"Start":"12:45.595 ","End":"12:49.270","Text":"I could multiply both sides by y here,"},{"Start":"12:49.270 ","End":"12:51.610","Text":"and get 1 equals,"},{"Start":"12:51.610 ","End":"12:53.635","Text":"y times 2 is 2y,"},{"Start":"12:53.635 ","End":"12:57.115","Text":"and then I could divide by 2 and get this."},{"Start":"12:57.115 ","End":"12:58.950","Text":"Anyway, this is the answer for y,"},{"Start":"12:58.950 ","End":"13:01.330","Text":"and as usual I like to be formal,"},{"Start":"13:01.330 ","End":"13:04.990","Text":"just to write the solution, we should redirect the word."},{"Start":"13:04.990 ","End":"13:10.688","Text":"The solution is x equals 1,"},{"Start":"13:10.688 ","End":"13:13.845","Text":"and y equals a half."},{"Start":"13:13.845 ","End":"13:16.240","Text":"A splash of color won\u0027t hurt."},{"Start":"13:16.240 ","End":"13:17.440","Text":"That\u0027s it."},{"Start":"13:17.440 ","End":"13:18.595","Text":"We\u0027re done with this clip,"},{"Start":"13:18.595 ","End":"13:20.660","Text":"more in the next."}],"ID":5012},{"Watched":false,"Name":"Not-Quite-Linear in Two Unknowns (continued)","Duration":"7m 3s","ChapterTopicVideoID":5026,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"Here we are again, it\u0027s a continuation."},{"Start":"00:03.060 ","End":"00:06.930","Text":"Let me just indicate that we\u0027re just continuing the previous clip."},{"Start":"00:06.930 ","End":"00:10.410","Text":"Give a couple of more examples and then we\u0027ll be done."},{"Start":"00:10.410 ","End":"00:15.300","Text":"First 1. Here\u0027s the example I wrote it out already."},{"Start":"00:15.300 ","End":"00:19.080","Text":"If time and what\u0027s wrong with this?"},{"Start":"00:19.080 ","End":"00:21.060","Text":"Why is it not linear?"},{"Start":"00:21.060 ","End":"00:27.600","Text":"The x, y thing and then you can\u0027t have x y just like you can\u0027t have x^2 or"},{"Start":"00:27.600 ","End":"00:31.080","Text":"square root of x in a linear equation just has to be numbers"},{"Start":"00:31.080 ","End":"00:35.350","Text":"times x numbers times y and numbers."},{"Start":"00:35.350 ","End":"00:37.265","Text":"It\u0027s not linear."},{"Start":"00:37.265 ","End":"00:41.675","Text":"But let\u0027s see what happens if we try and simplify the first equation."},{"Start":"00:41.675 ","End":"00:45.380","Text":"I\u0027m just going to simplify this over here."},{"Start":"00:45.380 ","End":"00:49.070","Text":"Let\u0027s multiply out, expand brackets."},{"Start":"00:49.070 ","End":"00:53.480","Text":"We get xy and then x times 3 is"},{"Start":"00:53.480 ","End":"01:01.900","Text":"3x minus y equals xy minus 1."},{"Start":"01:02.840 ","End":"01:12.740","Text":"Look xy cancels if I subtract xy from both sides this goes, and this goes."},{"Start":"01:12.740 ","End":"01:17.870","Text":"Now, I can just put this 1 over here."},{"Start":"01:17.870 ","End":"01:21.395","Text":"Well, I\u0027ll tell you what, I\u0027ll just write it in its place above."},{"Start":"01:21.395 ","End":"01:26.195","Text":"What we have left is 3x minus y equals minus 1 so the first 1 is"},{"Start":"01:26.195 ","End":"01:35.640","Text":"3x minus y equals minus 1."},{"Start":"01:36.320 ","End":"01:39.590","Text":"Now we\u0027re back to our usual situation."},{"Start":"01:39.590 ","End":"01:41.860","Text":"Basically, that\u0027s all the idea of this exercise to"},{"Start":"01:41.860 ","End":"01:43.970","Text":"show you that even if there is a non-linear thing,"},{"Start":"01:43.970 ","End":"01:46.025","Text":"if you expand, it might cancel out."},{"Start":"01:46.025 ","End":"01:50.965","Text":"But let\u0027s just solve it anyway for practice so that I could just leave it at that."},{"Start":"01:50.965 ","End":"01:55.700","Text":"You know what? I\u0027m going to do it by the substitution method and not by elimination."},{"Start":"01:55.700 ","End":"02:00.534","Text":"We\u0027ve been doing too much elimination. Let\u0027s see."},{"Start":"02:00.534 ","End":"02:07.240","Text":"How about we take x and isolate it from this 1 so we get from this 1,"},{"Start":"02:07.240 ","End":"02:12.340","Text":"x equals plus y plus 1."},{"Start":"02:12.340 ","End":"02:17.635","Text":"Now that I have x, I have to substitute it but has to be in the other 1."},{"Start":"02:17.635 ","End":"02:23.839","Text":"I\u0027m going to substitute it in this 1 here."},{"Start":"02:24.300 ","End":"02:29.060","Text":"That\u0027s a bit of a mess. Looks better."},{"Start":"02:29.060 ","End":"02:35.125","Text":"I\u0027m looking at this 1 that\u0027s colored in orange and putting x is y plus 1,"},{"Start":"02:35.125 ","End":"02:38.230","Text":"so 3 and then instead of x,"},{"Start":"02:38.230 ","End":"02:45.735","Text":"I\u0027m putting y plus 1 and the rest of it minus y equals minus 1."},{"Start":"02:45.735 ","End":"02:48.195","Text":"1 equation in y,"},{"Start":"02:48.195 ","End":"02:54.475","Text":"3y plus 3 minus y equals minus 1."},{"Start":"02:54.475 ","End":"02:56.270","Text":"Variables to the left,"},{"Start":"02:56.270 ","End":"02:57.890","Text":"numbers to the right."},{"Start":"02:57.890 ","End":"03:04.380","Text":"Let\u0027s see 3y minus y is 2y,"},{"Start":"03:04.380 ","End":"03:05.974","Text":"on the other side,"},{"Start":"03:05.974 ","End":"03:09.890","Text":"I got minus 1 and then the minus 3 comes over,"},{"Start":"03:09.890 ","End":"03:14.980","Text":"so it\u0027s minus 4 and y equals minus 2."},{"Start":"03:14.980 ","End":"03:22.370","Text":"That\u0027s y and then what you do is you substitute this in the other 1 that\u0027s"},{"Start":"03:22.370 ","End":"03:29.400","Text":"the x and so now I get that x equals minus 2 plus 1,"},{"Start":"03:29.400 ","End":"03:35.520","Text":"x equals, instead of y I put minus 2 plus 1."},{"Start":"03:35.520 ","End":"03:41.385","Text":"Then minus 2 plus 1 is minus 1 so x is minus 1."},{"Start":"03:41.385 ","End":"03:43.040","Text":"That\u0027s x, that\u0027s y,"},{"Start":"03:43.040 ","End":"03:45.935","Text":"I\u0027ll skip the curly braces part."},{"Start":"03:45.935 ","End":"03:49.350","Text":"Let\u0027s just do the last example."},{"Start":"03:49.870 ","End":"03:54.590","Text":"Here we go. Here it is, I wrote it out."},{"Start":"03:54.590 ","End":"03:56.585","Text":"Once again, we have xy,"},{"Start":"03:56.585 ","End":"04:02.725","Text":"which means that it\u0027s already not linear and this time the xy is not going to cancel."},{"Start":"04:02.725 ","End":"04:05.449","Text":"But the obvious thing to do is to simplify"},{"Start":"04:05.449 ","End":"04:09.220","Text":"the second equation so let\u0027s see what we get here."},{"Start":"04:09.220 ","End":"04:16.975","Text":"Multiplying out y times 3 times x. I\u0027ll change the order and call it 3x y."},{"Start":"04:16.975 ","End":"04:27.310","Text":"Y times 2 is 2y with a minus and it\u0027s still equal to 24."},{"Start":"04:28.610 ","End":"04:35.310","Text":"What I have here is I have basically 2 quantities."},{"Start":"04:35.310 ","End":"04:43.370","Text":"I have x y here and here and I also have y."},{"Start":"04:43.370 ","End":"04:52.670","Text":"Now referring to these 2 equations and xy is just like another variable."},{"Start":"04:52.670 ","End":"04:53.870","Text":"Just treat it as a whole,"},{"Start":"04:53.870 ","End":"04:56.660","Text":"just like we did with 1 over x and 1 over y."},{"Start":"04:56.660 ","End":"05:01.040","Text":"Just think of it like maybe this is x and this is y."},{"Start":"05:01.040 ","End":"05:03.500","Text":"We have a choice of techniques."},{"Start":"05:03.500 ","End":"05:06.020","Text":"I think the most obvious thing to do is again,"},{"Start":"05:06.020 ","End":"05:11.645","Text":"to use substitution and not elimination because we already have what xy is."},{"Start":"05:11.645 ","End":"05:13.235","Text":"If I put it in here,"},{"Start":"05:13.235 ","End":"05:15.650","Text":"that should be z so let\u0027s do that."},{"Start":"05:15.650 ","End":"05:20.875","Text":"Let\u0027s put xy from here,"},{"Start":"05:20.875 ","End":"05:24.660","Text":"I\u0027m going to put it into here."},{"Start":"05:24.660 ","End":"05:30.560","Text":"What do we get? We get 3 instead of xy,"},{"Start":"05:30.560 ","End":"05:36.470","Text":"I\u0027m writing 12 times 12 because there\u0027s a times here,"},{"Start":"05:36.470 ","End":"05:40.260","Text":"of course and then minus 2y=24."},{"Start":"05:42.250 ","End":"05:47.305","Text":"Let\u0027s do this, 36 minus 2y=24,"},{"Start":"05:47.305 ","End":"05:50.160","Text":"bring the 36 to the other side."},{"Start":"05:50.160 ","End":"05:57.645","Text":"Minus 2y=24 minus 36 is minus 12."},{"Start":"05:57.645 ","End":"06:01.485","Text":"Minus 2y is minus 12 divide by minus 2,"},{"Start":"06:01.485 ","End":"06:04.740","Text":"y is minus 12 over minus 2."},{"Start":"06:04.740 ","End":"06:11.095","Text":"Minus over minus is plus is 6 so now that I have that y is 6,"},{"Start":"06:11.095 ","End":"06:13.715","Text":"I can substitute it somewhere."},{"Start":"06:13.715 ","End":"06:16.430","Text":"The most obvious place,"},{"Start":"06:16.430 ","End":"06:23.630","Text":"the simplest place would be to go right to the first 1 here."},{"Start":"06:23.630 ","End":"06:32.190","Text":"I\u0027ll put y=6 here and then we\u0027ll get x times y"},{"Start":"06:32.190 ","End":"06:40.695","Text":"is 6 times 6 equals 12 and now I divide both sides by 6,"},{"Start":"06:40.695 ","End":"06:45.030","Text":"and I\u0027ve got x=12 over 6,"},{"Start":"06:45.030 ","End":"06:49.740","Text":"which is 2 and so we\u0027ve solved this,"},{"Start":"06:49.740 ","End":"06:59.610","Text":"that x=2 and y=6."},{"Start":"06:59.610 ","End":"07:03.250","Text":"Then we\u0027re done with this topic."}],"ID":5013},{"Watched":false,"Name":"Exercise 1","Duration":"3m 47s","ChapterTopicVideoID":5012,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.730","Text":"Here we have a system of 2 equations and 2 unknowns."},{"Start":"00:05.730 ","End":"00:11.100","Text":"But this is a bit different this time because instead of x and y,"},{"Start":"00:11.100 ","End":"00:13.575","Text":"we have 1/x and 1/y."},{"Start":"00:13.575 ","End":"00:14.730","Text":"Let\u0027s take a look at that."},{"Start":"00:14.730 ","End":"00:17.350","Text":"Let me just slightly rewrite this so you\u0027ll see it."},{"Start":"00:17.350 ","End":"00:21.560","Text":"The first equation is 3 times"},{"Start":"00:21.560 ","End":"00:27.800","Text":"1/x plus 1 times 1/y equals 4."},{"Start":"00:27.800 ","End":"00:37.190","Text":"The second one is 5 times 1/x minus 1 times 1/y also equals 4."},{"Start":"00:37.190 ","End":"00:38.735","Text":"If I highlight it,"},{"Start":"00:38.735 ","End":"00:44.530","Text":"here I have 1/x and here I have 1/x,"},{"Start":"00:44.530 ","End":"00:46.090","Text":"and here I have 1/y,"},{"Start":"00:46.090 ","End":"00:49.040","Text":"and here I have 1/y."},{"Start":"00:49.040 ","End":"00:52.100","Text":"It\u0027s little bit like having x and y,"},{"Start":"00:52.100 ","End":"00:54.950","Text":"but instead of x and y it\u0027s 1/x and 1/y,"},{"Start":"00:54.950 ","End":"00:59.930","Text":"the same techniques almost apply here also."},{"Start":"00:59.930 ","End":"01:04.955","Text":"We\u0027re going to treat this like an elimination, not a substitution."},{"Start":"01:04.955 ","End":"01:08.930","Text":"Elimination because to take advantage of"},{"Start":"01:08.930 ","End":"01:13.295","Text":"the opportunity that I have the same thing here and here with a plus and with a minus."},{"Start":"01:13.295 ","End":"01:16.820","Text":"What I want to do is like an arithmetic to add"},{"Start":"01:16.820 ","End":"01:22.130","Text":"these equations and get the results over here."},{"Start":"01:22.130 ","End":"01:27.215","Text":"3 times 1/x plus 5 times 1/x,"},{"Start":"01:27.215 ","End":"01:30.500","Text":"like 3 apples and 5 apples is 8 apples,"},{"Start":"01:30.500 ","End":"01:33.545","Text":"so it\u0027s 8 times 1/x."},{"Start":"01:33.545 ","End":"01:38.090","Text":"Now, plus this, minus this is nothing,"},{"Start":"01:38.090 ","End":"01:40.490","Text":"I just don\u0027t write anything here."},{"Start":"01:40.490 ","End":"01:44.610","Text":"4 plus 4 is 8."},{"Start":"01:44.890 ","End":"01:47.690","Text":"Now I have 8 something,"},{"Start":"01:47.690 ","End":"01:49.685","Text":"8 of this thing is equal to 8."},{"Start":"01:49.685 ","End":"01:52.100","Text":"If I divide both sides by 8,"},{"Start":"01:52.100 ","End":"01:56.945","Text":"I get that the thing itself is just equal to 1."},{"Start":"01:56.945 ","End":"01:59.550","Text":"Now if (1/x)=1,"},{"Start":"02:00.490 ","End":"02:04.325","Text":"then I claim that x=1."},{"Start":"02:04.325 ","End":"02:10.055","Text":"To see this, you could just multiply both sides of the equation by x."},{"Start":"02:10.055 ","End":"02:13.670","Text":"If we multiply both sides of the equation by x,"},{"Start":"02:13.670 ","End":"02:15.665","Text":"I get x/x is 1,"},{"Start":"02:15.665 ","End":"02:18.055","Text":"x times 1 is x."},{"Start":"02:18.055 ","End":"02:23.505","Text":"1=x means x=1. Same thing."},{"Start":"02:23.505 ","End":"02:25.705","Text":"We have 1 of them."},{"Start":"02:25.705 ","End":"02:27.625","Text":"I\u0027ll underline it here."},{"Start":"02:27.625 ","End":"02:30.580","Text":"Now I need to get the other variable."},{"Start":"02:30.580 ","End":"02:34.645","Text":"This time, I\u0027m going to use substitution,"},{"Start":"02:34.645 ","End":"02:37.510","Text":"that\u0027s what we do when we have 1 variable and it doesn\u0027t"},{"Start":"02:37.510 ","End":"02:40.555","Text":"matter in which of these two I substitute."},{"Start":"02:40.555 ","End":"02:43.285","Text":"Let\u0027s just take the first one because it\u0027s first."},{"Start":"02:43.285 ","End":"02:47.440","Text":"Put x=1 into here."},{"Start":"02:47.440 ","End":"02:57.845","Text":"Now I will get 3/1 plus 1/y equals 4."},{"Start":"02:57.845 ","End":"03:00.565","Text":"Now, what is 3/1?"},{"Start":"03:00.565 ","End":"03:02.125","Text":"It\u0027s just 3."},{"Start":"03:02.125 ","End":"03:12.120","Text":"Already I can take the 3 over to the other side and say 1/y equals 4 minus 3 is 1."},{"Start":"03:12.120 ","End":"03:15.740","Text":"If I multiply both sides by y,"},{"Start":"03:15.740 ","End":"03:19.265","Text":"similar to what I did here, I get 1=y."},{"Start":"03:19.265 ","End":"03:21.740","Text":"In other words, y=1."},{"Start":"03:21.740 ","End":"03:26.750","Text":"Or you could think 1 over what gives me 1, 1/1 is 1."},{"Start":"03:26.750 ","End":"03:29.270","Text":"I\u0027ll underline this one. Now I\u0027ve got this and this."},{"Start":"03:29.270 ","End":"03:32.180","Text":"I just want to collect them and"},{"Start":"03:32.180 ","End":"03:40.445","Text":"say x=1 and y=1 is our solution."},{"Start":"03:40.445 ","End":"03:43.460","Text":"I like to highlight our solution."},{"Start":"03:43.460 ","End":"03:45.960","Text":"There it is and we\u0027re done."}],"ID":5014},{"Watched":false,"Name":"Exercise 2","Duration":"5m 3s","ChapterTopicVideoID":5013,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"Here we have another system of equations,"},{"Start":"00:03.360 ","End":"00:06.345","Text":"it\u0027s two equations in two unknowns, x and y."},{"Start":"00:06.345 ","End":"00:10.635","Text":"But it\u0027s another one of those where we have x and y on the denominator."},{"Start":"00:10.635 ","End":"00:16.725","Text":"The thing to do is to rewrite this slightly as 3 times"},{"Start":"00:16.725 ","End":"00:23.790","Text":"1/x plus 3 times 1/y equals 2."},{"Start":"00:23.790 ","End":"00:29.090","Text":"Here, just taking the numerators out and putting them at the side."},{"Start":"00:29.090 ","End":"00:35.953","Text":"9 times 1/x minus"},{"Start":"00:35.953 ","End":"00:39.750","Text":"4 times 1/y equals"},{"Start":"00:39.750 ","End":"00:41.545","Text":"minus 7."},{"Start":"00:41.545 ","End":"00:43.940","Text":"It\u0027s just like usual,"},{"Start":"00:43.940 ","End":"00:45.380","Text":"except instead of x and y,"},{"Start":"00:45.380 ","End":"00:47.405","Text":"we have 1/x,"},{"Start":"00:47.405 ","End":"00:51.555","Text":"that\u0027s here and here,"},{"Start":"00:51.555 ","End":"00:54.550","Text":"and here we have 1/y."},{"Start":"00:56.990 ","End":"00:59.640","Text":"Proceed as usual,"},{"Start":"00:59.640 ","End":"01:04.625","Text":"just have to decide whether we want to go for substitution or elimination."},{"Start":"01:04.625 ","End":"01:08.390","Text":"It seems to me that elimination is the choice here because we"},{"Start":"01:08.390 ","End":"01:12.755","Text":"can\u0027t easily get one of these in terms of the others."},{"Start":"01:12.755 ","End":"01:16.160","Text":"Even once you\u0027ve decided that it\u0027s elimination,"},{"Start":"01:16.160 ","End":"01:21.920","Text":"you still have to decide how to go about what to multiply this and this by."},{"Start":"01:21.920 ","End":"01:24.005","Text":"Well, if I look at 3 and 4,"},{"Start":"01:24.005 ","End":"01:26.600","Text":"there\u0027s nothing I can multiply one by to get the other."},{"Start":"01:26.600 ","End":"01:28.580","Text":"But in the case of 3 and 9 there is,"},{"Start":"01:28.580 ","End":"01:30.110","Text":"so I can save a step."},{"Start":"01:30.110 ","End":"01:32.345","Text":"If I multiply the top one,"},{"Start":"01:32.345 ","End":"01:35.330","Text":"well not by 3 but by minus 3."},{"Start":"01:35.330 ","End":"01:39.500","Text":"I\u0027m saying is if I multiply the whole top equation by minus 3,"},{"Start":"01:39.500 ","End":"01:41.960","Text":"the first term here we\u0027ll have minus 9/x,"},{"Start":"01:41.960 ","End":"01:43.820","Text":"and here we\u0027ll have plus 9/x."},{"Start":"01:43.820 ","End":"01:45.665","Text":"That will do nicely for addition."},{"Start":"01:45.665 ","End":"01:49.205","Text":"Let\u0027s do that. I\u0027ll write this new one over here."},{"Start":"01:49.205 ","End":"01:53.435","Text":"Well, actually I wanted to multiply this minus 3 by the 1 over here."},{"Start":"01:53.435 ","End":"01:59.770","Text":"So minus 3 times 3 is minus 9 1/x."},{"Start":"01:59.770 ","End":"02:05.250","Text":"Minus 3 times 3 again is minus 9 1/y."},{"Start":"02:05.250 ","End":"02:09.090","Text":"Minus 3 times 2 is minus 6."},{"Start":"02:09.090 ","End":"02:17.520","Text":"The second one just as is 9 times one over x minus 4 times 1/y equals minus 7."},{"Start":"02:17.520 ","End":"02:20.445","Text":"Now I\u0027m going to add these two,"},{"Start":"02:20.445 ","End":"02:26.325","Text":"like arithmetic problem with addition."},{"Start":"02:26.325 ","End":"02:31.640","Text":"Minus 9 of these plus 9 of these is nothing so I don\u0027t write anything here."},{"Start":"02:31.640 ","End":"02:34.820","Text":"Minus 9 and minus 4,"},{"Start":"02:34.820 ","End":"02:40.795","Text":"they strengthen each other two negatives, minus 13 1/y."},{"Start":"02:40.795 ","End":"02:45.290","Text":"Minus 6 and minus 7 is minus 13."},{"Start":"02:45.290 ","End":"02:48.485","Text":"How lucky we are we get minus 13 twice."},{"Start":"02:48.485 ","End":"02:51.455","Text":"If we divide by minus 13,"},{"Start":"02:51.455 ","End":"02:55.915","Text":"we\u0027ve got 1/y is equal to 1."},{"Start":"02:55.915 ","End":"03:01.340","Text":"We\u0027ve seen this thing before that this immediately leads to y equals 1."},{"Start":"03:01.340 ","End":"03:08.700","Text":"We either multiply both sides by y or you take the reciprocal of both sides,"},{"Start":"03:08.700 ","End":"03:12.635","Text":"or you just ask if 1 over what is 1 and the answer is 1."},{"Start":"03:12.635 ","End":"03:15.605","Text":"That\u0027s one of them solved. Now we need x."},{"Start":"03:15.605 ","End":"03:20.690","Text":"We just take either one of these and I\u0027ll go for the first one because"},{"Start":"03:20.690 ","End":"03:26.450","Text":"it\u0027s smaller numbers and I\u0027ll substitute y into it."},{"Start":"03:26.450 ","End":"03:32.855","Text":"So I\u0027m going to take this one and put y equals 1 into this top equation."},{"Start":"03:32.855 ","End":"03:34.460","Text":"I\u0027ll write it over here."},{"Start":"03:34.460 ","End":"03:42.575","Text":"I\u0027ve got 3 times 1/x plus 3 times 1/y,"},{"Start":"03:42.575 ","End":"03:46.220","Text":"which is 1/1,"},{"Start":"03:46.220 ","End":"03:48.145","Text":"is equal to 2."},{"Start":"03:48.145 ","End":"03:57.375","Text":"Now, 3 times 1, I didn\u0027t say that right."},{"Start":"03:57.375 ","End":"04:00.330","Text":"I mean, 1/1 is 1 and 1 times 3 is 3,"},{"Start":"04:00.330 ","End":"04:02.895","Text":"so this whole thing is just 3."},{"Start":"04:02.895 ","End":"04:13.960","Text":"What I get is 3/x is equal to 2 minus 3 is minus 1."},{"Start":"04:13.970 ","End":"04:20.225","Text":"See if we got some room here."},{"Start":"04:20.225 ","End":"04:23.605","Text":"I have multiplied both sides by x,"},{"Start":"04:23.605 ","End":"04:32.655","Text":"and then I\u0027ll get 3 is equal to minus x and if 3 is minus x,"},{"Start":"04:32.655 ","End":"04:37.070","Text":"then x is minus 3 that\u0027s easy to see."},{"Start":"04:37.070 ","End":"04:41.075","Text":"Just switch sides and then divide by minus 1."},{"Start":"04:41.075 ","End":"04:43.342","Text":"We would get minus 3 equals x,"},{"Start":"04:43.342 ","End":"04:44.770","Text":"that x is minus 3."},{"Start":"04:44.770 ","End":"04:46.145","Text":"That\u0027s the second one."},{"Start":"04:46.145 ","End":"04:48.545","Text":"Let me just collect these two together."},{"Start":"04:48.545 ","End":"04:54.860","Text":"What I have is x equals minus"},{"Start":"04:54.860 ","End":"05:02.880","Text":"3 and y equals 1 and highlight this and we\u0027re done."}],"ID":5015},{"Watched":false,"Name":"Exercise 3","Duration":"4m 44s","ChapterTopicVideoID":5014,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.145","Text":"Here we have another system of 2 equations in 2 unknowns."},{"Start":"00:05.145 ","End":"00:10.245","Text":"This time it\u0027s a hybrid because the x appears regular,"},{"Start":"00:10.245 ","End":"00:13.125","Text":"but we have 1 over y instead of y,"},{"Start":"00:13.125 ","End":"00:16.005","Text":"a mixture of things we\u0027ve been doing."},{"Start":"00:16.005 ","End":"00:24.195","Text":"But that\u0027s, let\u0027s just write this as 4 times x,"},{"Start":"00:24.195 ","End":"00:29.655","Text":"and I like to put the number out in front minus 7 times 1 over y,"},{"Start":"00:29.655 ","End":"00:37.140","Text":"because 1 over y is what I want to isolate and emphasize as another variable."},{"Start":"00:37.140 ","End":"00:41.700","Text":"Here, 5x plus twice,"},{"Start":"00:41.700 ","End":"00:45.270","Text":"1 over y equals 7."},{"Start":"00:45.270 ","End":"00:47.820","Text":"It\u0027s like I have 2 variables,"},{"Start":"00:47.820 ","End":"00:50.055","Text":"1 of them is x,"},{"Start":"00:50.055 ","End":"00:58.770","Text":"and I\u0027ll do this in 1 color of the x and the other variable y, 1 over y."},{"Start":"00:58.770 ","End":"01:02.505","Text":"Other than that, it\u0027s just it was x and y,"},{"Start":"01:02.505 ","End":"01:11.015","Text":"so we\u0027ll do this system of equations by the elimination method."},{"Start":"01:11.015 ","End":"01:13.820","Text":"There is no easy substitution and even the elimination,"},{"Start":"01:13.820 ","End":"01:16.590","Text":"we\u0027re going to need 2 multiplications."},{"Start":"01:16.810 ","End":"01:18.950","Text":"Not clear which is which,"},{"Start":"01:18.950 ","End":"01:21.995","Text":"but maybe because one\u0027s minus and one\u0027s plus,"},{"Start":"01:21.995 ","End":"01:24.740","Text":"then I only have to multiply it by positive quantities."},{"Start":"01:24.740 ","End":"01:28.175","Text":"Let\u0027s multiply the top one by 2 and the bottom one by 7,"},{"Start":"01:28.175 ","End":"01:29.900","Text":"so I\u0027ll just indicate that this,"},{"Start":"01:29.900 ","End":"01:31.580","Text":"I\u0027m going to multiply by 2,"},{"Start":"01:31.580 ","End":"01:33.620","Text":"this whole equation, and this one,"},{"Start":"01:33.620 ","End":"01:36.610","Text":"I\u0027m going to multiply by 7."},{"Start":"01:36.610 ","End":"01:38.755","Text":"If I do that,"},{"Start":"01:38.755 ","End":"01:43.340","Text":"then the first one becomes everything\u0027s multiplied by 2, so instead of 4x,"},{"Start":"01:43.340 ","End":"01:48.350","Text":"it\u0027s 8x, and instead of minus 7, it\u0027s minus 14,"},{"Start":"01:48.350 ","End":"01:50.680","Text":"but it\u0027s still 1 over y,"},{"Start":"01:50.680 ","End":"01:54.025","Text":"and here minus 6."},{"Start":"01:54.025 ","End":"01:57.245","Text":"The second one, if I multiply it by 7,"},{"Start":"01:57.245 ","End":"02:06.840","Text":"becomes 35x, and then plus 14,"},{"Start":"02:07.210 ","End":"02:10.045","Text":"not y, 1 over y."},{"Start":"02:10.045 ","End":"02:15.655","Text":"That\u0027s better, is equal to 7 times 7 is 49."},{"Start":"02:15.655 ","End":"02:19.070","Text":"Now we arranged it so that if we add them, they would cancel,"},{"Start":"02:19.070 ","End":"02:20.960","Text":"so let me just take the sum of these 2,"},{"Start":"02:20.960 ","End":"02:25.185","Text":"I won\u0027t bother with the plus and the addition line,"},{"Start":"02:25.185 ","End":"02:28.530","Text":"8x plus 35 x,"},{"Start":"02:28.530 ","End":"02:31.140","Text":"8 and 35 is 43,"},{"Start":"02:31.140 ","End":"02:36.320","Text":"so that\u0027s 43x minus something plus the same thing is nothing,"},{"Start":"02:36.320 ","End":"02:39.115","Text":"I\u0027m not going to write anything even."},{"Start":"02:39.115 ","End":"02:41.685","Text":"Minus 6 plus 49,"},{"Start":"02:41.685 ","End":"02:45.600","Text":"49 minus 6 is 43,"},{"Start":"02:45.600 ","End":"02:47.820","Text":"same number here and here,"},{"Start":"02:47.820 ","End":"02:49.350","Text":"so let\u0027s divide by it,"},{"Start":"02:49.350 ","End":"02:52.395","Text":"we get straightaway that x=1,"},{"Start":"02:52.395 ","End":"02:55.645","Text":"that\u0027s x, we still have to find y."},{"Start":"02:55.645 ","End":"02:59.855","Text":"We substitute in either one of these,"},{"Start":"02:59.855 ","End":"03:02.090","Text":"You know what, we\u0027ll go with the lower one,"},{"Start":"03:02.090 ","End":"03:04.610","Text":"doesn\u0027t have any negatives, I like that better."},{"Start":"03:04.610 ","End":"03:06.830","Text":"Put x=1 here,"},{"Start":"03:06.830 ","End":"03:08.540","Text":"I\u0027m to going put a big arrow here,"},{"Start":"03:08.540 ","End":"03:10.790","Text":"but we\u0027re going over here with x,"},{"Start":"03:10.790 ","End":"03:16.850","Text":"and then I\u0027ll just put an arrow like this and say x=1."},{"Start":"03:16.850 ","End":"03:21.765","Text":"We have that 5 times 1,"},{"Start":"03:21.765 ","End":"03:26.365","Text":"I\u0027ll like to write the thing I substituted in brackets to emphasize,"},{"Start":"03:26.365 ","End":"03:33.395","Text":"plus 2 over y=7,1 equation with 1 variable y."},{"Start":"03:33.395 ","End":"03:36.475","Text":"Let\u0027s see. What we get next is,"},{"Start":"03:36.475 ","End":"03:38.870","Text":"I\u0027ll continue over here,"},{"Start":"03:39.470 ","End":"03:43.810","Text":"5 times 1 is 5 and if I bring it to the other side, it\u0027s negative,"},{"Start":"03:43.810 ","End":"03:51.470","Text":"so I get that 2 over y=7 minus 5 is 2."},{"Start":"03:51.510 ","End":"03:55.615","Text":"If I multiply both sides by y,"},{"Start":"03:55.615 ","End":"04:01.310","Text":"I get 2= 2y."},{"Start":"04:01.310 ","End":"04:03.945","Text":"If I then divide by 2,"},{"Start":"04:03.945 ","End":"04:06.975","Text":"I get 1=y,"},{"Start":"04:06.975 ","End":"04:10.935","Text":"but I won\u0027t write 1=y, I\u0027ll write y=1."},{"Start":"04:10.935 ","End":"04:12.480","Text":"That\u0027s another thing about equality,"},{"Start":"04:12.480 ","End":"04:14.480","Text":"you can change sides if 2 things are equal,"},{"Start":"04:14.480 ","End":"04:16.920","Text":"they\u0027re equal the other way around too."},{"Start":"04:17.890 ","End":"04:24.979","Text":"What to do now is just collect these 2 together and just write it in some corner."},{"Start":"04:24.979 ","End":"04:33.760","Text":"Let\u0027s see, here\u0027s a good spot, that x=1,"},{"Start":"04:33.760 ","End":"04:35.130","Text":"y=1, that\u0027s the solution,"},{"Start":"04:35.130 ","End":"04:37.820","Text":"a solution is a pair of values and I, as usual,"},{"Start":"04:37.820 ","End":"04:43.440","Text":"I\u0027m going to highlight it and declare that we are done."}],"ID":5016},{"Watched":false,"Name":"Exercise 4","Duration":"3m 41s","ChapterTopicVideoID":5015,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:06.225","Text":"Here, I have another system of 2 equations and 2 unknowns,"},{"Start":"00:06.225 ","End":"00:09.270","Text":"and I\u0027m thinking which strategy should I use."},{"Start":"00:09.270 ","End":"00:15.060","Text":"Suddenly, I could say that x equals plus y plus 2 and substitute,"},{"Start":"00:15.060 ","End":"00:17.100","Text":"but the top one is a bit of a mess,"},{"Start":"00:17.100 ","End":"00:19.380","Text":"and I have the feeling that in any event,"},{"Start":"00:19.380 ","End":"00:22.590","Text":"it would be good to organize the first equation."},{"Start":"00:22.590 ","End":"00:26.265","Text":"Let\u0027s do that and then we can still decide to do substitution if we want."},{"Start":"00:26.265 ","End":"00:28.425","Text":"I\u0027m doing this tidying up here."},{"Start":"00:28.425 ","End":"00:31.245","Text":"Let\u0027s open the brackets, have xy."},{"Start":"00:31.245 ","End":"00:34.470","Text":"Now xy is not a good sign, like x squared,"},{"Start":"00:34.470 ","End":"00:38.385","Text":"so I\u0027m hoping that somehow it\u0027ll cancel. We\u0027ll see."},{"Start":"00:38.385 ","End":"00:41.400","Text":"x times 2 is 2x,"},{"Start":"00:41.400 ","End":"00:43.290","Text":"and then plus y,"},{"Start":"00:43.290 ","End":"00:49.365","Text":"and then on this side equals xy minus 5."},{"Start":"00:49.365 ","End":"00:54.780","Text":"Here\u0027s the equality. Here\u0027s xy and here\u0027s xy on both sides,"},{"Start":"00:54.780 ","End":"00:58.275","Text":"so what we can do is just cancel it."},{"Start":"00:58.275 ","End":"01:01.950","Text":"What we\u0027re really doing is subtracting xy from both sides,"},{"Start":"01:01.950 ","End":"01:04.880","Text":"or we can say bring the xy over here and make it negative."},{"Start":"01:04.880 ","End":"01:08.255","Text":"Either way, they both vanish from the face of the earth."},{"Start":"01:08.255 ","End":"01:12.600","Text":"What we\u0027re left with is something fairly normal."},{"Start":"01:12.600 ","End":"01:15.980","Text":"Let\u0027s just collect like terms,"},{"Start":"01:15.980 ","End":"01:17.270","Text":"variables on the left,"},{"Start":"01:17.270 ","End":"01:20.660","Text":"numbers on the right, and let\u0027s see."},{"Start":"01:20.660 ","End":"01:24.800","Text":"Here we have 2x plus y,"},{"Start":"01:24.800 ","End":"01:31.395","Text":"that\u0027s nothing, and then is equal to minus 5."},{"Start":"01:31.395 ","End":"01:33.300","Text":"That\u0027s in pretty normal form."},{"Start":"01:33.300 ","End":"01:36.175","Text":"Then we have the second equation here."},{"Start":"01:36.175 ","End":"01:41.565","Text":"This is my number 2,"},{"Start":"01:41.565 ","End":"01:43.350","Text":"and this is number 1."},{"Start":"01:43.350 ","End":"01:45.360","Text":"Oops. Yes, number 1."},{"Start":"01:45.360 ","End":"01:47.280","Text":"I\u0027m not used to Roman letters."},{"Start":"01:47.280 ","End":"01:51.105","Text":"This is like a variation of number 1."},{"Start":"01:51.105 ","End":"01:56.045","Text":"Really, these 2 are the ones that we\u0027re going to be considering."},{"Start":"01:56.045 ","End":"01:58.620","Text":"Now, looking at it,"},{"Start":"01:59.110 ","End":"02:01.372","Text":"hard to say which is better,"},{"Start":"02:01.372 ","End":"02:02.920","Text":"substitution or elimination,"},{"Start":"02:02.920 ","End":"02:05.660","Text":"but because we have a minus y and a plus y,"},{"Start":"02:05.660 ","End":"02:07.760","Text":"it begs for elimination."},{"Start":"02:07.760 ","End":"02:10.840","Text":"I\u0027m going to write number 2 over here again."},{"Start":"02:10.840 ","End":"02:13.275","Text":"Let\u0027s see. What do we have?"},{"Start":"02:13.275 ","End":"02:16.905","Text":"x minus y equals 2."},{"Start":"02:16.905 ","End":"02:21.195","Text":"Now, this is my system of equations so I\u0027m going to add the 2."},{"Start":"02:21.195 ","End":"02:24.400","Text":"2x plus x is 3x,"},{"Start":"02:24.400 ","End":"02:29.130","Text":"y and minus y annihilate each other, nothing left."},{"Start":"02:29.130 ","End":"02:32.150","Text":"Is this a minus?"},{"Start":"02:32.150 ","End":"02:33.470","Text":"Yeah, this is a minus."},{"Start":"02:33.470 ","End":"02:38.045","Text":"Minus 5 plus 2 is minus 3."},{"Start":"02:38.045 ","End":"02:40.010","Text":"If I divide by 3,"},{"Start":"02:40.010 ","End":"02:44.245","Text":"I get that x is equal to minus 1."},{"Start":"02:44.245 ","End":"02:49.570","Text":"Then when we have x, we substitute it in 1 of the equations."},{"Start":"02:50.180 ","End":"02:53.265","Text":"I think the easiest is this one."},{"Start":"02:53.265 ","End":"02:55.050","Text":"This one here is the same thing."},{"Start":"02:55.050 ","End":"02:56.460","Text":"Put x equals 1,"},{"Start":"02:56.460 ","End":"02:58.470","Text":"we\u0027ll put it into here."},{"Start":"02:58.470 ","End":"03:02.765","Text":"If I put minus 1 for x in here,"},{"Start":"03:02.765 ","End":"03:10.640","Text":"what I get is x is minus 1 and I still have minus y is equal to 2."},{"Start":"03:10.640 ","End":"03:13.010","Text":"Let\u0027s see. Variables on the left,"},{"Start":"03:13.010 ","End":"03:14.675","Text":"so I keep the minus y,"},{"Start":"03:14.675 ","End":"03:16.040","Text":"numbers on the right,"},{"Start":"03:16.040 ","End":"03:19.455","Text":"2 plus 1 is 3,"},{"Start":"03:19.455 ","End":"03:21.330","Text":"and if minus y is 3,"},{"Start":"03:21.330 ","End":"03:23.650","Text":"then y is minus 3."},{"Start":"03:23.650 ","End":"03:28.415","Text":"I have both of them and I\u0027ll just write that out."},{"Start":"03:28.415 ","End":"03:33.780","Text":"That our answer is a pair of numbers, x and y,"},{"Start":"03:33.780 ","End":"03:35.520","Text":"x is minus 1,"},{"Start":"03:35.520 ","End":"03:38.220","Text":"y is minus 3,"},{"Start":"03:38.220 ","End":"03:41.440","Text":"and we are done."}],"ID":5017},{"Watched":false,"Name":"Exercise 5","Duration":"3m 15s","ChapterTopicVideoID":5016,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"In this exercise, we have a system of equations,"},{"Start":"00:03.540 ","End":"00:06.270","Text":"it\u0027s 2 equations and 2 unknowns,"},{"Start":"00:06.270 ","End":"00:13.570","Text":"x and y, and we have to solve them or it, the system."},{"Start":"00:13.570 ","End":"00:17.010","Text":"Once again, we\u0027ve seen this before, we had an xy,"},{"Start":"00:17.010 ","End":"00:22.020","Text":"which is an unwanted element and we would like to somehow get rid of this xy."},{"Start":"00:22.020 ","End":"00:25.080","Text":"In the previous exercise,"},{"Start":"00:25.080 ","End":"00:26.910","Text":"it just canceled from both sides."},{"Start":"00:26.910 ","End":"00:29.190","Text":"Here we have to do a little bit more work."},{"Start":"00:29.190 ","End":"00:32.520","Text":"Let\u0027s open the brackets in the second equation."},{"Start":"00:32.520 ","End":"00:34.470","Text":"You know what I\u0027ll label the equations,"},{"Start":"00:34.470 ","End":"00:36.360","Text":"let me call this one, a 1,"},{"Start":"00:36.360 ","End":"00:39.065","Text":"I\u0027ll use Roman numerals and this one number 2."},{"Start":"00:39.065 ","End":"00:45.255","Text":"So let\u0027s expand number 2 and see what we get."},{"Start":"00:45.255 ","End":"00:47.330","Text":"Y times 3 times x,"},{"Start":"00:47.330 ","End":"00:51.740","Text":"I can change the order and write this as 3xy."},{"Start":"00:51.740 ","End":"00:54.445","Text":"I do this because I wanted to look more like this."},{"Start":"00:54.445 ","End":"01:01.080","Text":"Y times -4 is -4, y=20."},{"Start":"01:01.080 ","End":"01:05.470","Text":"Now here I have 3xy and here I only have 1xy."},{"Start":"01:05.470 ","End":"01:12.895","Text":"But if I multiply the first equation by 3 or better still by -3,"},{"Start":"01:12.895 ","End":"01:16.880","Text":"then I\u0027ll be able to add equations and get rid of this, xy."},{"Start":"01:16.880 ","End":"01:22.595","Text":"Let\u0027s multiply the first equation by -3."},{"Start":"01:22.595 ","End":"01:24.371","Text":"I\u0027ll write it down here,"},{"Start":"01:24.371 ","End":"01:29.450","Text":"-3 times xy is -3xy."},{"Start":"01:29.450 ","End":"01:32.285","Text":"There\u0027s nothing else on the left-hand side,"},{"Start":"01:32.285 ","End":"01:37.565","Text":"and -3 times 20 is -60."},{"Start":"01:37.565 ","End":"01:43.685","Text":"Now I\u0027m in the position of being able to add these 2 and getting rid of this xy."},{"Start":"01:43.685 ","End":"01:52.160","Text":"This plus this is nothing -4y plus nothing is -4y,"},{"Start":"01:52.160 ","End":"01:58.945","Text":"and 20 plus -60 is -40."},{"Start":"01:58.945 ","End":"02:04.245","Text":"Now if I just divide by -4 on both sides,"},{"Start":"02:04.245 ","End":"02:10.884","Text":"I\u0027ll get that y=-40 over -4 is 10 positive."},{"Start":"02:10.884 ","End":"02:14.365","Text":"We\u0027ve already found one of the variables y."},{"Start":"02:14.365 ","End":"02:20.200","Text":"Now, all we have to do is substitute this value of y in one of these equations,"},{"Start":"02:20.200 ","End":"02:22.240","Text":"and the first one,"},{"Start":"02:22.240 ","End":"02:23.590","Text":"it looks simplest to me."},{"Start":"02:23.590 ","End":"02:29.130","Text":"Let\u0027s put it into the first one, into here."},{"Start":"02:29.130 ","End":"02:38.430","Text":"What I get is x times y is 10=20."},{"Start":"02:38.430 ","End":"02:40.814","Text":"X times 10 is 20."},{"Start":"02:40.814 ","End":"02:44.210","Text":"Already you get in your heads, you can see that 2 times 10 is 20."},{"Start":"02:44.210 ","End":"02:45.950","Text":"But if we do it properly,"},{"Start":"02:45.950 ","End":"02:47.975","Text":"we divide both sides by 10,"},{"Start":"02:47.975 ","End":"02:58.290","Text":"or we put the 10 into the denominator of the other side and get x is 20/10, which is 2."},{"Start":"02:58.290 ","End":"03:01.309","Text":"Now we have x and now we have y."},{"Start":"03:01.309 ","End":"03:10.290","Text":"All we have to do is just write it neatly as a solution and say x=2 and y=10,"},{"Start":"03:10.290 ","End":"03:14.850","Text":"and perhaps highlight it because this is the answer and we\u0027re done."}],"ID":5018},{"Watched":false,"Name":"Exercise 6","Duration":"3m 41s","ChapterTopicVideoID":5017,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:05.265","Text":"Here again, we have 2 equations in 2 unknowns,"},{"Start":"00:05.265 ","End":"00:08.950","Text":"x and y, and we want to solve this system."},{"Start":"00:09.410 ","End":"00:15.060","Text":"As before, we said that x y is an undesirable element,"},{"Start":"00:15.060 ","End":"00:16.410","Text":"if you want to call it that,"},{"Start":"00:16.410 ","End":"00:19.455","Text":"and we\u0027d like to get rid of it somehow."},{"Start":"00:19.455 ","End":"00:21.720","Text":"Now if I look here,"},{"Start":"00:21.720 ","End":"00:26.475","Text":"it appears with coefficient minus 4 and here with coefficient plus 1."},{"Start":"00:26.475 ","End":"00:30.495","Text":"If I were to multiply the second equation by 4,"},{"Start":"00:30.495 ","End":"00:32.850","Text":"then I\u0027d have a minus 4 and a plus 4 and then I"},{"Start":"00:32.850 ","End":"00:35.160","Text":"could add and get rid of it. That\u0027s what we\u0027ll do."},{"Start":"00:35.160 ","End":"00:40.650","Text":"We\u0027ll multiply the second equation by plus 4."},{"Start":"00:40.650 ","End":"00:45.040","Text":"I just put a plus there and it will be okay."},{"Start":"00:45.230 ","End":"00:49.470","Text":"Let\u0027s see, 6 times 4 is 24,"},{"Start":"00:49.470 ","End":"00:56.734","Text":"x plus 4xy equals minus 80,"},{"Start":"00:56.734 ","End":"00:58.760","Text":"multiply everything by 4."},{"Start":"00:58.760 ","End":"01:02.720","Text":"Now if I look at this equation,"},{"Start":"01:02.720 ","End":"01:05.370","Text":"I can add these 2 and this term,"},{"Start":"01:05.370 ","End":"01:07.740","Text":"and this term will cancel. Let\u0027s do that."},{"Start":"01:07.740 ","End":"01:15.500","Text":"5x plus 24x is 29x."},{"Start":"01:15.500 ","End":"01:17.600","Text":"This with this cancels, of course,"},{"Start":"01:17.600 ","End":"01:19.480","Text":"that\u0027s why we multiply this by 4."},{"Start":"01:19.480 ","End":"01:21.130","Text":"There\u0027s nothing here."},{"Start":"01:21.130 ","End":"01:29.670","Text":"What we\u0027re left with is the right-hand side where we have 22 minus 80,"},{"Start":"01:29.830 ","End":"01:34.820","Text":"and this is minus 58."},{"Start":"01:34.820 ","End":"01:39.255","Text":"To get x, we just have to divide by 29,"},{"Start":"01:39.255 ","End":"01:43.810","Text":"or if you like, put the 29 in the denominator when we have a product,"},{"Start":"01:43.810 ","End":"01:45.515","Text":"we turned it into a division."},{"Start":"01:45.515 ","End":"01:48.990","Text":"So minus 58 over 29."},{"Start":"01:48.990 ","End":"01:52.560","Text":"The minus stays, but 58 over 29 is 2,"},{"Start":"01:52.560 ","End":"01:54.510","Text":"so we have minus 2."},{"Start":"01:54.510 ","End":"01:56.865","Text":"We\u0027ve got one of the 2 variables,"},{"Start":"01:56.865 ","End":"01:59.910","Text":"we\u0027ve got x, now we need to find y."},{"Start":"01:59.910 ","End":"02:05.020","Text":"We just substitute in either one of them, whichever looks easier."},{"Start":"02:05.020 ","End":"02:08.450","Text":"Second one looks easier to me."},{"Start":"02:08.450 ","End":"02:12.060","Text":"It doesn\u0027t really matter, but I\u0027m going to go for the second one."},{"Start":"02:12.060 ","End":"02:15.055","Text":"I\u0027m going to put the minus 2 in there,"},{"Start":"02:15.055 ","End":"02:18.555","Text":"and then get, I\u0027m just copying here across,"},{"Start":"02:18.555 ","End":"02:24.395","Text":"6x is 6 times minus 2 and then plus"},{"Start":"02:24.395 ","End":"02:31.335","Text":"xy minus 2 times y equals minus 20."},{"Start":"02:31.335 ","End":"02:34.150","Text":"Again, I just copied this equation,"},{"Start":"02:34.150 ","End":"02:36.355","Text":"but wherever I see x,"},{"Start":"02:36.355 ","End":"02:39.685","Text":"I put minus 2 in parentheses."},{"Start":"02:39.685 ","End":"02:41.800","Text":"Now we can evaluate this."},{"Start":"02:41.800 ","End":"02:44.520","Text":"We just have one equation, y,"},{"Start":"02:44.520 ","End":"02:47.430","Text":"6 times minus 2 is minus 12,"},{"Start":"02:47.430 ","End":"02:52.689","Text":"minus 2 times y minus 2y equals minus 20."},{"Start":"02:52.689 ","End":"02:55.000","Text":"As usual, variables on the left,"},{"Start":"02:55.000 ","End":"02:56.570","Text":"numbers on the right."},{"Start":"02:56.570 ","End":"03:01.455","Text":"Minus 2y equals minus 20 plus 12,"},{"Start":"03:01.455 ","End":"03:02.985","Text":"because it switch sides,"},{"Start":"03:02.985 ","End":"03:10.140","Text":"and this is equal to minus 8."},{"Start":"03:10.140 ","End":"03:15.615","Text":"I\u0027ll copy this. Then just divide by minus 2."},{"Start":"03:15.615 ","End":"03:20.250","Text":"We get that y equals minus 8 over minus 2."},{"Start":"03:20.250 ","End":"03:21.930","Text":"Minus over minus is plus,"},{"Start":"03:21.930 ","End":"03:24.000","Text":"so the answer is 4."},{"Start":"03:24.000 ","End":"03:28.490","Text":"Now all I have to do is just to write the solution nicely and"},{"Start":"03:28.490 ","End":"03:32.720","Text":"say that the solution is that x equals and y equals,"},{"Start":"03:32.720 ","End":"03:37.720","Text":"let\u0027s see, x is minus 2, y is 4."},{"Start":"03:37.720 ","End":"03:42.010","Text":"I\u0027d like to highlight, and we\u0027re done."}],"ID":5019},{"Watched":false,"Name":"Exercise 7","Duration":"5m 48s","ChapterTopicVideoID":5018,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.755","Text":"In this exercise, we have a system of 2 nonlinear equations in y and x."},{"Start":"00:07.755 ","End":"00:10.365","Text":"Let me, first of all, copy it."},{"Start":"00:10.365 ","End":"00:13.800","Text":"We have a typical scenario where we have"},{"Start":"00:13.800 ","End":"00:19.335","Text":"the same variable y in terms of x in 2 different ways."},{"Start":"00:19.335 ","End":"00:22.410","Text":"The idea, in this case,"},{"Start":"00:22.410 ","End":"00:24.705","Text":"I call it cut out the middleman."},{"Start":"00:24.705 ","End":"00:28.425","Text":"Get rid of y by saying if y equals this and y equals this,"},{"Start":"00:28.425 ","End":"00:31.900","Text":"then we can compare the right-hand sides."},{"Start":"00:32.360 ","End":"00:35.865","Text":"What we get is that"},{"Start":"00:35.865 ","End":"00:45.175","Text":"x^2+6x-7 from the right-hand side here is equal to 4x+8 from here."},{"Start":"00:45.175 ","End":"00:49.490","Text":"Now we\u0027ll try and solve this for x."},{"Start":"00:49.490 ","End":"00:52.370","Text":"It looks like it\u0027s going to be a quadratic."},{"Start":"00:52.370 ","End":"00:58.960","Text":"Let\u0027s move all the terms over to the left-hand side. We get"},{"Start":"01:10.180 ","End":"01:13.750","Text":"x^2+6x-4x is +2x and -7-8 is -15=0."},{"Start":"01:13.750 ","End":"01:16.645","Text":"I don\u0027t want to spend time,"},{"Start":"01:16.645 ","End":"01:20.720","Text":"solving quadratic equations, which you already know how to do."},{"Start":"01:20.720 ","End":"01:23.375","Text":"I\u0027ll factorize this for you."},{"Start":"01:23.375 ","End":"01:28.070","Text":"This turns out to be x+5,"},{"Start":"01:28.070 ","End":"01:33.155","Text":"x-3, and you can just check it."},{"Start":"01:33.155 ","End":"01:38.990","Text":"X times x is x^2+5x-3x is 2x-15."},{"Start":"01:38.990 ","End":"01:47.735","Text":"2x is either x=-5 or x=3."},{"Start":"01:47.735 ","End":"01:52.310","Text":"Because product is 0 means either one of them must be 0,"},{"Start":"01:52.310 ","End":"01:55.190","Text":"at least one of them, so we get this."},{"Start":"01:55.190 ","End":"02:00.485","Text":"But that\u0027s not all because the solution requires both x and y."},{"Start":"02:00.485 ","End":"02:03.520","Text":"Let\u0027s take each case separately."},{"Start":"02:03.520 ","End":"02:07.949","Text":"If x is equal to,"},{"Start":"02:07.949 ","End":"02:09.060","Text":"let\u0027s say the 3,"},{"Start":"02:09.060 ","End":"02:10.820","Text":"we\u0027ll take that one first."},{"Start":"02:10.820 ","End":"02:14.945","Text":"What we do then is we substitute it."},{"Start":"02:14.945 ","End":"02:17.900","Text":"You can substitute it in either."},{"Start":"02:17.900 ","End":"02:23.300","Text":"I will choose the second one because it looks simpler."},{"Start":"02:23.300 ","End":"02:27.060","Text":"It\u0027s linear, it\u0027s easier."},{"Start":"02:27.190 ","End":"02:29.780","Text":"In this case, we get"},{"Start":"02:29.780 ","End":"02:36.750","Text":"that y=4 times 3+8"},{"Start":"02:37.430 ","End":"02:43.000","Text":"and that is equal to 12+8 which is 20."},{"Start":"02:43.220 ","End":"02:45.600","Text":"One solution"},{"Start":"02:45.600 ","End":"02:54.250","Text":"is x=3,"},{"Start":"02:54.250 ","End":"02:55.440","Text":"y=20."},{"Start":"02:55.440 ","End":"02:57.660","Text":"Now let\u0027s go for the other solution."},{"Start":"02:57.660 ","End":"03:02.020","Text":"We let x=-5."},{"Start":"03:02.290 ","End":"03:06.470","Text":"Once again, I can substitute in either."},{"Start":"03:06.470 ","End":"03:08.555","Text":"I should get the same."},{"Start":"03:08.555 ","End":"03:12.804","Text":"Let\u0027s do it in the second one."},{"Start":"03:12.804 ","End":"03:17.333","Text":"Y=4 times"},{"Start":"03:17.333 ","End":"03:24.570","Text":"-5+8 which is -20+8,"},{"Start":"03:24.570 ","End":"03:28.665","Text":"that\u0027s -12 and so our second solution"},{"Start":"03:28.665 ","End":"03:33.040","Text":"is x=-5"},{"Start":"03:33.040 ","End":"03:39.420","Text":"and y=-12."},{"Start":"03:39.420 ","End":"03:41.520","Text":"Let me highlight the solutions."},{"Start":"03:41.520 ","End":"03:46.414","Text":"Here\u0027s one solution and here\u0027s the other solution."},{"Start":"03:46.414 ","End":"03:50.735","Text":"Now, once in a while, ideally,"},{"Start":"03:50.735 ","End":"03:57.379","Text":"every time we should check and verify that these really are solutions,"},{"Start":"03:57.379 ","End":"03:58.835","Text":"each solution should be checked."},{"Start":"03:58.835 ","End":"04:00.320","Text":"I\u0027ll tell you when I\u0027ll compromise."},{"Start":"04:00.320 ","End":"04:02.779","Text":"I\u0027ll check that the first one is a solution."},{"Start":"04:02.779 ","End":"04:06.590","Text":"What we do is we see that this pair of x,"},{"Start":"04:06.590 ","End":"04:09.845","Text":"y satisfies the original system."},{"Start":"04:09.845 ","End":"04:12.800","Text":"Let\u0027s check the first."},{"Start":"04:12.800 ","End":"04:16.760","Text":"The first one, y is 20,"},{"Start":"04:16.760 ","End":"04:22.010","Text":"and I\u0027ll write an equal with a question mark because that\u0027s what I\u0027m checking."},{"Start":"04:22.010 ","End":"04:26.040","Text":"Is it equal to 3^2+6 times 3-7?"},{"Start":"04:31.400 ","End":"04:36.740","Text":"Even if it is, I also have to have the other one to be equal."},{"Start":"04:36.740 ","End":"04:45.590","Text":"I have to also have the 20=4 times 3+8."},{"Start":"04:45.590 ","End":"04:51.860","Text":"They both have to be true in order for this to be a solution."},{"Start":"04:51.860 ","End":"04:54.120","Text":"Now this one we\u0027ve already checked."},{"Start":"04:54.120 ","End":"04:57.405","Text":"Can we just look at this line here? We\u0027ve checked that."},{"Start":"04:57.405 ","End":"05:01.680","Text":"Here I can say fine and here let\u0027s see."},{"Start":"05:01.680 ","End":"05:04.005","Text":"Left-hand side is 20."},{"Start":"05:04.005 ","End":"05:09.810","Text":"It\u0027s explicit and right-hand side 3^2 is 9,"},{"Start":"05:09.810 ","End":"05:13.320","Text":"6 times 3=18,"},{"Start":"05:13.320 ","End":"05:20.340","Text":"9+18-7, 27-7 is 20."},{"Start":"05:20.340 ","End":"05:21.795","Text":"This is checked."},{"Start":"05:21.795 ","End":"05:27.560","Text":"That means that this pair of solutions has been verified."},{"Start":"05:27.560 ","End":"05:32.980","Text":"I leave it to you to verify the other one."},{"Start":"05:32.980 ","End":"05:37.370","Text":"We can say it\u0027s optional to verify,"},{"Start":"05:37.370 ","End":"05:38.600","Text":"but it\u0027s a good idea."},{"Start":"05:38.600 ","End":"05:41.090","Text":"It\u0027s an easy way to catch a mistake you might have"},{"Start":"05:41.090 ","End":"05:44.525","Text":"made in an exam or even a homework assignment."},{"Start":"05:44.525 ","End":"05:47.730","Text":"We\u0027re done with this one."}],"ID":5020},{"Watched":false,"Name":"Exercise 8","Duration":"5m 29s","ChapterTopicVideoID":5019,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this exercise, we have a non linear system,"},{"Start":"00:03.270 ","End":"00:05.220","Text":"2 equations and 2 unknowns,"},{"Start":"00:05.220 ","End":"00:11.090","Text":"x and y. I can see a couple of ways we could approach this."},{"Start":"00:11.090 ","End":"00:15.210","Text":"One thing we could do is look at the first one and say y is given in terms"},{"Start":"00:15.210 ","End":"00:20.340","Text":"of x and then substitute y into the second equation."},{"Start":"00:20.340 ","End":"00:23.280","Text":"A variant of this would be to say,"},{"Start":"00:23.280 ","End":"00:24.990","Text":"well, in the second equation,"},{"Start":"00:24.990 ","End":"00:26.790","Text":"if I brought the 4x over,"},{"Start":"00:26.790 ","End":"00:31.455","Text":"I\u0027d have y in terms of x both here and here and compare the right hand sides."},{"Start":"00:31.455 ","End":"00:33.060","Text":"There\u0027s no very much difference."},{"Start":"00:33.060 ","End":"00:37.590","Text":"I\u0027ll go with the second approach,"},{"Start":"00:37.590 ","End":"00:41.535","Text":"just copy the equations over here."},{"Start":"00:41.535 ","End":"00:44.360","Text":"The second one, like I said,"},{"Start":"00:44.360 ","End":"00:49.415","Text":"I\u0027ll rewrite as y equals 4x minus 2,"},{"Start":"00:49.415 ","End":"00:51.965","Text":"by bringing this over to the other side."},{"Start":"00:51.965 ","End":"00:54.470","Text":"Now I have y equals and y equals,"},{"Start":"00:54.470 ","End":"00:57.779","Text":"so I compare the right hand side,"},{"Start":"00:57.779 ","End":"01:06.115","Text":"which gives me just x. I get x^2 plus 3x minus 2 = 4x minus 2."},{"Start":"01:06.115 ","End":"01:08.882","Text":"It\u0027s a quadratic, I just need to arrange it,"},{"Start":"01:08.882 ","End":"01:10.325","Text":"put everything on the left."},{"Start":"01:10.325 ","End":"01:15.205","Text":"We\u0027ve got x^2 plus 3x minus 4x is minus x."},{"Start":"01:15.205 ","End":"01:20.595","Text":"The minus 2 with the minus 2 cancels, equals 0."},{"Start":"01:20.595 ","End":"01:23.645","Text":"This is easy to factorize."},{"Start":"01:23.645 ","End":"01:28.370","Text":"Just take x out, x,(x-1)=0."},{"Start":"01:28.370 ","End":"01:32.645","Text":"When a product is 0, then one or the other must be 0."},{"Start":"01:32.645 ","End":"01:38.735","Text":"We get that x=0 or x=1."},{"Start":"01:38.735 ","End":"01:41.855","Text":"Then we need to find for each of these,"},{"Start":"01:41.855 ","End":"01:44.540","Text":"what is the corresponding y."},{"Start":"01:44.540 ","End":"01:48.830","Text":"If we have x=0,"},{"Start":"01:48.830 ","End":"01:54.915","Text":"then we substitute this in either one,"},{"Start":"01:54.915 ","End":"01:57.040","Text":"could be this one."},{"Start":"01:58.640 ","End":"02:02.855","Text":"This one\u0027s a good one for substituting, it\u0027s pretty simple."},{"Start":"02:02.855 ","End":"02:10.560","Text":"If x is 0, then y equals 4 times 0 minus 2,"},{"Start":"02:10.560 ","End":"02:13.320","Text":"which is minus 2."},{"Start":"02:13.320 ","End":"02:16.815","Text":"I\u0027ve got one solution,"},{"Start":"02:16.815 ","End":"02:22.005","Text":"x= 0, y=-2."},{"Start":"02:22.005 ","End":"02:27.400","Text":"I like to put it in a little curly brackets because the solution is a pair x and y."},{"Start":"02:27.400 ","End":"02:29.765","Text":"Now, the other solution,"},{"Start":"02:29.765 ","End":"02:33.620","Text":"which we get from taking x=1,"},{"Start":"02:33.620 ","End":"02:35.645","Text":"once again, I\u0027ll use this,"},{"Start":"02:35.645 ","End":"02:39.260","Text":"to substitute, it is the most convenient of them all."},{"Start":"02:39.260 ","End":"02:45.810","Text":"I get that y equals 4 times 1 minus 2,"},{"Start":"02:45.810 ","End":"02:49.740","Text":"and that comes out to be 4 minus 2 is 2,"},{"Start":"02:49.740 ","End":"02:59.000","Text":"and so the other solution is x=1, y=2."},{"Start":"02:59.000 ","End":"03:00.770","Text":"I like to highlight the solutions here,"},{"Start":"03:00.770 ","End":"03:04.580","Text":"solution 1 and here solution 2."},{"Start":"03:04.580 ","End":"03:12.080","Text":"Then this is the optional part but recommended is to verify solutions by"},{"Start":"03:12.080 ","End":"03:19.295","Text":"substituting in the original equations up here."},{"Start":"03:19.295 ","End":"03:23.135","Text":"Let\u0027s see. If x=0,"},{"Start":"03:23.135 ","End":"03:26.240","Text":"then on the first one,"},{"Start":"03:26.240 ","End":"03:29.040","Text":"and y is minus 2,"},{"Start":"03:29.040 ","End":"03:32.946","Text":"I get minus 2 equals question mark,"},{"Start":"03:32.946 ","End":"03:35.765","Text":"question mark because I\u0027m verifying it,"},{"Start":"03:35.765 ","End":"03:40.830","Text":"0^2 plus 3 times 0 minus 2."},{"Start":"03:40.830 ","End":"03:45.800","Text":"I also have to make sure that the other one holds to that y,"},{"Start":"03:45.800 ","End":"03:47.570","Text":"which is minus 2,"},{"Start":"03:47.570 ","End":"03:50.930","Text":"minus 4 times x,"},{"Start":"03:50.930 ","End":"03:59.250","Text":"which is 0, equals question mark, minus 2."},{"Start":"03:59.250 ","End":"04:01.530","Text":"Now, let\u0027s look here,"},{"Start":"04:01.530 ","End":"04:04.230","Text":"I have left hand side is minus 2,"},{"Start":"04:04.230 ","End":"04:08.590","Text":"right hand side, 0 plus 0 minus 2, so yes, this is equal."},{"Start":"04:09.710 ","End":"04:14.400","Text":"Let\u0027s see if the other one, minus 2,"},{"Start":"04:14.400 ","End":"04:17.490","Text":"minus 4 times 0 is just minus 2,"},{"Start":"04:17.490 ","End":"04:19.500","Text":"which is the same as minus 2, so yes."},{"Start":"04:19.500 ","End":"04:21.935","Text":"When both of them hold true,"},{"Start":"04:21.935 ","End":"04:25.895","Text":"then I can say that this solution is verified."},{"Start":"04:25.895 ","End":"04:29.540","Text":"Let\u0027s try the other one."},{"Start":"04:29.540 ","End":"04:32.630","Text":"I won\u0027t always be verifying both,"},{"Start":"04:32.630 ","End":"04:35.980","Text":"but let\u0027s just do it this time."},{"Start":"04:36.710 ","End":"04:39.260","Text":"In the original equations,"},{"Start":"04:39.260 ","End":"04:40.640","Text":"y is 2,"},{"Start":"04:40.640 ","End":"04:45.210","Text":"so 2 equals question mark,"},{"Start":"04:45.710 ","End":"04:51.120","Text":"1^2 plus 3 times 1 minus 2,"},{"Start":"04:51.120 ","End":"04:53.495","Text":"and I\u0027ll just write the other one already,"},{"Start":"04:53.495 ","End":"04:55.580","Text":"and y minus 4x,"},{"Start":"04:55.580 ","End":"05:03.870","Text":"2 minus 4x equals minus 2 question mark."},{"Start":"05:03.870 ","End":"05:11.550","Text":"Let\u0027s see. 1^2 plus 3 times 1 is 1 plus 3 is 4,"},{"Start":"05:11.550 ","End":"05:13.810","Text":"minus 2 is 2, so yes."},{"Start":"05:14.770 ","End":"05:20.770","Text":"2 minus 4 is minus 2, yes, and so this one is yes."},{"Start":"05:23.090 ","End":"05:29.830","Text":"We\u0027ve verified them both and we are done."}],"ID":5021},{"Watched":false,"Name":"Exercise 9","Duration":"6m 20s","ChapterTopicVideoID":5020,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.490","Text":"In this exercise, we have a non-linear system of equations,"},{"Start":"00:05.490 ","End":"00:09.000","Text":"two equations and two unknowns, x and y."},{"Start":"00:09.000 ","End":"00:12.180","Text":"Well, the first one is linear and the second one is not,"},{"Start":"00:12.180 ","End":"00:14.325","Text":"but it\u0027s still nonlinear,"},{"Start":"00:14.325 ","End":"00:24.690","Text":"and the best way to solve this one is to substitute y in terms of x in this equation."},{"Start":"00:24.690 ","End":"00:25.770","Text":"Because in the first one,"},{"Start":"00:25.770 ","End":"00:28.500","Text":"I\u0027ve given explicitly what y is in terms of x,"},{"Start":"00:28.500 ","End":"00:29.760","Text":"and if I put it here,"},{"Start":"00:29.760 ","End":"00:35.565","Text":"I\u0027ll get an equation all in x. I\u0027ll just copy these down here."},{"Start":"00:35.565 ","End":"00:38.880","Text":"I said if I substitute y into here,"},{"Start":"00:38.880 ","End":"00:42.075","Text":"in fact, y only appears in this place here,"},{"Start":"00:42.075 ","End":"00:48.000","Text":"then I\u0027ll get x^2/2 plus 2,"},{"Start":"00:48.000 ","End":"00:54.850","Text":"and then y is 1 minus 3x^2=2."},{"Start":"00:54.850 ","End":"00:57.875","Text":"Now I have just one unknown x."},{"Start":"00:57.875 ","End":"01:07.935","Text":"Let\u0027s expand brackets and we get x^2/2 plus 2,"},{"Start":"01:07.935 ","End":"01:12.079","Text":"and I\u0027m using the formula for a minus b squared,"},{"Start":"01:12.079 ","End":"01:14.945","Text":"which is a^2 minus 2ab."},{"Start":"01:14.945 ","End":"01:21.995","Text":"That\u0027s 6x. Don\u0027t need this 2."},{"Start":"01:21.995 ","End":"01:27.565","Text":"Plus 9^2=2."},{"Start":"01:27.565 ","End":"01:29.690","Text":"Let\u0027s get rid of fractions."},{"Start":"01:29.690 ","End":"01:32.345","Text":"Multiply both sides by 2,"},{"Start":"01:32.345 ","End":"01:35.900","Text":"and also bring this over here."},{"Start":"01:35.900 ","End":"01:41.370","Text":"I end up with x^2 plus,"},{"Start":"01:41.370 ","End":"01:43.800","Text":"now if I\u0027m multiplying by 2,"},{"Start":"01:43.800 ","End":"01:47.175","Text":"then this makes this 2 into a 4."},{"Start":"01:47.175 ","End":"01:50.850","Text":"I\u0027ve got 4 times 1 is 4,"},{"Start":"01:50.850 ","End":"01:58.755","Text":"4 times 6 is 24 minus 24x and 4 times 9 is 36."},{"Start":"01:58.755 ","End":"02:02.460","Text":"This also by 2, which is 4."},{"Start":"02:02.460 ","End":"02:05.495","Text":"Now I have a quadratic."},{"Start":"02:05.495 ","End":"02:08.300","Text":"Just need to arrange it a bit. Let\u0027s see."},{"Start":"02:08.300 ","End":"02:10.295","Text":"Let\u0027s take the x^s first."},{"Start":"02:10.295 ","End":"02:16.510","Text":"So x^2 and 36x^2 is 37x^2."},{"Start":"02:16.510 ","End":"02:22.230","Text":"Next, the x\u0027s only have here minus 24x,"},{"Start":"02:22.580 ","End":"02:28.530","Text":"and numbers, 4 and minus 4 cancel."},{"Start":"02:28.530 ","End":"02:31.870","Text":"This equal 0."},{"Start":"02:32.200 ","End":"02:36.635","Text":"I don\u0027t have a constant term means that I can factorize,"},{"Start":"02:36.635 ","End":"02:45.420","Text":"x times 37x minus 24=0."},{"Start":"02:45.550 ","End":"02:48.475","Text":"More space here."},{"Start":"02:48.475 ","End":"02:51.315","Text":"When a product is 0,"},{"Start":"02:51.315 ","End":"02:54.785","Text":"1 or other of the factors has to be 0."},{"Start":"02:54.785 ","End":"02:58.050","Text":"We get that either x=0."},{"Start":"02:58.660 ","End":"03:03.260","Text":"The other possibility is that the second factor is 0,"},{"Start":"03:03.260 ","End":"03:08.790","Text":"which gives us that x is 24/37."},{"Start":"03:11.830 ","End":"03:18.830","Text":"Of course, we found x. We also have to find y because the solution is a pair of x and y."},{"Start":"03:18.830 ","End":"03:20.935","Text":"Let\u0027s take the first one."},{"Start":"03:20.935 ","End":"03:23.640","Text":"What we need to do is substitute,"},{"Start":"03:23.640 ","End":"03:25.670","Text":"and we can choose either,"},{"Start":"03:25.670 ","End":"03:28.520","Text":"but it\u0027s preferable to go with"},{"Start":"03:28.520 ","End":"03:33.200","Text":"the first one because that gives us y explicitly in terms of x."},{"Start":"03:33.200 ","End":"03:41.270","Text":"If x is 0, then we get that y=1 minus 3x,"},{"Start":"03:41.270 ","End":"03:45.725","Text":"which is 1 minus 3 times 0,"},{"Start":"03:45.725 ","End":"03:50.150","Text":"and that is equal to1."},{"Start":"03:50.150 ","End":"03:52.940","Text":"If I write it,"},{"Start":"03:52.940 ","End":"03:55.970","Text":"I like to use the curly braces."},{"Start":"03:55.970 ","End":"03:58.520","Text":"I say x=0,"},{"Start":"03:58.520 ","End":"04:04.175","Text":"y=1 is one solution."},{"Start":"04:04.175 ","End":"04:06.200","Text":"In the other case,"},{"Start":"04:06.200 ","End":"04:08.270","Text":"we get that y is equal to,"},{"Start":"04:08.270 ","End":"04:10.055","Text":"again, 1 minus 3x,"},{"Start":"04:10.055 ","End":"04:17.545","Text":"which is 1 minus 3 times 24/37."},{"Start":"04:17.545 ","End":"04:20.500","Text":"An exercise in fractions."},{"Start":"04:20.500 ","End":"04:25.505","Text":"Let\u0027s put everything over a common denominator, 37,"},{"Start":"04:25.505 ","End":"04:29.555","Text":"and then one is just 37/37,"},{"Start":"04:29.555 ","End":"04:31.070","Text":"and here we have minus,"},{"Start":"04:31.070 ","End":"04:37.530","Text":"let\u0027s see, 3 times 24 is 72."},{"Start":"04:39.270 ","End":"04:46.305","Text":"That equals minus 35/ 37."},{"Start":"04:46.305 ","End":"04:48.155","Text":"If I write it,"},{"Start":"04:48.155 ","End":"04:50.180","Text":"the solution as a pair,"},{"Start":"04:50.180 ","End":"04:56.375","Text":"I get x=24/ 37"},{"Start":"04:56.375 ","End":"05:03.526","Text":"and y equals minus 35/37."},{"Start":"05:03.526 ","End":"05:05.740","Text":"Why don\u0027t I just highlight them a bit."},{"Start":"05:05.740 ","End":"05:09.967","Text":"The solution, there is solution 1 and there is solution 2."},{"Start":"05:09.967 ","End":"05:12.370","Text":"Basically we\u0027re done."},{"Start":"05:12.370 ","End":"05:18.339","Text":"But sometimes I like to verify the solutions by substituting in the original."},{"Start":"05:18.339 ","End":"05:20.260","Text":"I\u0027ll just verify this one."},{"Start":"05:20.260 ","End":"05:22.780","Text":"I don\u0027t want to do a lot of work with fractions."},{"Start":"05:22.780 ","End":"05:24.790","Text":"Let\u0027s see if this one works."},{"Start":"05:24.790 ","End":"05:27.540","Text":"We have to plug it in both."},{"Start":"05:27.540 ","End":"05:29.185","Text":"In the first one,"},{"Start":"05:29.185 ","End":"05:39.670","Text":"I have to ask if y equals 1 minus 3x is 1 equal to 1 minus 3 times 0?"},{"Start":"05:39.740 ","End":"05:49.030","Text":"Then I have to check if 0^2/2 plus 2y^2,"},{"Start":"05:50.270 ","End":"05:55.450","Text":"if this equals 2?"},{"Start":"05:55.450 ","End":"05:57.118","Text":"In both cases,"},{"Start":"05:57.118 ","End":"06:00.755","Text":"the answer is yes because this is 1=1,"},{"Start":"06:00.755 ","End":"06:06.905","Text":"yes, and here\u0027s 0 plus 2 times 1^2 is 2 is equal to 2."},{"Start":"06:06.905 ","End":"06:10.040","Text":"This one is verified."},{"Start":"06:10.040 ","End":"06:13.160","Text":"I\u0027ll leave it at that. I don\u0027t want to verify."},{"Start":"06:13.160 ","End":"06:15.230","Text":"The other one is too much work with fractions."},{"Start":"06:15.230 ","End":"06:17.165","Text":"I\u0027ll leave that up to you if you want to do it,"},{"Start":"06:17.165 ","End":"06:20.400","Text":"and we\u0027ll just stop here."}],"ID":5022},{"Watched":false,"Name":"Exercise 10","Duration":"8m 4s","ChapterTopicVideoID":5021,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.340","Text":"Here we have a non-linear system of 2 equations and 2 unknowns,"},{"Start":"00:05.340 ","End":"00:07.005","Text":"x, and y,"},{"Start":"00:07.005 ","End":"00:14.490","Text":"and I don\u0027t have 1 variable expressed in terms of the other,"},{"Start":"00:14.490 ","End":"00:18.870","Text":"which is an ideal situation but if I look at the first one,"},{"Start":"00:18.870 ","End":"00:21.720","Text":"I see that without any trouble I could get y in"},{"Start":"00:21.720 ","End":"00:25.065","Text":"terms of x and then substitute in the second."},{"Start":"00:25.065 ","End":"00:28.335","Text":"So let me just copy down here,"},{"Start":"00:28.335 ","End":"00:30.655","Text":"and let\u0027s do that."},{"Start":"00:30.655 ","End":"00:32.705","Text":"From the first one,"},{"Start":"00:32.705 ","End":"00:38.420","Text":"I can get y explicitly in terms of x as 2x minus 1,"},{"Start":"00:38.420 ","End":"00:41.674","Text":"just brought the 2x over to the other side."},{"Start":"00:41.674 ","End":"00:45.245","Text":"Now that I have that I could substitute y,"},{"Start":"00:45.245 ","End":"00:47.325","Text":"which only appears here,"},{"Start":"00:47.325 ","End":"00:50.450","Text":"and then we\u0027ll get from the second equation,"},{"Start":"00:50.450 ","End":"00:54.890","Text":"x^2 plus 2y are"},{"Start":"00:54.890 ","End":"01:00.760","Text":"replaced by (2x-1)^2 = 3."},{"Start":"01:00.760 ","End":"01:03.050","Text":"Now I have an equation and just one"},{"Start":"01:03.050 ","End":"01:05.329","Text":"unknown and it looks like it\u0027s going to be a quadratic."},{"Start":"01:05.329 ","End":"01:08.490","Text":"We just need to rearrange it a bit."},{"Start":"01:09.520 ","End":"01:15.830","Text":"So x^2 plus. Now, let\u0027s do 2 operations at once."},{"Start":"01:15.830 ","End":"01:20.415","Text":"We\u0027ll square this and I\u0027ll just remind you of the formula that"},{"Start":"01:20.415 ","End":"01:28.250","Text":"(a-b)^2 is a^2 minus 2ab plus b^2."},{"Start":"01:28.250 ","End":"01:31.340","Text":"I\u0027ll do that here, but I\u0027ll also double each term from the 2."},{"Start":"01:31.340 ","End":"01:34.775","Text":"(2x)^2 is 4x^2."},{"Start":"01:34.775 ","End":"01:38.320","Text":"I don\u0027t write 4x^2, I write 8x^2."},{"Start":"01:38.320 ","End":"01:43.855","Text":"The middle term would be minus 4x,"},{"Start":"01:43.855 ","End":"01:46.940","Text":"but I double it so it\u0027s minus 8x."},{"Start":"01:46.940 ","End":"01:52.635","Text":"The last term 1 squared also double that is plus 2,"},{"Start":"01:52.635 ","End":"01:54.230","Text":"then I have equals 3,"},{"Start":"01:54.230 ","End":"01:57.830","Text":"so I will write it as minus 3 equals 0."},{"Start":"01:57.830 ","End":"02:06.835","Text":"Now, just to collect like terms and still the minority of the x^2 first,"},{"Start":"02:06.835 ","End":"02:12.865","Text":"1x^2 plus 8x^2 is 9x^2."},{"Start":"02:12.865 ","End":"02:18.690","Text":"Then the x\u0027s I only have from here minus 8x."},{"Start":"02:19.130 ","End":"02:27.025","Text":"Numbers, 2 minus 3 is minus 1 is equal to 0."},{"Start":"02:27.025 ","End":"02:29.989","Text":"Now you know how to solve quadratic equations."},{"Start":"02:29.989 ","End":"02:31.460","Text":"I don\u0027t want to waste time with that."},{"Start":"02:31.460 ","End":"02:33.375","Text":"I\u0027ll just give you the answer."},{"Start":"02:33.375 ","End":"02:35.080","Text":"There are 2 answers,"},{"Start":"02:35.080 ","End":"02:42.710","Text":"x equals 1 or x equals minus 1/9."},{"Start":"02:42.780 ","End":"02:45.400","Text":"You could check by substitution."},{"Start":"02:45.400 ","End":"02:48.220","Text":"Well, let\u0027s just check the easy one, x equals 1."},{"Start":"02:48.220 ","End":"02:51.700","Text":"So 9 times 1 squared is 9 minus 8 minus"},{"Start":"02:51.700 ","End":"02:56.330","Text":"1 here that\u0027s 0 and check if the other one works too."},{"Start":"02:56.330 ","End":"02:59.370","Text":"At this point,"},{"Start":"02:59.370 ","End":"03:04.915","Text":"the easiest thing to do to get y in each case is to substitute here."},{"Start":"03:04.915 ","End":"03:08.365","Text":"Because here we have explicitly y in terms of x."},{"Start":"03:08.365 ","End":"03:12.970","Text":"If I plug in x equals 1 into this,"},{"Start":"03:12.970 ","End":"03:16.295","Text":"I get y equals twice 1 minus 1,"},{"Start":"03:16.295 ","End":"03:19.215","Text":"which is also 1,"},{"Start":"03:19.215 ","End":"03:24.470","Text":"and if I plug in minus 1/9,"},{"Start":"03:24.470 ","End":"03:32.015","Text":"then I\u0027ll get that y equals minus 2/9 minus 1,"},{"Start":"03:32.015 ","End":"03:34.070","Text":"1 is 99,"},{"Start":"03:34.070 ","End":"03:38.390","Text":"so I get minus 11/9."},{"Start":"03:38.390 ","End":"03:41.910","Text":"So y equals minus 11/9."},{"Start":"03:42.160 ","End":"03:45.830","Text":"I\u0027ll leave it as an improper fraction."},{"Start":"03:45.830 ","End":"03:55.270","Text":"We could also write it if you want to as minus 1 and 2/9 as a mixed number."},{"Start":"03:55.270 ","End":"03:57.320","Text":"So we have 2 solutions."},{"Start":"03:57.320 ","End":"04:01.820","Text":"The first solution is x equals 1,"},{"Start":"04:01.820 ","End":"04:04.670","Text":"y equals 1,"},{"Start":"04:04.670 ","End":"04:07.175","Text":"and the second solution,"},{"Start":"04:07.175 ","End":"04:11.990","Text":"x equals minus 1/9,"},{"Start":"04:11.990 ","End":"04:18.665","Text":"y equals minus 1 and 2/9."},{"Start":"04:18.665 ","End":"04:21.450","Text":"Highlight the solutions."},{"Start":"04:21.490 ","End":"04:24.695","Text":"We are done but as you know,"},{"Start":"04:24.695 ","End":"04:27.800","Text":"I like to also verify the solutions."},{"Start":"04:27.800 ","End":"04:32.450","Text":"It\u0027s a good habit and often you can detect errors,"},{"Start":"04:32.450 ","End":"04:38.040","Text":"so let\u0027s do that and we\u0027ll substitute in the original."},{"Start":"04:38.040 ","End":"04:44.120","Text":"For this one, I have to check 2 things that y minus 2x is 1."},{"Start":"04:44.120 ","End":"04:47.940","Text":"In other words, that y,"},{"Start":"04:47.940 ","End":"04:51.585","Text":"which is 1 minus twice x,"},{"Start":"04:51.585 ","End":"04:53.415","Text":"which is 1,"},{"Start":"04:53.415 ","End":"04:58.865","Text":"but it\u0027s equal to the question mark that\u0027s what I\u0027m verifying minus 1."},{"Start":"04:58.865 ","End":"05:02.765","Text":"The other thing I have to check is if x^2,"},{"Start":"05:02.765 ","End":"05:05.015","Text":"which is 1^2,"},{"Start":"05:05.015 ","End":"05:11.220","Text":"plus twice y is also 1^2,"},{"Start":"05:11.220 ","End":"05:14.010","Text":"is this equal to 3?"},{"Start":"05:14.010 ","End":"05:16.995","Text":"In this case, pretty clear,"},{"Start":"05:16.995 ","End":"05:22.420","Text":"1 minus 2 is minus 1 and 1 plus 2 is 3,"},{"Start":"05:22.810 ","End":"05:27.245","Text":"which means that this solution is verified."},{"Start":"05:27.245 ","End":"05:30.550","Text":"For the other one, we need to do a bit of fractions,"},{"Start":"05:30.550 ","End":"05:32.210","Text":"and I\u0027ve changed my mind."},{"Start":"05:32.210 ","End":"05:36.950","Text":"I want to write this as an improper fraction,"},{"Start":"05:36.950 ","End":"05:38.900","Text":"this form rather than this."},{"Start":"05:38.900 ","End":"05:41.345","Text":"Let\u0027s see what do we have to check."},{"Start":"05:41.345 ","End":"05:43.070","Text":"From the first equation,"},{"Start":"05:43.070 ","End":"05:48.585","Text":"I need to check that minus 11 over 9,"},{"Start":"05:48.585 ","End":"05:53.670","Text":"minus twice, minus 1/9,"},{"Start":"05:53.670 ","End":"05:57.580","Text":"y minus 2x is minus 1."},{"Start":"06:00.530 ","End":"06:02.610","Text":"I\u0027ll leave a bit of space."},{"Start":"06:02.610 ","End":"06:08.615","Text":"And the next thing I have to check is that x^2 plus 2y^2"},{"Start":"06:08.615 ","End":"06:18.765","Text":"minus 1/9 squared plus twice minus 11 over 9 squared."},{"Start":"06:18.765 ","End":"06:22.275","Text":"Is this equal to 3?"},{"Start":"06:22.275 ","End":"06:24.030","Text":"Well, let\u0027s see."},{"Start":"06:24.030 ","End":"06:25.580","Text":"In the first case I have,"},{"Start":"06:25.580 ","End":"06:28.560","Text":"if I put everything over 9,"},{"Start":"06:30.170 ","End":"06:33.525","Text":"I get minus 11,"},{"Start":"06:33.525 ","End":"06:42.840","Text":"and then minus 2 times minus 1 is plus 2/9."},{"Start":"06:42.840 ","End":"06:47.445","Text":"This is equal to minus 9 over 9."},{"Start":"06:47.445 ","End":"06:51.460","Text":"This is equal to minus 1."},{"Start":"06:52.430 ","End":"06:56.805","Text":"This is okay. As for this one,"},{"Start":"06:56.805 ","End":"06:59.050","Text":"we\u0027ll do the left-hand side."},{"Start":"06:59.050 ","End":"07:02.310","Text":"Since everything is squared here,"},{"Start":"07:03.010 ","End":"07:06.025","Text":"I can ignore the minuses."},{"Start":"07:06.025 ","End":"07:11.665","Text":"1/9 squared is 1/81."},{"Start":"07:11.665 ","End":"07:15.105","Text":"In fact, we\u0027re going to put everything over 81."},{"Start":"07:15.105 ","End":"07:18.000","Text":"The next one is,"},{"Start":"07:18.000 ","End":"07:21.195","Text":"11 squared is 121."},{"Start":"07:21.195 ","End":"07:30.210","Text":"121 times 2,"},{"Start":"07:30.210 ","End":"07:39.090","Text":"this gives us twice a 121 is 242, which is 243/81."},{"Start":"07:39.090 ","End":"07:40.445","Text":"If you do the division,"},{"Start":"07:40.445 ","End":"07:42.650","Text":"you\u0027ll see that this is equal to 3."},{"Start":"07:42.650 ","End":"07:46.510","Text":"I mean, just look 81 times 3 is 243."},{"Start":"07:46.510 ","End":"07:48.825","Text":"That\u0027s okay too."},{"Start":"07:48.825 ","End":"07:53.070","Text":"The second one has been verified also."},{"Start":"07:53.070 ","End":"07:56.170","Text":"Let\u0027s say, if you have the time,"},{"Start":"07:56.170 ","End":"08:00.690","Text":"I recommend verifying solutions."}],"ID":5023},{"Watched":false,"Name":"Exercise 11","Duration":"7m 54s","ChapterTopicVideoID":5022,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.410","Text":"Here we have a system of nonlinear equations,"},{"Start":"00:04.410 ","End":"00:07.635","Text":"2 equations and 2 unknowns, x and y."},{"Start":"00:07.635 ","End":"00:11.250","Text":"The usual way to solve is to try and"},{"Start":"00:11.250 ","End":"00:15.195","Text":"isolate 1 variable in terms of the other and substitute."},{"Start":"00:15.195 ","End":"00:19.170","Text":"In this case, the first equation"},{"Start":"00:19.170 ","End":"00:23.070","Text":"is easier to work with to get y in terms of x or the other way around."},{"Start":"00:23.070 ","End":"00:24.990","Text":"They\u0027re both equally difficult,"},{"Start":"00:24.990 ","End":"00:27.720","Text":"so let\u0027s copy it over here."},{"Start":"00:27.720 ","End":"00:31.665","Text":"I\u0027ll get y in terms of x."},{"Start":"00:31.665 ","End":"00:39.990","Text":"From here it\u0027s easy to see that y=4 over x,"},{"Start":"00:39.990 ","End":"00:42.230","Text":"but of course, x mustn\u0027t be 0."},{"Start":"00:42.230 ","End":"00:45.800","Text":"If at the end we find that x=0,"},{"Start":"00:45.800 ","End":"00:47.980","Text":"then we\u0027ll have to disqualify it,"},{"Start":"00:47.980 ","End":"00:51.375","Text":"but that probably won\u0027t happen."},{"Start":"00:51.375 ","End":"00:52.815","Text":"Anyway, we\u0027ll see."},{"Start":"00:52.815 ","End":"00:55.235","Text":"Now that we have y in terms of x,"},{"Start":"00:55.235 ","End":"01:01.010","Text":"we then substitute y into the second equation."},{"Start":"01:01.010 ","End":"01:03.035","Text":"In fact, the y just appears here,"},{"Start":"01:03.035 ","End":"01:08.990","Text":"so we get 25x^2 plus 4 and instead of y,"},{"Start":"01:08.990 ","End":"01:14.525","Text":"I write (4/x)^2 =100."},{"Start":"01:14.525 ","End":"01:18.120","Text":"Now I have an equation and just 1 variable x,"},{"Start":"01:18.120 ","End":"01:21.580","Text":"so let\u0027s proceed to solve that."},{"Start":"01:22.030 ","End":"01:25.535","Text":"I\u0027d like to get rid of this denominator here,"},{"Start":"01:25.535 ","End":"01:29.105","Text":"so let\u0027s multiply both sides by x^2."},{"Start":"01:29.105 ","End":"01:36.680","Text":"Then we have 25x^4 and then plus 4 times"},{"Start":"01:36.680 ","End":"01:41.870","Text":"4^2 is 4 times 4 times 4"},{"Start":"01:41.870 ","End":"01:47.135","Text":"is 64 over x^2,"},{"Start":"01:47.135 ","End":"01:48.875","Text":"but times x^2,"},{"Start":"01:48.875 ","End":"01:57.675","Text":"so it\u0027s just 64 and that is equal to 100x^2."},{"Start":"01:57.675 ","End":"02:01.930","Text":"Now let\u0027s just rewrite it in order"},{"Start":"02:02.120 ","End":"02:10.540","Text":"25x^4 minus a 100x^2 plus 64=0."},{"Start":"02:10.540 ","End":"02:13.070","Text":"This is not a quadratic equation,"},{"Start":"02:13.070 ","End":"02:14.470","Text":"it\u0027s a degree 4,"},{"Start":"02:14.470 ","End":"02:17.140","Text":"but the odd powers are missing,"},{"Start":"02:17.140 ","End":"02:22.820","Text":"so the standard trick is a substitution where we substitute that"},{"Start":"02:22.820 ","End":"02:30.215","Text":"t=x^2 and then we get a quadratic in t because x^4=(x^2)^2,"},{"Start":"02:30.215 ","End":"02:37.950","Text":"so what we get is 25t^2 minus 100x^2 is t plus"},{"Start":"02:37.950 ","End":"02:46.730","Text":"64 =0 and now it\u0027s a quadratic in t. I know you know how to solve quadratic equations,"},{"Start":"02:46.730 ","End":"02:49.970","Text":"so I\u0027ll just give you the answer."},{"Start":"02:49.970 ","End":"02:51.440","Text":"There\u0027s 2 answers."},{"Start":"02:51.440 ","End":"02:57.380","Text":"1 is that t=16/5."},{"Start":"02:57.380 ","End":"03:07.710","Text":"I prefer it as an improper fraction in this case and the other possibility is t=4/5,"},{"Start":"03:08.810 ","End":"03:13.805","Text":"and that\u0027s t. Now we have to get back to x."},{"Start":"03:13.805 ","End":"03:15.485","Text":"So since t=x^2,"},{"Start":"03:15.485 ","End":"03:21.810","Text":"we have that x^2= 16/5 or"},{"Start":"03:21.810 ","End":"03:29.500","Text":"x^2=4/5 and now to get to x,"},{"Start":"03:29.500 ","End":"03:32.244","Text":"we have to take the square root plus or minus."},{"Start":"03:32.244 ","End":"03:35.125","Text":"So actually we have 4 possibilities for x."},{"Start":"03:35.125 ","End":"03:39.390","Text":"For x, it could be taking the square root,"},{"Start":"03:39.390 ","End":"03:43.320","Text":"it could be 4 over the square root of 5,"},{"Start":"03:43.320 ","End":"03:47.175","Text":"could be -4 over the square root of 5."},{"Start":"03:47.175 ","End":"03:51.870","Text":"Could be square root of 2 is 2 over square root of"},{"Start":"03:51.870 ","End":"03:57.460","Text":"5 and it could be -2 over the square root of 5,"},{"Start":"03:57.590 ","End":"04:02.290","Text":"so that\u0027s already 4 possibilities."},{"Start":"04:02.290 ","End":"04:07.525","Text":"Let\u0027s see what we get for y in each 1 of these cases."},{"Start":"04:07.525 ","End":"04:10.315","Text":"To find y for each of these,"},{"Start":"04:10.315 ","End":"04:12.610","Text":"we have it here,"},{"Start":"04:12.610 ","End":"04:14.620","Text":"y in terms of x,"},{"Start":"04:14.620 ","End":"04:20.380","Text":"so let\u0027s see what we get in each case and I\u0027m corresponding them,"},{"Start":"04:20.380 ","End":"04:22.895","Text":"each x with its y."},{"Start":"04:22.895 ","End":"04:25.630","Text":"If x is 4 over root 5,"},{"Start":"04:25.630 ","End":"04:30.555","Text":"then for example, write it,"},{"Start":"04:30.555 ","End":"04:35.970","Text":"y is 4 over 4 over root 5,"},{"Start":"04:35.970 ","End":"04:37.930","Text":"and then when you divide by a fraction,"},{"Start":"04:37.930 ","End":"04:40.315","Text":"you multiply by the inverse fraction,"},{"Start":"04:40.315 ","End":"04:45.465","Text":"so y comes out to be just the square root of 5."},{"Start":"04:45.465 ","End":"04:48.675","Text":"It\u0027s 4 root 5 over 4."},{"Start":"04:48.675 ","End":"04:53.195","Text":"In the second case,"},{"Start":"04:53.195 ","End":"04:56.180","Text":"clearly, this is minus,"},{"Start":"04:56.180 ","End":"04:58.085","Text":"I\u0027m just going to get minus."},{"Start":"04:58.085 ","End":"05:00.960","Text":"Well, try and get organized here."},{"Start":"05:00.960 ","End":"05:02.355","Text":"Okay. For this x,"},{"Start":"05:02.355 ","End":"05:06.740","Text":"we said that root 5 would be the corresponding 1."},{"Start":"05:06.740 ","End":"05:11.780","Text":"In this case, negative would just make the thing negative,"},{"Start":"05:11.780 ","End":"05:15.185","Text":"so it\u0027s going to be minus root 5."},{"Start":"05:15.185 ","End":"05:20.300","Text":"In this case, I will get that y equals 4 over x,"},{"Start":"05:20.300 ","End":"05:25.065","Text":"which is 4 over 2 over root 5."},{"Start":"05:25.065 ","End":"05:28.560","Text":"This comes out to be root 5 goes to the top,"},{"Start":"05:28.560 ","End":"05:29.940","Text":"but I still have 4 over 2,"},{"Start":"05:29.940 ","End":"05:32.910","Text":"which is 2, so it\u0027s 2 root 5."},{"Start":"05:32.910 ","End":"05:34.620","Text":"I\u0027m going to write that here,"},{"Start":"05:34.620 ","End":"05:38.565","Text":"2 root 5 and in the last case,"},{"Start":"05:38.565 ","End":"05:41.450","Text":"similarly, if I make x negative,"},{"Start":"05:41.450 ","End":"05:43.535","Text":"then y just comes out negative."},{"Start":"05:43.535 ","End":"05:50.870","Text":"I have -2 root 5."},{"Start":"05:50.870 ","End":"05:53.170","Text":"We have 4 pairs."},{"Start":"05:53.170 ","End":"05:55.590","Text":"I won\u0027t write them all,"},{"Start":"05:55.590 ","End":"06:00.330","Text":"but the first pair is x equals, y equals,"},{"Start":"06:00.330 ","End":"06:03.435","Text":"x is 4 over root 5,"},{"Start":"06:03.435 ","End":"06:09.740","Text":"y is root 5 and for each of these,"},{"Start":"06:09.740 ","End":"06:11.315","Text":"I get a solution."},{"Start":"06:11.315 ","End":"06:12.590","Text":"I\u0027m just saving time,"},{"Start":"06:12.590 ","End":"06:13.865","Text":"I won\u0027t write them all,"},{"Start":"06:13.865 ","End":"06:15.770","Text":"but these correspond,"},{"Start":"06:15.770 ","End":"06:17.525","Text":"this with this, this with this,"},{"Start":"06:17.525 ","End":"06:21.270","Text":"each 1 with its corresponding."},{"Start":"06:23.000 ","End":"06:27.410","Text":"Put some lines, so we\u0027d see that this goes with this."},{"Start":"06:27.410 ","End":"06:30.665","Text":"It\u0027s not that I can mix and match 1 of these and 1 of these."},{"Start":"06:30.665 ","End":"06:36.590","Text":"Okay. I\u0027ve got 4 of these possibilities and let me just check,"},{"Start":"06:36.590 ","End":"06:39.000","Text":"verify 1 of them,"},{"Start":"06:39.000 ","End":"06:42.110","Text":"say this 1, that it really is a solution."},{"Start":"06:42.110 ","End":"06:45.185","Text":"I need to see if xy=4,"},{"Start":"06:45.185 ","End":"06:53.850","Text":"so xy is 4 over root 5 times root"},{"Start":"06:53.850 ","End":"06:57.655","Text":"5 and this equals 4."},{"Start":"06:57.655 ","End":"07:02.240","Text":"So this 1 is first equation satisfied and the second 1,"},{"Start":"07:02.240 ","End":"07:08.210","Text":"let\u0027s see, 25x^2 plus 4y^2."},{"Start":"07:08.210 ","End":"07:10.730","Text":"Let\u0027s see if we can get it to be a 100,"},{"Start":"07:10.730 ","End":"07:19.200","Text":"is 25x^2 is 16 over 5"},{"Start":"07:19.810 ","End":"07:25.040","Text":"and then 4y^2,"},{"Start":"07:25.040 ","End":"07:27.310","Text":"y^2 is just 5."},{"Start":"07:27.310 ","End":"07:29.490","Text":"Let\u0027s see what we get."},{"Start":"07:29.490 ","End":"07:37.350","Text":"25 over 5 is 5 times 16 is 80 plus 20."},{"Start":"07:37.350 ","End":"07:39.885","Text":"This is equal to a 100."},{"Start":"07:39.885 ","End":"07:43.110","Text":"So yes, it works also."},{"Start":"07:43.110 ","End":"07:48.230","Text":"So this pair is verified and similarly for the other 3 which I\u0027ve"},{"Start":"07:48.230 ","End":"07:54.940","Text":"just left for you to complete and we are done."}],"ID":5024},{"Watched":false,"Name":"Exercise 12","Duration":"9m 44s","ChapterTopicVideoID":5023,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"Here we have a system of nonlinear equations,"},{"Start":"00:03.390 ","End":"00:05.220","Text":"2 equations in 2 unknowns,"},{"Start":"00:05.220 ","End":"00:06.720","Text":"x and y,"},{"Start":"00:06.720 ","End":"00:13.260","Text":"and the strategy, as usual is to try and get 1 variable in terms of the other."},{"Start":"00:13.260 ","End":"00:17.610","Text":"The first equation is most amenable for that."},{"Start":"00:17.610 ","End":"00:21.970","Text":"There\u0027s equations down here and"},{"Start":"00:22.100 ","End":"00:25.470","Text":"I\u0027ll use this to write y in terms of x. I"},{"Start":"00:25.470 ","End":"00:28.395","Text":"could have done at x in terms of y doesn\u0027t really matter."},{"Start":"00:28.395 ","End":"00:33.765","Text":"From here I get y equals 6 over x,"},{"Start":"00:33.765 ","End":"00:37.039","Text":"just makes sure that the end that x doesn\u0027t come out 0."},{"Start":"00:37.039 ","End":"00:41.195","Text":"Otherwise something is wrong because we can\u0027t divide by 0."},{"Start":"00:41.195 ","End":"00:45.530","Text":"Then we substitute this y into the second equation,"},{"Start":"00:45.530 ","End":"00:47.585","Text":"and then we\u0027ll get something just in x."},{"Start":"00:47.585 ","End":"00:52.880","Text":"What we get in fact is that x^2 over 2 plus,"},{"Start":"00:52.880 ","End":"00:56.120","Text":"instead of y, I put 6 over x."},{"Start":"00:56.120 ","End":"01:06.780","Text":"What I get is 6 over x^2 over 3 equals 5."},{"Start":"01:06.780 ","End":"01:11.830","Text":"Now this is just 1 equation in 1 unknown x,"},{"Start":"01:11.830 ","End":"01:14.270","Text":"let\u0027s see if we can simplify it."},{"Start":"01:14.270 ","End":"01:18.730","Text":"What I\u0027m going to do is I\u0027ll do 2 steps in 1."},{"Start":"01:18.730 ","End":"01:23.450","Text":"I\u0027ll multiply both sides by x^2 in order to get rid of the fraction and I"},{"Start":"01:23.450 ","End":"01:28.310","Text":"will also cancel the numbers. Let\u0027s see."},{"Start":"01:28.310 ","End":"01:31.170","Text":"Multiplying by X^2."},{"Start":"01:31.420 ","End":"01:37.610","Text":"Perhaps I\u0027ll just indicate that the multiplying by X^2 both sides and"},{"Start":"01:37.610 ","End":"01:43.560","Text":"then I\u0027ve got x^4 over 2 plus,"},{"Start":"01:43.560 ","End":"01:50.190","Text":"now the 6 over x^2 is 6^2 over x^2 but the x^2 cancels,"},{"Start":"01:50.190 ","End":"01:55.020","Text":"so it\u0027s just 6^2 over 3."},{"Start":"01:55.020 ","End":"01:57.910","Text":"This is equal to 5x^2."},{"Start":"01:58.040 ","End":"02:03.470","Text":"Now I\u0027m going to bring everything to the left-hand side"},{"Start":"02:03.470 ","End":"02:10.030","Text":"and slightly rearrange. Let\u0027s see."},{"Start":"02:20.720 ","End":"02:22.850","Text":"Well, you know what,"},{"Start":"02:22.850 ","End":"02:24.335","Text":"I\u0027ll leave that, sorry."},{"Start":"02:24.335 ","End":"02:28.355","Text":"We\u0027ll just rearrange a bit."},{"Start":"02:28.355 ","End":"02:33.350","Text":"Got x^4 over 2 and then minus"},{"Start":"02:33.350 ","End":"02:41.620","Text":"5x^2 and then plus 6^2 over 3 is 36 over 3,"},{"Start":"02:41.620 ","End":"02:45.560","Text":"which is 12 equals 0."},{"Start":"02:45.560 ","End":"02:50.915","Text":"Now I\u0027ll just get rid of that fraction by multiplying by 2 so"},{"Start":"02:50.915 ","End":"02:59.000","Text":"x^4 minus 10x^2 plus 24 equals 0."},{"Start":"02:59.000 ","End":"03:02.840","Text":"As a standard trick here when you have a degree 4 equation,"},{"Start":"03:02.840 ","End":"03:05.030","Text":"but there\u0027s no x cubed or x,"},{"Start":"03:05.030 ","End":"03:06.830","Text":"the powers are missing,"},{"Start":"03:06.830 ","End":"03:09.740","Text":"then the trick is to substitute some latter,"},{"Start":"03:09.740 ","End":"03:15.290","Text":"say t to be X^2 and that will give us a quadratic in t,"},{"Start":"03:15.290 ","End":"03:19.400","Text":"as you\u0027ll see, x^4 is just t^2."},{"Start":"03:19.400 ","End":"03:21.855","Text":"X^2 is t,"},{"Start":"03:21.855 ","End":"03:26.585","Text":"10^2 minus 10t plus 24 equals 0."},{"Start":"03:26.585 ","End":"03:28.685","Text":"Straightforward quadratic."},{"Start":"03:28.685 ","End":"03:32.285","Text":"But since you know how to solve quadratics,"},{"Start":"03:32.285 ","End":"03:36.965","Text":"I\u0027ll just save the time I\u0027m giving you the answer right away."},{"Start":"03:36.965 ","End":"03:45.340","Text":"That t is either equal to 4 or t is equal to 6."},{"Start":"03:45.950 ","End":"03:52.445","Text":"Let\u0027s just check one of them to see that I haven\u0027t done something grossly wrong."},{"Start":"03:52.445 ","End":"03:54.845","Text":"Let\u0027s see, 4^2 is 16,"},{"Start":"03:54.845 ","End":"03:59.165","Text":"16 minus 40 plus 24,"},{"Start":"03:59.165 ","End":"04:02.155","Text":"the pluses are 16 and 24, which is 40,"},{"Start":"04:02.155 ","End":"04:05.650","Text":"works similarly, take my word for it."},{"Start":"04:05.650 ","End":"04:07.795","Text":"These are the 2 solutions."},{"Start":"04:07.795 ","End":"04:12.440","Text":"That\u0027s t. What about x?"},{"Start":"04:13.550 ","End":"04:16.275","Text":"I\u0027ll continue over here."},{"Start":"04:16.275 ","End":"04:25.060","Text":"What I get is that either X^2 equals 4 or X^2 equals 6."},{"Start":"04:25.060 ","End":"04:29.110","Text":"Now, I\u0027ve got 4 possibilities for x,"},{"Start":"04:29.110 ","End":"04:34.250","Text":"because x could equal one of the following."},{"Start":"04:34.250 ","End":"04:38.430","Text":"It could be square root of 4 plus or minus,"},{"Start":"04:38.430 ","End":"04:47.625","Text":"so that\u0027s 2 or minus 2 and then plus or minus the square root of 6, 4 possibilities."},{"Start":"04:47.625 ","End":"04:51.365","Text":"Now, each x has its corresponding y."},{"Start":"04:51.365 ","End":"04:55.650","Text":"We just substitute in this here"},{"Start":"04:55.650 ","End":"05:03.850","Text":"and the corresponding y in each case is if I substitute here,"},{"Start":"05:03.850 ","End":"05:08.750","Text":"6 over 2 is 3."},{"Start":"05:08.750 ","End":"05:13.840","Text":"6 over minus 2 is minus 3."},{"Start":"05:13.840 ","End":"05:18.605","Text":"6 over root 6 is just root 6."},{"Start":"05:18.605 ","End":"05:20.050","Text":"Hope you\u0027re okay with that."},{"Start":"05:20.050 ","End":"05:23.720","Text":"If not, let\u0027s say 6 over root 6."},{"Start":"05:23.970 ","End":"05:27.465","Text":"There is several ways of doing this."},{"Start":"05:27.465 ","End":"05:29.930","Text":"One of them is just by saying,"},{"Start":"05:29.930 ","End":"05:35.770","Text":"because root 6 times root 6 is equal to 6,"},{"Start":"05:35.770 ","End":"05:39.605","Text":"since square root of anything times itself is the original number"},{"Start":"05:39.605 ","End":"05:43.965","Text":"and I divide both sides by root 6, I get this."},{"Start":"05:43.965 ","End":"05:45.815","Text":"This gives us straightaway."},{"Start":"05:45.815 ","End":"05:47.420","Text":"In other way of doing it,"},{"Start":"05:47.420 ","End":"05:50.930","Text":"you could multiply top and bottom by root 6 over root"},{"Start":"05:50.930 ","End":"05:54.710","Text":"6 and then you\u0027ll get a 6 in the denominator, which will cancel."},{"Start":"05:54.710 ","End":"05:57.810","Text":"Any of that, this is the answer,"},{"Start":"05:57.810 ","End":"06:03.240","Text":"6 over root 6 is just root 6."},{"Start":"06:03.240 ","End":"06:08.130","Text":"6 over minus root 6 is just"},{"Start":"06:08.130 ","End":"06:14.830","Text":"minus 6 over minus root 6 is just minus root 6."},{"Start":"06:15.370 ","End":"06:18.680","Text":"Sometimes I write the answers with curly braces."},{"Start":"06:18.680 ","End":"06:20.330","Text":"This time I\u0027ll just do it differently."},{"Start":"06:20.330 ","End":"06:23.120","Text":"I\u0027ll say there\u0027s 4 possibilities for x, y."},{"Start":"06:23.120 ","End":"06:26.405","Text":"We have possibility of 2,"},{"Start":"06:26.405 ","End":"06:32.250","Text":"3 or we have minus 2, minus 3,"},{"Start":"06:32.250 ","End":"06:35.340","Text":"or square root of 6,"},{"Start":"06:35.340 ","End":"06:37.380","Text":"square root of 6,"},{"Start":"06:37.380 ","End":"06:40.725","Text":"or minus square root of 6,"},{"Start":"06:40.725 ","End":"06:42.885","Text":"minus square root of 6."},{"Start":"06:42.885 ","End":"06:50.420","Text":"4 possible answers and I could end here,"},{"Start":"06:50.420 ","End":"06:57.960","Text":"or we could verify at least part of these by substituting in the original equation."},{"Start":"06:59.500 ","End":"07:03.320","Text":"At least they\u0027ll verify the first one. We\u0027ll see."},{"Start":"07:03.320 ","End":"07:05.750","Text":"Tell you what, I\u0027ll verify all of them."},{"Start":"07:05.750 ","End":"07:09.590","Text":"Let\u0027s see if we can make it organized in a way."},{"Start":"07:09.590 ","End":"07:13.280","Text":"I have to check 2 equations for each of these."},{"Start":"07:13.280 ","End":"07:15.680","Text":"But on the right-hand side I have a constant."},{"Start":"07:15.680 ","End":"07:18.620","Text":"I have an idea."},{"Start":"07:18.620 ","End":"07:20.405","Text":"I\u0027ll do it in a table."},{"Start":"07:20.405 ","End":"07:23.585","Text":"Make a table."},{"Start":"07:23.585 ","End":"07:27.605","Text":"What I want to do is check what xy is,"},{"Start":"07:27.605 ","End":"07:31.190","Text":"and then see that in each case it comes out to be 6."},{"Start":"07:31.190 ","End":"07:34.850","Text":"Then we\u0027ll also check what is x^2 over"},{"Start":"07:34.850 ","End":"07:41.040","Text":"2 plus y^2 over 3 and hope that we get 5 in each case."},{"Start":"07:41.040 ","End":"07:42.750","Text":"Let\u0027s see. In this case,"},{"Start":"07:42.750 ","End":"07:45.645","Text":"xy is just 2 times 3,"},{"Start":"07:45.645 ","End":"07:47.145","Text":"which is 6,"},{"Start":"07:47.145 ","End":"07:48.375","Text":"and that\u0027s what we want."},{"Start":"07:48.375 ","End":"07:50.845","Text":"This one is okay."},{"Start":"07:50.845 ","End":"07:53.435","Text":"I\u0027ll continue with the column."},{"Start":"07:53.435 ","End":"08:00.555","Text":"Minus 2 times minus 3 is 6 also, okay."},{"Start":"08:00.555 ","End":"08:05.175","Text":"Here, x times y is root 6 times root 6,"},{"Start":"08:05.175 ","End":"08:08.970","Text":"which is of course 6, okay."},{"Start":"08:08.970 ","End":"08:13.890","Text":"Minus root 6 times minus root 6,"},{"Start":"08:13.890 ","End":"08:19.210","Text":"minus and minus is plus and root 6 times root 6 is 6, checked."},{"Start":"08:19.210 ","End":"08:23.660","Text":"Now let\u0027s try the other one which is going to be a bit more involved,"},{"Start":"08:23.660 ","End":"08:25.865","Text":"x^2 over 2,"},{"Start":"08:25.865 ","End":"08:29.690","Text":"x^2 is 4 over 2,"},{"Start":"08:29.690 ","End":"08:34.200","Text":"and y^2 is 9 over 3."},{"Start":"08:34.200 ","End":"08:35.535","Text":"What does that equal?"},{"Start":"08:35.535 ","End":"08:36.990","Text":"4 over 2 is 2,"},{"Start":"08:36.990 ","End":"08:38.550","Text":"9 over 3 is 3,"},{"Start":"08:38.550 ","End":"08:40.245","Text":"2 plus 3,"},{"Start":"08:40.245 ","End":"08:43.620","Text":"without a calculator is 5."},{"Start":"08:43.620 ","End":"08:47.150","Text":"Good. We\u0027re hoping to get 5 in each of these cases."},{"Start":"08:47.150 ","End":"08:49.415","Text":"Now let\u0027s see."},{"Start":"08:49.415 ","End":"08:56.045","Text":"In this case we have negative 2 squared,"},{"Start":"08:56.045 ","End":"09:00.320","Text":"which is 4. Same thing."},{"Start":"09:00.320 ","End":"09:03.580","Text":"Because of the squared,"},{"Start":"09:03.710 ","End":"09:06.295","Text":"instead of x minus x,"},{"Start":"09:06.295 ","End":"09:08.980","Text":"it\u0027s going to work so we\u0027re fine here too."},{"Start":"09:08.980 ","End":"09:10.450","Text":"What about this?"},{"Start":"09:10.450 ","End":"09:12.265","Text":"We need x^2 over 2,"},{"Start":"09:12.265 ","End":"09:15.705","Text":"X^2 is 6 over 2."},{"Start":"09:15.705 ","End":"09:18.870","Text":"Here, y squared is also 6."},{"Start":"09:18.870 ","End":"09:20.760","Text":"This time it\u0027s over 3."},{"Start":"09:20.760 ","End":"09:22.695","Text":"What is this equal to?"},{"Start":"09:22.695 ","End":"09:24.585","Text":"3 plus 2,"},{"Start":"09:24.585 ","End":"09:27.640","Text":"which is 5, check."},{"Start":"09:27.640 ","End":"09:32.330","Text":"Same thing with the minus minus because it\u0027s squared and squared,"},{"Start":"09:32.330 ","End":"09:37.890","Text":"this one is going to work also so all 4 solutions have been verified"},{"Start":"09:37.890 ","End":"09:44.370","Text":"so we are really in good shape solved and verified. That\u0027s it."}],"ID":5025},{"Watched":false,"Name":"Exercise 13","Duration":"8m 30s","ChapterTopicVideoID":5024,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"Here we have a system of nonlinear equations,"},{"Start":"00:03.990 ","End":"00:06.960","Text":"two equations and two unknowns,"},{"Start":"00:06.960 ","End":"00:09.465","Text":"our favorites, x and y."},{"Start":"00:09.465 ","End":"00:14.535","Text":"In this case, we already have one variable in terms of the other."},{"Start":"00:14.535 ","End":"00:19.005","Text":"Let\u0027s use that and substitute y into the second equation."},{"Start":"00:19.005 ","End":"00:21.990","Text":"First, I\u0027ll just copy here."},{"Start":"00:21.990 ","End":"00:25.830","Text":"Then I\u0027ll plug y from here into the second equation,"},{"Start":"00:25.830 ","End":"00:27.600","Text":"y is just over here."},{"Start":"00:27.600 ","End":"00:33.105","Text":"So what we get is 3x^2 minus and instead of y,"},{"Start":"00:33.105 ","End":"00:35.670","Text":"1 minus 2x^2,"},{"Start":"00:35.670 ","End":"00:42.280","Text":"but it\u0027s squared over 3 equals 3."},{"Start":"00:42.760 ","End":"00:45.955","Text":"To compute the numerator here,"},{"Start":"00:45.955 ","End":"00:49.880","Text":"we\u0027ll use one of our favorite formulas."},{"Start":"00:49.880 ","End":"00:56.630","Text":"a minus b squared is a^2 minus 2ab plus b^2."},{"Start":"00:56.630 ","End":"00:58.745","Text":"I\u0027ll do that here, but at the same time,"},{"Start":"00:58.745 ","End":"01:03.365","Text":"I\u0027ll also multiply everything by 3 to get rid of this fraction."},{"Start":"01:03.365 ","End":"01:09.165","Text":"Here we have 9x^2 minus."},{"Start":"01:09.165 ","End":"01:13.170","Text":"Then we\u0027ll use this formula in a moment."},{"Start":"01:13.480 ","End":"01:19.010","Text":"Also this we multiply by 3 and that makes it 9,"},{"Start":"01:19.010 ","End":"01:21.649","Text":"but I\u0027ll write it on the other side."},{"Start":"01:21.649 ","End":"01:24.140","Text":"All I\u0027m missing now is this bit,"},{"Start":"01:24.140 ","End":"01:26.390","Text":"which is the expansion of this."},{"Start":"01:26.390 ","End":"01:31.370","Text":"That is, let\u0027s see, a is 1 and a^2 is 1 minus 2ab,"},{"Start":"01:31.370 ","End":"01:33.890","Text":"b is the other part."},{"Start":"01:33.890 ","End":"01:42.890","Text":"So we get minus 4x^2 and b^2 is 2^2 is 4,"},{"Start":"01:42.890 ","End":"01:47.080","Text":"and (x^2)^2 is x to the fourth."},{"Start":"01:47.080 ","End":"01:49.880","Text":"It\u0027s a degree 4 equation."},{"Start":"01:49.880 ","End":"01:53.435","Text":"Let\u0027s just organize and collect."},{"Start":"01:53.435 ","End":"01:59.675","Text":"What we have and because it\u0027s a minus here,"},{"Start":"01:59.675 ","End":"02:03.920","Text":"I\u0027ll just throw everything to the right-hand side."},{"Start":"02:03.920 ","End":"02:07.710","Text":"Or I\u0027ll just reverse signs over here."},{"Start":"02:10.660 ","End":"02:16.900","Text":"I\u0027ll just put everything on the right-hand side, 0 equals."},{"Start":"02:16.900 ","End":"02:19.675","Text":"Let\u0027s take them in order."},{"Start":"02:19.675 ","End":"02:21.475","Text":"x to the fourth terms,"},{"Start":"02:21.475 ","End":"02:24.575","Text":"here I have minus 4x to the fourth."},{"Start":"02:24.575 ","End":"02:27.875","Text":"On the other side, it\u0027s 4x^4."},{"Start":"02:27.875 ","End":"02:30.065","Text":"Now let\u0027s see what do we have with x^2?"},{"Start":"02:30.065 ","End":"02:34.130","Text":"We have 9x^2 minus, minus, plus."},{"Start":"02:34.130 ","End":"02:40.645","Text":"So it\u0027s got 13x^2 and on the other side, it\u0027s minus 13x^2."},{"Start":"02:40.645 ","End":"02:47.650","Text":"Then just constants minus 1 minus 9 is minus 10."},{"Start":"02:47.650 ","End":"02:50.960","Text":"On the other side, it\u0027s plus 10."},{"Start":"02:51.370 ","End":"02:58.255","Text":"Now we use our usual trick of substituting t"},{"Start":"02:58.255 ","End":"03:02.330","Text":"for x^2 and we get"},{"Start":"03:02.330 ","End":"03:09.740","Text":"4t^2 minus 13t plus 10 equals 0."},{"Start":"03:09.740 ","End":"03:15.140","Text":"It\u0027s a quadratic. As usual,"},{"Start":"03:15.140 ","End":"03:17.480","Text":"I\u0027ll just give you the answers."},{"Start":"03:17.480 ","End":"03:27.420","Text":"The answers are that t is equal to 2 or 5 over 4."},{"Start":"03:27.950 ","End":"03:31.560","Text":"Now we need to get back from t to x."},{"Start":"03:31.560 ","End":"03:33.160","Text":"Since t is x^2,"},{"Start":"03:33.160 ","End":"03:36.065","Text":"we just take plus or minus the square root of each."},{"Start":"03:36.065 ","End":"03:39.005","Text":"For x, we have four possibilities."},{"Start":"03:39.005 ","End":"03:44.590","Text":"We have the square root of 2 minus the square root of 2,"},{"Start":"03:44.590 ","End":"03:47.360","Text":"and then plus or minus the square root of this,"},{"Start":"03:47.360 ","End":"03:52.080","Text":"which is the square root of 5 over the square root of 4,"},{"Start":"03:52.080 ","End":"03:53.810","Text":"which is 2,"},{"Start":"03:53.810 ","End":"03:58.515","Text":"or minus the square root of 5 over 2."},{"Start":"03:58.515 ","End":"04:00.710","Text":"For each of these,"},{"Start":"04:00.710 ","End":"04:04.400","Text":"I need to find the corresponding y."},{"Start":"04:04.400 ","End":"04:09.870","Text":"Let\u0027s see, y would be in each case."},{"Start":"04:10.110 ","End":"04:16.465","Text":"I might need to do some side calculations if x is the square root of 2."},{"Start":"04:16.465 ","End":"04:18.730","Text":"Let\u0027s see if we can do it in our heads."},{"Start":"04:18.730 ","End":"04:26.215","Text":"Then I am substituting in here gives me y in terms of x."},{"Start":"04:26.215 ","End":"04:30.305","Text":"So if x is square root of 2, x^2 is 2,"},{"Start":"04:30.305 ","End":"04:32.505","Text":"1 minus 4,"},{"Start":"04:32.505 ","End":"04:36.095","Text":"that gives me minus 3."},{"Start":"04:36.095 ","End":"04:41.390","Text":"The next one will give me exactly the same because it appears as x^2,"},{"Start":"04:41.390 ","End":"04:44.730","Text":"so I\u0027ll also get minus 3."},{"Start":"04:44.730 ","End":"04:48.125","Text":"If x is root 5 over 2,"},{"Start":"04:48.125 ","End":"04:53.665","Text":"actually I could say that x^2 is t. It\u0027s 5 over 4."},{"Start":"04:53.665 ","End":"04:56.865","Text":"1 minus twice 5 over 4."},{"Start":"04:56.865 ","End":"05:02.530","Text":"Let\u0027s see. 1 minus twice 5 over 4."},{"Start":"05:04.250 ","End":"05:07.050","Text":"2 into 4 cancels."},{"Start":"05:07.050 ","End":"05:12.460","Text":"It\u0027s 1 minus 5 over 2."},{"Start":"05:12.490 ","End":"05:20.240","Text":"That\u0027s altogether 2 minus 5 over 2 is minus 3 over 2 I make it,"},{"Start":"05:20.240 ","End":"05:25.105","Text":"so minus 3 over 2."},{"Start":"05:25.105 ","End":"05:28.550","Text":"Again, because x appears only as x^2 here,"},{"Start":"05:28.550 ","End":"05:32.405","Text":"same thing again, minus 3 over 2."},{"Start":"05:32.405 ","End":"05:37.910","Text":"That gives us four possibilities for x, y."},{"Start":"05:37.910 ","End":"05:41.135","Text":"Let\u0027s see what those four possibilities are."},{"Start":"05:41.135 ","End":"05:47.645","Text":"I could have square root of 2 minus 3."},{"Start":"05:47.645 ","End":"05:51.755","Text":"Each one goes with its corresponding one."},{"Start":"05:51.755 ","End":"05:57.260","Text":"I could have minus square root of 2 minus 3."},{"Start":"05:57.260 ","End":"06:04.695","Text":"I could have square root of 5 over 2 minus 3 over 2,"},{"Start":"06:04.695 ","End":"06:06.285","Text":"and the fourth and last,"},{"Start":"06:06.285 ","End":"06:08.955","Text":"minus root 5 over 2,"},{"Start":"06:08.955 ","End":"06:11.880","Text":"also minus 3 over 2."},{"Start":"06:11.880 ","End":"06:14.650","Text":"Four solutions."},{"Start":"06:14.760 ","End":"06:17.635","Text":"Sometimes I like to check,"},{"Start":"06:17.635 ","End":"06:20.485","Text":"verify the solutions in the original."},{"Start":"06:20.485 ","End":"06:25.360","Text":"Because we got y from x using the first equation,"},{"Start":"06:25.360 ","End":"06:28.525","Text":"it\u0027s unlikely that we\u0027ll fail on the first."},{"Start":"06:28.525 ","End":"06:36.715","Text":"Let\u0027s just see that they all satisfy that second equation and because x appears as x^2,"},{"Start":"06:36.715 ","End":"06:44.620","Text":"all I want to do is for this pair to check what is 3x^2 minus y^2 over"},{"Start":"06:44.620 ","End":"06:53.610","Text":"3 and see if it comes out to be 3. x^2 here is 2 in both cases,"},{"Start":"06:53.610 ","End":"06:57.465","Text":"so we\u0027ve got 3 times 2 minus,"},{"Start":"06:57.465 ","End":"07:02.760","Text":"and y^2 is 9 over 3."},{"Start":"07:02.760 ","End":"07:10.587","Text":"We\u0027ve got 6 minus 3 and 6 minus 3 is certainly 3,"},{"Start":"07:10.587 ","End":"07:14.045","Text":"so we\u0027re okay with this and with this."},{"Start":"07:14.045 ","End":"07:17.225","Text":"As for these, let\u0027s see,"},{"Start":"07:17.225 ","End":"07:25.680","Text":"3x^2 minus y^2 over 3. x^2 is t,"},{"Start":"07:25.680 ","End":"07:29.730","Text":"it\u0027s 5 over 4."},{"Start":"07:29.730 ","End":"07:40.030","Text":"So I\u0027ve got 3 times 5 over 4 and y^2 here is 9 over 4."},{"Start":"07:45.200 ","End":"07:51.015","Text":"But also divided by 3."},{"Start":"07:51.015 ","End":"07:53.994","Text":"I\u0027ll put the 3 on the bottom."},{"Start":"07:53.994 ","End":"07:56.440","Text":"What we get is the first one,"},{"Start":"07:56.440 ","End":"08:03.540","Text":"3 times 5 over 4 is 15 over 4."},{"Start":"08:03.540 ","End":"08:06.312","Text":"The second one,"},{"Start":"08:06.312 ","End":"08:09.405","Text":"3 goes into 9 three times,"},{"Start":"08:09.405 ","End":"08:12.760","Text":"so it\u0027s 15 over 4 minus 3 over 4."},{"Start":"08:12.760 ","End":"08:17.085","Text":"That is equal to 12 over 4,"},{"Start":"08:17.085 ","End":"08:20.295","Text":"and 12 over 4 is certainly equal to 3."},{"Start":"08:20.295 ","End":"08:23.370","Text":"These two are also verified."},{"Start":"08:23.370 ","End":"08:30.880","Text":"Yeah, these are the four solutions. We\u0027re done."}],"ID":5026},{"Watched":false,"Name":"Exercise 14","Duration":"4m 31s","ChapterTopicVideoID":5025,"CourseChapterTopicPlaylistID":3872,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"Here we have a system of nonlinear equations,"},{"Start":"00:05.280 ","End":"00:07.770","Text":"2 equations in 2 unknowns,"},{"Start":"00:07.770 ","End":"00:09.850","Text":"x, and y."},{"Start":"00:11.090 ","End":"00:17.700","Text":"The easiest thing to do here would be to isolate y in terms of"},{"Start":"00:17.700 ","End":"00:24.315","Text":"x from the first equation and then substitute y into the second equation."},{"Start":"00:24.315 ","End":"00:29.025","Text":"Let me just copy the exercises over here."},{"Start":"00:29.025 ","End":"00:32.640","Text":"Like I said, we can write y in terms of x"},{"Start":"00:32.640 ","End":"00:41.715","Text":"by y=6 minus 4x^2 and then substitute y into the second equation."},{"Start":"00:41.715 ","End":"00:51.420","Text":"This will give us 3x^2 minus y is 6 minus 4x^2."},{"Start":"00:51.420 ","End":"00:57.030","Text":"But squared over 4 equals"},{"Start":"00:57.030 ","End":"01:04.560","Text":"2, minus 2=0."},{"Start":"01:04.560 ","End":"01:14.060","Text":"Let\u0027s multiply out and I\u0027m going to use the formula"},{"Start":"01:14.060 ","End":"01:24.230","Text":"that a minus b^2 is a^2 minus 2ab plus b^2."},{"Start":"01:24.230 ","End":"01:32.710","Text":"Then we get 3x^2 minus, over 4."},{"Start":"01:32.710 ","End":"01:34.690","Text":"Looking at the formula,"},{"Start":"01:34.690 ","End":"01:42.800","Text":"I see I get 36 minus twice 6 times 4 is 48x^2,"},{"Start":"01:43.560 ","End":"01:48.700","Text":"and then this 1^2 is plus 16x to"},{"Start":"01:48.700 ","End":"01:56.020","Text":"the fourth minus 2=0."},{"Start":"01:56.020 ","End":"02:03.885","Text":"What I suggest is dividing top and bottom by 4 here."},{"Start":"02:03.885 ","End":"02:07.470","Text":"This 4 will cancel here,"},{"Start":"02:07.470 ","End":"02:10.075","Text":"it goes into it 9 times."},{"Start":"02:10.075 ","End":"02:15.105","Text":"Here I\u0027ll get 12 and here I\u0027ll get 4."},{"Start":"02:15.105 ","End":"02:19.429","Text":"Then I don\u0027t have fractions and I\u0027ll just have to collect like terms."},{"Start":"02:19.429 ","End":"02:25.560","Text":"I think I\u0027ll just rewrite it without all these crossing outs."},{"Start":"02:26.270 ","End":"02:30.660","Text":"3x^2, I can also apply the minus,"},{"Start":"02:30.660 ","End":"02:34.320","Text":"so it\u0027s minus 36 plus"},{"Start":"02:34.320 ","End":"02:42.550","Text":"12x^2 minus 4x to the fourth, minus 2=0."},{"Start":"02:42.550 ","End":"02:45.320","Text":"Now I\u0027ll collect like terms."},{"Start":"02:45.320 ","End":"02:49.475","Text":"Since I liked the leading term to be positive,"},{"Start":"02:49.475 ","End":"02:54.005","Text":"it\u0027s just the thing will reverse the signs of everything."},{"Start":"02:54.005 ","End":"02:56.600","Text":"Instead of minus 4x to the fourth,"},{"Start":"02:56.600 ","End":"02:59.225","Text":"I\u0027ll put plus 4x to the fourth."},{"Start":"02:59.225 ","End":"03:04.370","Text":"For x^2 I have 3 plus 12 is 15,"},{"Start":"03:04.370 ","End":"03:08.540","Text":"but I\u0027m reversing so it\u0027s minus 15x^2."},{"Start":"03:08.540 ","End":"03:18.510","Text":"Constants I would have minus 38 but it\u0027s plus 38=0."},{"Start":"03:18.510 ","End":"03:25.300","Text":"At this point, we do our usual trick of substituting t equals x^2,"},{"Start":"03:25.300 ","End":"03:27.550","Text":"and then we get a quadratic in T,"},{"Start":"03:27.550 ","End":"03:31.180","Text":"which is 4t^2 minus 15t"},{"Start":"03:31.180 ","End":"03:37.810","Text":"plus 38=0."},{"Start":"03:37.810 ","End":"03:42.595","Text":"It actually turns out that this quadratic equation has no solutions."},{"Start":"03:42.595 ","End":"03:47.650","Text":"One way to see this is just by using the formula or in"},{"Start":"03:47.650 ","End":"03:53.229","Text":"fact just looking at the b^2 minus 4ac part of the quadratic."},{"Start":"03:53.229 ","End":"03:55.150","Text":"Where this is our a,"},{"Start":"03:55.150 ","End":"03:57.160","Text":"this is b minus 15,"},{"Start":"03:57.160 ","End":"04:02.485","Text":"and c then, b^2 is 225."},{"Start":"04:02.485 ","End":"04:07.425","Text":"For 4ac is 4 times 4 times 38,"},{"Start":"04:07.425 ","End":"04:12.860","Text":"which comes out to be 608 which is certainly negative,"},{"Start":"04:12.860 ","End":"04:17.975","Text":"and so there is no square root and so there\u0027s no solution."},{"Start":"04:17.975 ","End":"04:20.720","Text":"If there is no solution for t,"},{"Start":"04:20.720 ","End":"04:25.085","Text":"then there\u0027s no solution for x and no solution in general."},{"Start":"04:25.085 ","End":"04:31.200","Text":"That can happen sometimes and so we\u0027re done."}],"ID":5027}],"Thumbnail":null,"ID":3872},{"Name":"Equations with Roots","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Equations with Roots","Duration":"10m 32s","ChapterTopicVideoID":5028,"CourseChapterTopicPlaylistID":3873,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.885","Text":"Starting a new topic, equations with roots."},{"Start":"00:03.885 ","End":"00:08.715","Text":"Actually the only kind of roots we\u0027re going be considering are square roots."},{"Start":"00:08.715 ","End":"00:12.315","Text":"I\u0027ll add the word square,"},{"Start":"00:12.315 ","End":"00:15.135","Text":"cube roots and other stuff don\u0027t belong here."},{"Start":"00:15.135 ","End":"00:18.990","Text":"These are so common that they deserve a topic on their own."},{"Start":"00:18.990 ","End":"00:21.030","Text":"When I say equations with square roots,"},{"Start":"00:21.030 ","End":"00:24.090","Text":"I mean that the variable is under the square root."},{"Start":"00:24.090 ","End":"00:28.245","Text":"For example, let\u0027s take the square root"},{"Start":"00:28.245 ","End":"00:35.070","Text":"of 3x minus 2=4."},{"Start":"00:35.070 ","End":"00:39.110","Text":"You can see that the variable x is under the square root sign."},{"Start":"00:39.110 ","End":"00:44.810","Text":"Now, the usual technique that we use to solve equations with"},{"Start":"00:44.810 ","End":"00:51.290","Text":"square roots is the concept or the technique of squaring both sides."},{"Start":"00:51.290 ","End":"00:54.245","Text":"I\u0027ll write that. What we want to do,"},{"Start":"00:54.245 ","End":"00:55.685","Text":"maybe not right away,"},{"Start":"00:55.685 ","End":"01:02.570","Text":"maybe after some tidying up that we square both sides of the equation."},{"Start":"01:02.570 ","End":"01:05.690","Text":"That\u0027s clear sides because if 2 things are equal,"},{"Start":"01:05.690 ","End":"01:07.475","Text":"then their squares are equal."},{"Start":"01:07.475 ","End":"01:10.685","Text":"In this case, if we squared both sides,"},{"Start":"01:10.685 ","End":"01:13.670","Text":"then on the right-hand side and doing that because it\u0027s easier,"},{"Start":"01:13.670 ","End":"01:18.455","Text":"4^2 is 16 and the square root of something,"},{"Start":"01:18.455 ","End":"01:19.685","Text":"if I square it,"},{"Start":"01:19.685 ","End":"01:24.070","Text":"just drops the square root, 3x minus 2."},{"Start":"01:24.070 ","End":"01:25.830","Text":"I hope this is clear."},{"Start":"01:25.830 ","End":"01:34.865","Text":"For example, if I take the square root of 9 and then I square it,"},{"Start":"01:34.865 ","End":"01:36.680","Text":"I get back to 9."},{"Start":"01:36.680 ","End":"01:40.880","Text":"This is always the case because that\u0027s what the square root means,"},{"Start":"01:40.880 ","End":"01:44.735","Text":"that something which when squared gives me back what I originally had."},{"Start":"01:44.735 ","End":"01:50.915","Text":"Bearing that in mind and now we have a regular equation without square roots."},{"Start":"01:50.915 ","End":"01:59.585","Text":"Just solve it, 3x=18 and x=6."},{"Start":"01:59.585 ","End":"02:01.790","Text":"This would be the answer."},{"Start":"02:01.790 ","End":"02:05.900","Text":"But I like to check the answer that it\u0027s correct,"},{"Start":"02:05.900 ","End":"02:09.725","Text":"which is always good practice with equations. Let\u0027s see."},{"Start":"02:09.725 ","End":"02:12.980","Text":"Let\u0027s plug this in the original equation and"},{"Start":"02:12.980 ","End":"02:17.470","Text":"check if the square root of 3 times now instead of x,"},{"Start":"02:17.470 ","End":"02:20.330","Text":"I put 6 minus 2,"},{"Start":"02:20.330 ","End":"02:22.850","Text":"is it equal to, now I have to put a question mark."},{"Start":"02:22.850 ","End":"02:25.135","Text":"That\u0027s what I\u0027m checking. Is it equal to 4?"},{"Start":"02:25.135 ","End":"02:28.470","Text":"Well, 3 times 6 is 18,"},{"Start":"02:28.470 ","End":"02:31.820","Text":"18 minus 2 is 16 and indeed,"},{"Start":"02:31.820 ","End":"02:34.280","Text":"the square root of 16=4,"},{"Start":"02:34.280 ","End":"02:36.605","Text":"I would say yes, definitely so."},{"Start":"02:36.605 ","End":"02:41.044","Text":"This really is the solution to the equation."},{"Start":"02:41.044 ","End":"02:45.140","Text":"Now, although it\u0027s true that it\u0027s a good practice in general,"},{"Start":"02:45.140 ","End":"02:48.260","Text":"it turns out that this checking in the original is particularly"},{"Start":"02:48.260 ","End":"02:51.610","Text":"important for equations with square roots and we\u0027ll see."},{"Start":"02:51.610 ","End":"02:55.070","Text":"This was a simple example and it seems everything\u0027s straightforward,"},{"Start":"02:55.070 ","End":"02:57.035","Text":"you have a square root, you square both sides."},{"Start":"02:57.035 ","End":"03:01.100","Text":"But things that are a little bit more complicated as we\u0027ll see in the following examples,"},{"Start":"03:01.100 ","End":"03:07.700","Text":"there are various nuances and pitfalls and we shall see 1 important thing that"},{"Start":"03:07.700 ","End":"03:14.480","Text":"I must say about square roots is that what\u0027s under the square root must be non-negative."},{"Start":"03:14.480 ","End":"03:21.605","Text":"In this case, we would have to have in our answer that 3x minus 2 is bigger to =0."},{"Start":"03:21.605 ","End":"03:26.360","Text":"For example, if we came to the answer x=0,"},{"Start":"03:26.360 ","End":"03:33.020","Text":"we would have to rule it out right away because 3 times 0 minus 2 is not greater or =0."},{"Start":"03:33.020 ","End":"03:36.590","Text":"But sometimes this is called the domain of definition and sometimes"},{"Start":"03:36.590 ","End":"03:40.370","Text":"we even explicitly translate it to what it means in terms of x."},{"Start":"03:40.370 ","End":"03:42.347","Text":"In this case, bringing the 2 over,"},{"Start":"03:42.347 ","End":"03:47.330","Text":"x would have to be bigger or to =2/3 for this equation to"},{"Start":"03:47.330 ","End":"03:52.540","Text":"even make sense because we can\u0027t take square roots of negative numbers."},{"Start":"03:52.540 ","End":"03:54.020","Text":"We\u0027ll get more into that later."},{"Start":"03:54.020 ","End":"03:58.025","Text":"But just bear in mind that what\u0027s under the square root sign cannot be negative."},{"Start":"03:58.025 ","End":"04:01.265","Text":"Let\u0027s go on to our next example."},{"Start":"04:01.265 ","End":"04:04.160","Text":"I\u0027d like to keep the next example on the same page,"},{"Start":"04:04.160 ","End":"04:08.540","Text":"so I\u0027m going to just erase this and write it down."},{"Start":"04:08.540 ","End":"04:15.035","Text":"The square root of x plus 6=x."},{"Start":"04:15.035 ","End":"04:18.245","Text":"Up till now, things seemed very smooth and straightforward."},{"Start":"04:18.245 ","End":"04:21.830","Text":"This example will introduce 1 of the difficulties or"},{"Start":"04:21.830 ","End":"04:27.230","Text":"delicacies or nuances that we can encounter with equations with square roots."},{"Start":"04:27.230 ","End":"04:31.760","Text":"We again, proceed naively and square both sides."},{"Start":"04:31.760 ","End":"04:33.275","Text":"I don\u0027t even have to write it again."},{"Start":"04:33.275 ","End":"04:35.540","Text":"Sometimes instead of writing the whole thing,"},{"Start":"04:35.540 ","End":"04:37.370","Text":"I might say something like,"},{"Start":"04:37.370 ","End":"04:40.850","Text":"let\u0027s take both sides squared. That\u0027s a shorthand."},{"Start":"04:40.850 ","End":"04:44.520","Text":"Anyway, we get that x plus 6,"},{"Start":"04:44.520 ","End":"04:47.550","Text":"when we square this, is equal to x^2."},{"Start":"04:47.550 ","End":"04:49.955","Text":"If I bring everything to 1 side,"},{"Start":"04:49.955 ","End":"04:52.123","Text":"we get the quadratic equation,"},{"Start":"04:52.123 ","End":"04:56.720","Text":"x^2 minus x minus 6=0."},{"Start":"04:56.720 ","End":"04:59.460","Text":"I brought everything to the right-hand side and then switched sides."},{"Start":"04:59.460 ","End":"05:03.380","Text":"Now I don\u0027t want to waste time with solving quadratic equations,"},{"Start":"05:03.380 ","End":"05:05.839","Text":"so I\u0027m just going give you the answers,"},{"Start":"05:05.839 ","End":"05:10.300","Text":"that x=3 or minus 2."},{"Start":"05:10.300 ","End":"05:14.135","Text":"These are both solutions to this quadratic equation."},{"Start":"05:14.135 ","End":"05:18.740","Text":"Now, I mentioned before that with equations with square roots,"},{"Start":"05:18.740 ","End":"05:20.210","Text":"it\u0027s not just a good idea,"},{"Start":"05:20.210 ","End":"05:24.410","Text":"it\u0027s pretty much mandatory to check your solutions"},{"Start":"05:24.410 ","End":"05:28.865","Text":"in the original equation because something funny happens in the squaring process,"},{"Start":"05:28.865 ","End":"05:31.430","Text":"which I will explain later."},{"Start":"05:31.430 ","End":"05:36.230","Text":"Let\u0027s just see as if we were just checking to make sure we didn\u0027t make a mistake."},{"Start":"05:36.230 ","End":"05:38.185","Text":"We put in x=3,"},{"Start":"05:38.185 ","End":"05:42.965","Text":"so if I put in x=3 into the equation here,"},{"Start":"05:42.965 ","End":"05:48.780","Text":"I get the square root of 3 plus 6 equals,"},{"Start":"05:48.780 ","End":"05:50.390","Text":"and x is 3,"},{"Start":"05:50.390 ","End":"05:54.410","Text":"but I need a question mark because that\u0027s what I\u0027m checking."},{"Start":"05:54.410 ","End":"05:57.080","Text":"Before I even open this,"},{"Start":"05:57.080 ","End":"06:00.380","Text":"I just like to have the other one written down and this one,"},{"Start":"06:00.380 ","End":"06:07.189","Text":"I\u0027m also going to check and for this I\u0027m going to check if the square root of minus"},{"Start":"06:07.189 ","End":"06:14.215","Text":"2 plus 6 is this equal to minus 2?"},{"Start":"06:14.215 ","End":"06:16.145","Text":"I just put it in brackets for emphasis."},{"Start":"06:16.145 ","End":"06:18.875","Text":"I guess I don\u0027t really need that."},{"Start":"06:18.875 ","End":"06:20.810","Text":"Anyway, first things first,"},{"Start":"06:20.810 ","End":"06:23.810","Text":"3 plus 6 is 9 so I\u0027m asking,"},{"Start":"06:23.810 ","End":"06:26.720","Text":"is the square root of 9=3?"},{"Start":"06:26.720 ","End":"06:33.125","Text":"I answer yes and so x=3 is a valid solution and I\u0027m highlighting it."},{"Start":"06:33.125 ","End":"06:34.715","Text":"Now let\u0027s get onto the other one."},{"Start":"06:34.715 ","End":"06:38.435","Text":"This one, minus 2 plus 6 is 4,"},{"Start":"06:38.435 ","End":"06:45.115","Text":"so I\u0027m asking, is the square root of 4= to minus 2?"},{"Start":"06:45.115 ","End":"06:48.715","Text":"The answer is a definite no."},{"Start":"06:48.715 ","End":"06:53.695","Text":"People sometimes get confused about things and think there\u0027s 2 square roots,"},{"Start":"06:53.695 ","End":"06:55.230","Text":"plus and minus 2, but no."},{"Start":"06:55.230 ","End":"06:57.750","Text":"There\u0027s only 1 square root, 2,"},{"Start":"06:57.750 ","End":"07:01.600","Text":"4 and that is 2 and 2 is not equal to minus 2."},{"Start":"07:01.600 ","End":"07:04.325","Text":"What confuses people sometimes is that if I ask,"},{"Start":"07:04.325 ","End":"07:07.480","Text":"what is the answer to x^2=4?"},{"Start":"07:07.480 ","End":"07:08.985","Text":"Then it\u0027s 2 and minus 2."},{"Start":"07:08.985 ","End":"07:10.730","Text":"This thing squared is full,"},{"Start":"07:10.730 ","End":"07:12.500","Text":"but the square root always refers to"},{"Start":"07:12.500 ","End":"07:16.490","Text":"the positive square root and so the answer to this is no."},{"Start":"07:16.490 ","End":"07:20.180","Text":"Then I usually indicate this and in the exercises I\u0027ll do this,"},{"Start":"07:20.180 ","End":"07:24.050","Text":"I\u0027ll just put a red line through it or sometimes I even want to emphasize,"},{"Start":"07:24.050 ","End":"07:26.810","Text":"I put the whole cross through it."},{"Start":"07:26.810 ","End":"07:28.715","Text":"That\u0027s an overkill. Anyway,"},{"Start":"07:28.715 ","End":"07:31.610","Text":"this is yes and this is no."},{"Start":"07:31.610 ","End":"07:35.780","Text":"This demonstrates that we have to check when we get to"},{"Start":"07:35.780 ","End":"07:40.055","Text":"the end whether the solutions really are solutions."},{"Start":"07:40.055 ","End":"07:45.440","Text":"Now this minus 2 somehow crept in during the squaring process and this is"},{"Start":"07:45.440 ","End":"07:51.430","Text":"called an extraneous solution that\u0027s spelled extra and then N-E-O-U-S,"},{"Start":"07:51.430 ","End":"07:53.270","Text":"although I won\u0027t be using this word much."},{"Start":"07:53.270 ","End":"07:55.130","Text":"This is the correct word, but it\u0027s too technical."},{"Start":"07:55.130 ","End":"07:57.229","Text":"I\u0027ll just say an illegal solution,"},{"Start":"07:57.229 ","End":"07:59.960","Text":"a wrong solution, a bad solution,"},{"Start":"07:59.960 ","End":"08:01.895","Text":"invalid solution, or something."},{"Start":"08:01.895 ","End":"08:04.910","Text":"Now, I need to explain where this thing came from."},{"Start":"08:04.910 ","End":"08:07.590","Text":"Well, correction, I don\u0027t have to explain it."},{"Start":"08:07.590 ","End":"08:11.000","Text":"If you want, you can just take it on trust that when you square things,"},{"Start":"08:11.000 ","End":"08:13.985","Text":"extraneous solutions creep in."},{"Start":"08:13.985 ","End":"08:18.485","Text":"But I do want to give you some rationale behind this."},{"Start":"08:18.485 ","End":"08:22.220","Text":"1 way of explaining it is to say that when you square things,"},{"Start":"08:22.220 ","End":"08:26.420","Text":"it doesn\u0027t care about the sign plus or minus, it loses information."},{"Start":"08:26.420 ","End":"08:28.715","Text":"That\u0027s 1 way, but that\u0027s a bit vague."},{"Start":"08:28.715 ","End":"08:34.940","Text":"Another way of explaining it is to say to talk about I\u0027ll call it a buddy equation,"},{"Start":"08:34.940 ","End":"08:37.100","Text":"not a technical term just my term."},{"Start":"08:37.100 ","End":"08:41.735","Text":"Suppose I took this equation and took a similar equation,"},{"Start":"08:41.735 ","End":"08:43.820","Text":"square root of x plus 6,"},{"Start":"08:43.820 ","End":"08:46.805","Text":"but with a minus thrown in on 1 side."},{"Start":"08:46.805 ","End":"08:48.815","Text":"Suppose that this was my equation."},{"Start":"08:48.815 ","End":"08:51.230","Text":"When I square this equation,"},{"Start":"08:51.230 ","End":"08:54.050","Text":"I go about and say, it\u0027s got a square root, I square it."},{"Start":"08:54.050 ","End":"08:55.700","Text":"When I square it,"},{"Start":"08:55.700 ","End":"09:00.545","Text":"I get the same as if I had the original equation."},{"Start":"09:00.545 ","End":"09:02.870","Text":"When I look at this line,"},{"Start":"09:02.870 ","End":"09:05.615","Text":"am I solving this or am I solving this?"},{"Start":"09:05.615 ","End":"09:10.670","Text":"In actual fact, this thing which I called the buddy equation in informal term,"},{"Start":"09:10.670 ","End":"09:12.875","Text":"actually does satisfy,"},{"Start":"09:12.875 ","End":"09:15.454","Text":"the minus 2 satisfies this equation."},{"Start":"09:15.454 ","End":"09:18.320","Text":"Because if I put x=minus 2 here,"},{"Start":"09:18.320 ","End":"09:24.740","Text":"then I get the square root of minus 2 plus 6 is it equal to minus, minus 2?"},{"Start":"09:24.740 ","End":"09:27.500","Text":"In other words, is the square root of 4=2?"},{"Start":"09:27.500 ","End":"09:29.165","Text":"The answer is yes."},{"Start":"09:29.165 ","End":"09:30.920","Text":"This is a solution,"},{"Start":"09:30.920 ","End":"09:32.300","Text":"but not to this equation,"},{"Start":"09:32.300 ","End":"09:33.740","Text":"to the buddy equation,"},{"Start":"09:33.740 ","End":"09:35.240","Text":"the one with the opposite sign."},{"Start":"09:35.240 ","End":"09:39.349","Text":"The squaring loses that information of whether it\u0027s plus or minus."},{"Start":"09:39.349 ","End":"09:41.240","Text":"I\u0027ll leave it at that."},{"Start":"09:41.240 ","End":"09:44.165","Text":"You just accept the fact that when we square things,"},{"Start":"09:44.165 ","End":"09:47.180","Text":"wrong solutions, bad solutions,"},{"Start":"09:47.180 ","End":"09:51.140","Text":"non-solutions creep in and that we always have to check each"},{"Start":"09:51.140 ","End":"09:55.715","Text":"1 and say whether it is or whether it isn\u0027t."},{"Start":"09:55.715 ","End":"09:59.240","Text":"In this case, we had a quadratic equation and 2 solutions"},{"Start":"09:59.240 ","End":"10:02.600","Text":"and 1 of them was really a solution and 1 of them wasn\u0027t."},{"Start":"10:02.600 ","End":"10:04.160","Text":"But even with a quadratic,"},{"Start":"10:04.160 ","End":"10:06.830","Text":"they could both be, could neither of them be,"},{"Start":"10:06.830 ","End":"10:09.185","Text":"or recombination is possible."},{"Start":"10:09.185 ","End":"10:12.320","Text":"Really if I had to just stop the lesson here,"},{"Start":"10:12.320 ","End":"10:14.765","Text":"I would say, when you have equations with square roots,"},{"Start":"10:14.765 ","End":"10:20.735","Text":"always check each solution in the original equation and see if it works or not."},{"Start":"10:20.735 ","End":"10:23.690","Text":"If you just remember that, you\u0027ll be okay."},{"Start":"10:23.690 ","End":"10:28.460","Text":"Let\u0027s just take some more complicated examples and learn some more techniques."},{"Start":"10:28.460 ","End":"10:32.790","Text":"I\u0027ll take a break here and continue in the next clip."}],"ID":5028},{"Watched":false,"Name":"Equations with Roots (continued)","Duration":"10m 33s","ChapterTopicVideoID":5029,"CourseChapterTopicPlaylistID":3873,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.460","Text":"I\u0027m continuing where I left off in the previous clip,"},{"Start":"00:03.460 ","End":"00:07.695","Text":"so I\u0027ll write the word continued and there\u0027ll erase part of this."},{"Start":"00:07.695 ","End":"00:10.874","Text":"I\u0027ll just give a brief summary of what was last time"},{"Start":"00:10.874 ","End":"00:14.955","Text":"that the main tool we have in equations with square roots"},{"Start":"00:14.955 ","End":"00:18.000","Text":"is the tool of squaring both sides of"},{"Start":"00:18.000 ","End":"00:23.055","Text":"an equation after suitable arrangement if necessary, as we shall see."},{"Start":"00:23.055 ","End":"00:27.180","Text":"The thing to watch out for is after you\u0027ve squared an equation,"},{"Start":"00:27.180 ","End":"00:30.630","Text":"there is something called an extraneous solution,"},{"Start":"00:30.630 ","End":"00:33.690","Text":"these false solutions that can creep in,"},{"Start":"00:33.690 ","End":"00:35.160","Text":"in the process of squaring,"},{"Start":"00:35.160 ","End":"00:39.770","Text":"so it\u0027s very important that if there\u0027s any number of solutions, one or more,"},{"Start":"00:39.770 ","End":"00:42.950","Text":"you have to check each possible solution in"},{"Start":"00:42.950 ","End":"00:47.765","Text":"the original equation to see if it really is valid or not."},{"Start":"00:47.765 ","End":"00:50.930","Text":"In this case, we had a quadratic equation, 2 solutions,"},{"Start":"00:50.930 ","End":"00:54.350","Text":"and 1 of them turned out to be good and 1 of them was bad,"},{"Start":"00:54.350 ","End":"00:56.150","Text":"but anything could happen."},{"Start":"00:56.150 ","End":"01:00.050","Text":"I also want to say that there\u0027s 2 ways a solution can be bad,"},{"Start":"01:00.050 ","End":"01:04.970","Text":"it could be like here that we plug it in and the equality doesn\u0027t hold."},{"Start":"01:04.970 ","End":"01:08.105","Text":"I mean, 2 does not equal minus 2 in this case."},{"Start":"01:08.105 ","End":"01:10.820","Text":"There\u0027s another way it can go wrong and that is"},{"Start":"01:10.820 ","End":"01:13.570","Text":"it could be outside the domain of definition."},{"Start":"01:13.570 ","End":"01:14.870","Text":"In other words here,"},{"Start":"01:14.870 ","End":"01:17.165","Text":"since the square root has to be non negative,"},{"Start":"01:17.165 ","End":"01:23.810","Text":"x has to be bigger or equal to minus 6 because x plus 6 has to be bigger or equal to 0."},{"Start":"01:23.810 ","End":"01:25.190","Text":"If I got a solution,"},{"Start":"01:25.190 ","End":"01:30.215","Text":"just say for example I got a solution of minus 10, I didn\u0027t,"},{"Start":"01:30.215 ","End":"01:33.750","Text":"but let\u0027s just say, then I wouldn\u0027t even be able to substitute it,"},{"Start":"01:33.750 ","End":"01:37.850","Text":"never mind check equality because that would be square root of a negative number."},{"Start":"01:37.850 ","End":"01:39.290","Text":"There\u0027s 2 ways it can go wrong."},{"Start":"01:39.290 ","End":"01:43.160","Text":"Later on, we\u0027ll see examples of the other bad solution."},{"Start":"01:43.160 ","End":"01:46.310","Text":"But the important thing and I keep stressing is when you"},{"Start":"01:46.310 ","End":"01:50.225","Text":"have equations with square roots that you\u0027ve solved by squaring,"},{"Start":"01:50.225 ","End":"01:56.950","Text":"you have to check each solution to make sure it really satisfies the original equation."},{"Start":"01:56.950 ","End":"01:59.925","Text":"Having said that, I\u0027m going to clean the board,"},{"Start":"01:59.925 ","End":"02:02.055","Text":"here\u0027s the next example,"},{"Start":"02:02.055 ","End":"02:11.300","Text":"the square root of 4x plus 1 plus x is equal to 11."},{"Start":"02:11.300 ","End":"02:16.970","Text":"I\u0027ve introduced a slight difficulty because in the previous examples,"},{"Start":"02:16.970 ","End":"02:21.230","Text":"we had just the square root of something on 1 side."},{"Start":"02:21.230 ","End":"02:24.530","Text":"Here I have a square root plus something else."},{"Start":"02:24.530 ","End":"02:26.345","Text":"Now, there\u0027s only 1 square root,"},{"Start":"02:26.345 ","End":"02:28.850","Text":"1 term containing square root,"},{"Start":"02:28.850 ","End":"02:31.850","Text":"then I can put the other terms on the other side."},{"Start":"02:31.850 ","End":"02:41.155","Text":"In this case, what I\u0027ll get is the square root of 4x plus 1=11 minus x."},{"Start":"02:41.155 ","End":"02:44.135","Text":"Now I can proceed just like we did before,"},{"Start":"02:44.135 ","End":"02:46.940","Text":"this is really the main thing I\u0027m showing you in"},{"Start":"02:46.940 ","End":"02:50.630","Text":"this examples that if it isn\u0027t a square root and there are other terms,"},{"Start":"02:50.630 ","End":"02:52.250","Text":"move them over to the other side."},{"Start":"02:52.250 ","End":"02:53.810","Text":"But if we\u0027re already here,"},{"Start":"02:53.810 ","End":"02:55.660","Text":"let\u0027s continue solving it."},{"Start":"02:55.660 ","End":"02:58.140","Text":"I\u0027ll square both sides of the equation,"},{"Start":"02:58.140 ","End":"03:02.315","Text":"I indicate that I\u0027ll take the square of both sides,"},{"Start":"03:02.315 ","End":"03:09.520","Text":"and then here I get 4x plus 1 and here I get 11 minus x^2,"},{"Start":"03:09.520 ","End":"03:14.000","Text":"and in case you don\u0027t remember the binomial expansion,"},{"Start":"03:14.000 ","End":"03:15.755","Text":"here is a reminder,"},{"Start":"03:15.755 ","End":"03:22.090","Text":"and so this side becomes 11^2 is 121 minus twice 11 times x,"},{"Start":"03:22.090 ","End":"03:24.810","Text":"22x plus x^2,"},{"Start":"03:24.810 ","End":"03:27.140","Text":"this side is still the same as it was."},{"Start":"03:27.140 ","End":"03:30.785","Text":"Now, let\u0027s put everything on the right hand side,"},{"Start":"03:30.785 ","End":"03:32.490","Text":"but I\u0027m also going to switch sides,"},{"Start":"03:32.490 ","End":"03:34.530","Text":"so pretend this is the right hand side."},{"Start":"03:34.530 ","End":"03:36.030","Text":"We\u0027ve got x^2,"},{"Start":"03:36.030 ","End":"03:41.640","Text":"then we have minus 22x minus 4x is minus 26x,"},{"Start":"03:41.640 ","End":"03:48.125","Text":"and then we have 121 minus 1 is 120, and then it equals 0."},{"Start":"03:48.125 ","End":"03:51.560","Text":"Once again, a quadratic equation and I"},{"Start":"03:51.560 ","End":"03:55.280","Text":"don\u0027t want to waste time with solving quadratic equations,"},{"Start":"03:55.280 ","End":"04:03.330","Text":"so I\u0027ll tell you the solutions, x=6 and x=20."},{"Start":"04:03.330 ","End":"04:06.210","Text":"I\u0027ll tell you a secret, I didn\u0027t actually solve it,"},{"Start":"04:06.210 ","End":"04:09.620","Text":"there are actually Internet websites that have"},{"Start":"04:09.620 ","End":"04:14.060","Text":"a calculator to solve quadratic equations and if you look,"},{"Start":"04:14.060 ","End":"04:15.230","Text":"you\u0027ll find them,"},{"Start":"04:15.230 ","End":"04:16.865","Text":"but I don\u0027t want to encourage that."},{"Start":"04:16.865 ","End":"04:21.860","Text":"Anyway, you can pause and solve it manually and confirm that these are the solutions,"},{"Start":"04:21.860 ","End":"04:23.375","Text":"I\u0027m going to continue."},{"Start":"04:23.375 ","End":"04:28.430","Text":"The most important thing that I\u0027ve said about these equations which use"},{"Start":"04:28.430 ","End":"04:33.725","Text":"the squaring technique is that you have to check each one in the original equation."},{"Start":"04:33.725 ","End":"04:37.535","Text":"Let\u0027s check x=6 first."},{"Start":"04:37.535 ","End":"04:41.270","Text":"If I plug it in the original equation,"},{"Start":"04:41.270 ","End":"04:43.230","Text":"what I\u0027m asking is,"},{"Start":"04:43.230 ","End":"04:50.655","Text":"is the square root of 4 times 6 plus 1 plus 6,"},{"Start":"04:50.655 ","End":"04:53.669","Text":"is this equal to 11?"},{"Start":"04:53.669 ","End":"05:00.660","Text":"Let\u0027s see. 4 times 6 plus 1 is 25, that\u0027s 24 plus 1,"},{"Start":"05:00.660 ","End":"05:03.038","Text":"and the square root of 25 is 5,"},{"Start":"05:03.038 ","End":"05:04.425","Text":"and so I\u0027m asking,"},{"Start":"05:04.425 ","End":"05:08.910","Text":"is 5 plus 6=11?"},{"Start":"05:08.910 ","End":"05:11.265","Text":"I would say yes,"},{"Start":"05:11.265 ","End":"05:13.925","Text":"and so x equals 6 is good."},{"Start":"05:13.925 ","End":"05:15.880","Text":"Now let\u0027s go onto the next one."},{"Start":"05:15.880 ","End":"05:21.340","Text":"For x=20, I get square root of"},{"Start":"05:21.340 ","End":"05:27.575","Text":"4 times 20 plus 1 plus 20,"},{"Start":"05:27.575 ","End":"05:30.470","Text":"does this equal 11?"},{"Start":"05:30.470 ","End":"05:32.485","Text":"Well, let\u0027s see."},{"Start":"05:32.485 ","End":"05:36.840","Text":"4 times 20 plus 1 is 81,"},{"Start":"05:36.840 ","End":"05:40.020","Text":"square root of 81 is 9,"},{"Start":"05:40.020 ","End":"05:41.235","Text":"and now the question is,"},{"Start":"05:41.235 ","End":"05:46.050","Text":"is 9 plus 20 equal to 11?"},{"Start":"05:46.050 ","End":"05:50.150","Text":"In this case, the answer is negative, no, it isn\u0027t."},{"Start":"05:50.150 ","End":"05:54.800","Text":"Of course you notice that minus 9 plus 20 is 11,"},{"Start":"05:54.800 ","End":"05:57.185","Text":"but that\u0027s not what this is,"},{"Start":"05:57.185 ","End":"06:02.670","Text":"so I have to say that this is an extraneous solution, a bad,"},{"Start":"06:02.670 ","End":"06:04.890","Text":"illegal, whatever solution,"},{"Start":"06:04.890 ","End":"06:08.234","Text":"and I like to indicate that by just drawing a red line through it."},{"Start":"06:08.234 ","End":"06:10.760","Text":"It does not satisfy the equation,"},{"Start":"06:10.760 ","End":"06:13.240","Text":"the equality does not hold."},{"Start":"06:13.240 ","End":"06:15.450","Text":"Time to clear the board."},{"Start":"06:15.450 ","End":"06:17.595","Text":"Here\u0027s the next example."},{"Start":"06:17.595 ","End":"06:23.595","Text":"I\u0027ll take the square root of x^2 plus 5x plus"},{"Start":"06:23.595 ","End":"06:30.920","Text":"10 is equal to twice the square root of 5 minus x."},{"Start":"06:30.920 ","End":"06:35.720","Text":"There\u0027s a couple of things to note in this example we haven\u0027t had before."},{"Start":"06:35.720 ","End":"06:37.415","Text":"There\u0027s 2 square roots,"},{"Start":"06:37.415 ","End":"06:39.205","Text":"1 on each side."},{"Start":"06:39.205 ","End":"06:44.120","Text":"The good thing here is that the square root is of the whole thing,"},{"Start":"06:44.120 ","End":"06:46.055","Text":"there\u0027s no leftovers,"},{"Start":"06:46.055 ","End":"06:48.230","Text":"and on the right hand side,"},{"Start":"06:48.230 ","End":"06:51.680","Text":"a constant that multiplies the square root is not really a leftover,"},{"Start":"06:51.680 ","End":"06:54.535","Text":"it\u0027s not like a term with plus or minus."},{"Start":"06:54.535 ","End":"06:58.770","Text":"This is okay. This is as if we just had a square root."},{"Start":"06:58.770 ","End":"07:02.425","Text":"Usual technique of squaring both sides,"},{"Start":"07:02.425 ","End":"07:04.910","Text":"I\u0027ll take both sides to the power of 2 or square them,"},{"Start":"07:04.910 ","End":"07:09.840","Text":"and on the left I\u0027ll get x^2 plus 5x plus 10."},{"Start":"07:09.840 ","End":"07:11.280","Text":"On the right,"},{"Start":"07:11.280 ","End":"07:14.150","Text":"you need to know the formula from algebra that if I"},{"Start":"07:14.150 ","End":"07:17.059","Text":"have a times b and all that is squared,"},{"Start":"07:17.059 ","End":"07:18.832","Text":"then each one is squared separately."},{"Start":"07:18.832 ","End":"07:20.825","Text":"So 2^2 is 4,"},{"Start":"07:20.825 ","End":"07:23.975","Text":"the square root squared is just the thing itself,"},{"Start":"07:23.975 ","End":"07:26.318","Text":"but of course I need brackets,"},{"Start":"07:26.318 ","End":"07:32.090","Text":"and now I\u0027ll expand the brackets and bring stuff to the other side to collect like terms."},{"Start":"07:32.090 ","End":"07:34.070","Text":"We\u0027ve got x^2, let\u0027s see how many x?"},{"Start":"07:34.070 ","End":"07:37.640","Text":"5x from here, minus 4x becomes plus 4x,"},{"Start":"07:37.640 ","End":"07:41.025","Text":"so I have 9x, and here I have 20,"},{"Start":"07:41.025 ","End":"07:42.630","Text":"subtract it from the 10,"},{"Start":"07:42.630 ","End":"07:45.045","Text":"makes it minus 10 equals 0."},{"Start":"07:45.045 ","End":"07:49.190","Text":"Quadratic equation, I don\u0027t want to waste time solving quadratic equations,"},{"Start":"07:49.190 ","End":"07:57.165","Text":"so I\u0027ll tell you that the solutions are x equals 1 and x equals minus 10,"},{"Start":"07:57.165 ","End":"08:01.760","Text":"and you can pause and verify it for yourself or substitute."},{"Start":"08:01.760 ","End":"08:07.880","Text":"There are websites that have calculators to compute solutions to quadratic equations,"},{"Start":"08:07.880 ","End":"08:09.910","Text":"so I\u0027m just going to take it from here."},{"Start":"08:09.910 ","End":"08:12.200","Text":"Now, because we use the squaring,"},{"Start":"08:12.200 ","End":"08:15.980","Text":"we must check each one to see if it really is"},{"Start":"08:15.980 ","End":"08:20.450","Text":"a solution and not one of those extraneous false solutions."},{"Start":"08:20.450 ","End":"08:23.050","Text":"Let\u0027s take the 1 first,"},{"Start":"08:23.050 ","End":"08:27.405","Text":"so if x equals 1 and I substitute it here,"},{"Start":"08:27.405 ","End":"08:28.985","Text":"what I\u0027m asking is,"},{"Start":"08:28.985 ","End":"08:36.510","Text":"the square root of 1^2 is 1 plus 5 times 1 plus 10,"},{"Start":"08:36.510 ","End":"08:42.350","Text":"is this equal to twice the square root of 5 minus 1?"},{"Start":"08:42.350 ","End":"08:50.725","Text":"Well, this is the square root of 16 and this is twice the square root of 4,"},{"Start":"08:50.725 ","End":"08:52.560","Text":"so what I\u0027m asking is,"},{"Start":"08:52.560 ","End":"08:56.175","Text":"is 4 equal to 2 times 2?"},{"Start":"08:56.175 ","End":"08:57.855","Text":"The answer is yes,"},{"Start":"08:57.855 ","End":"09:00.605","Text":"we have validated this solution."},{"Start":"09:00.605 ","End":"09:03.560","Text":"Now, let\u0027s try the next one."},{"Start":"09:03.560 ","End":"09:09.835","Text":"Let\u0027s let x equals minus 10 this time and see what we get on the left."},{"Start":"09:09.835 ","End":"09:13.940","Text":"The square root minus 10 squared is still plus 100,"},{"Start":"09:13.940 ","End":"09:16.100","Text":"but that\u0027s minus 50,"},{"Start":"09:16.100 ","End":"09:18.928","Text":"and then we have plus 10,"},{"Start":"09:18.928 ","End":"09:26.535","Text":"and I want to know if this is equal to twice the square root of 5 minus minus 10."},{"Start":"09:26.535 ","End":"09:30.270","Text":"In other words, is the square root,"},{"Start":"09:30.270 ","End":"09:32.490","Text":"this comes out to be 60,"},{"Start":"09:32.490 ","End":"09:41.865","Text":"is the square root of 60 equal to twice the square root of this is 15?"},{"Start":"09:41.865 ","End":"09:45.960","Text":"The answer is yes, I\u0027ll show you why."},{"Start":"09:45.960 ","End":"09:48.470","Text":"We had a similar thing with the square root of 8,"},{"Start":"09:48.470 ","End":"09:51.475","Text":"we can break the square root of 60 up,"},{"Start":"09:51.475 ","End":"10:00.095","Text":"as the square root of 4 times 15 and then we take each piece separately,"},{"Start":"10:00.095 ","End":"10:01.370","Text":"square root of a product,"},{"Start":"10:01.370 ","End":"10:04.280","Text":"you just take the square root of each of the factors,"},{"Start":"10:04.280 ","End":"10:07.160","Text":"square root of 15 and you can see now that square root of 4 is"},{"Start":"10:07.160 ","End":"10:11.030","Text":"2 and so this really is equality and"},{"Start":"10:11.030 ","End":"10:17.870","Text":"so this one is also verified and here\u0027s an example where both of the solutions came out."},{"Start":"10:17.870 ","End":"10:20.480","Text":"Previously we had 1 yes and 1 no,"},{"Start":"10:20.480 ","End":"10:22.295","Text":"this time both are yes,"},{"Start":"10:22.295 ","End":"10:25.865","Text":"so this is a bit of an extra new technique or there\u0027s 2 square roots,"},{"Start":"10:25.865 ","End":"10:32.880","Text":"and I think we\u0027ll take a break now and continue in the next clip."}],"ID":5029},{"Watched":false,"Name":"Equations with Roots (continued2)","Duration":"11m 25s","ChapterTopicVideoID":5030,"CourseChapterTopicPlaylistID":3873,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"We\u0027re continuing equations with square root."},{"Start":"00:02.640 ","End":"00:05.160","Text":"I guess that\u0027s continued again."},{"Start":"00:05.160 ","End":"00:06.885","Text":"I\u0027ll clear the board."},{"Start":"00:06.885 ","End":"00:15.150","Text":"The example I\u0027ll take next is the square root of x minus 2 times the square root of"},{"Start":"00:15.150 ","End":"00:23.490","Text":"2x minus 2 equals the square root of 13 minus x^2."},{"Start":"00:23.490 ","End":"00:26.880","Text":"There\u0027s a couple of differences in this example."},{"Start":"00:26.880 ","End":"00:28.980","Text":"The first is that I have 3 square roots,"},{"Start":"00:28.980 ","End":"00:31.110","Text":"although it\u0027s a product not a sum."},{"Start":"00:31.110 ","End":"00:34.500","Text":"A product is actually easier than a sum."},{"Start":"00:34.500 ","End":"00:39.000","Text":"Second difference will be in the solution and you\u0027ll see."},{"Start":"00:39.000 ","End":"00:41.625","Text":"Well, I don\u0027t want to put a spoiler in."},{"Start":"00:41.625 ","End":"00:47.350","Text":"We do what we usually do is we take both sides and we square them,"},{"Start":"00:47.350 ","End":"00:50.390","Text":"and don\u0027t forget that when you have a product squared,"},{"Start":"00:50.390 ","End":"00:53.405","Text":"you square each piece separately."},{"Start":"00:53.405 ","End":"00:58.415","Text":"If I square this I get x minus 2 and I need brackets and the same here,"},{"Start":"00:58.415 ","End":"01:03.605","Text":"I square this the square root sign drops off but I still need brackets,"},{"Start":"01:03.605 ","End":"01:08.405","Text":"and here just get the 13 minus x^2."},{"Start":"01:08.405 ","End":"01:10.010","Text":"I don\u0027t need brackets."},{"Start":"01:10.010 ","End":"01:13.070","Text":"Expand this, let\u0027s see."},{"Start":"01:13.070 ","End":"01:15.530","Text":"x with 2x is 2x^2 squared."},{"Start":"01:15.530 ","End":"01:16.730","Text":"Let\u0027s collect x\u0027s,"},{"Start":"01:16.730 ","End":"01:21.070","Text":"I have a minus 4x from here and a minus 2x from here,"},{"Start":"01:21.070 ","End":"01:26.295","Text":"so I\u0027ve got minus 6x and this with this gives me plus 4,"},{"Start":"01:26.295 ","End":"01:28.500","Text":"still 13 minus x^2."},{"Start":"01:28.500 ","End":"01:32.075","Text":"Then if I bring this to the other side,2 minus minus 1,"},{"Start":"01:32.075 ","End":"01:36.320","Text":"so it\u0027s 3x^2, 2 minus minus 6x."},{"Start":"01:36.320 ","End":"01:38.630","Text":"The 13 comes over as minus 13,"},{"Start":"01:38.630 ","End":"01:41.485","Text":"so it\u0027s minus 9 equals 0."},{"Start":"01:41.485 ","End":"01:44.700","Text":"It\u0027s a good idea to divide by"},{"Start":"01:44.700 ","End":"01:51.410","Text":"a common factor if you can make smaller numbers and then divide both sides by 3,"},{"Start":"01:51.410 ","End":"01:57.035","Text":"and then we\u0027ll get x^2 minus 2x minus 3 equals 0."},{"Start":"01:57.035 ","End":"02:01.189","Text":"As usual I\u0027m not going to solve quadratic equations,"},{"Start":"02:01.189 ","End":"02:04.355","Text":"I\u0027m just going to give you the solutions."},{"Start":"02:04.355 ","End":"02:09.795","Text":"x could be 3 and x could be minus 1."},{"Start":"02:09.795 ","End":"02:13.680","Text":"We need to check each 1 in the original."},{"Start":"02:13.680 ","End":"02:20.855","Text":"First I\u0027m going to check the x equals 3 and see what we get."},{"Start":"02:20.855 ","End":"02:23.900","Text":"We\u0027ve got the square root of 3 minus 2."},{"Start":"02:23.900 ","End":"02:25.950","Text":"I\u0027m substituting in this 1,"},{"Start":"02:25.950 ","End":"02:30.870","Text":"times the square root of twice 3 minus 2,"},{"Start":"02:30.870 ","End":"02:39.250","Text":"and I\u0027m asking, is this equal to the square root of 13 minus 3^2?"},{"Start":"02:39.250 ","End":"02:41.930","Text":"Let\u0027s see what this gives me."},{"Start":"02:41.930 ","End":"02:46.180","Text":"Square root of 3 minus 2 is a square root of 1 is 1,"},{"Start":"02:46.180 ","End":"02:49.380","Text":"2 times 3 is 6 minus 2 is 4,"},{"Start":"02:49.380 ","End":"02:51.585","Text":"so square root of 4 is 2."},{"Start":"02:51.585 ","End":"02:56.770","Text":"Here I\u0027ve got the square root of 13 minus 9,"},{"Start":"02:56.770 ","End":"02:59.695","Text":"which is the square root of 4, which is 2."},{"Start":"02:59.695 ","End":"03:04.430","Text":"Yes, this is a solution and I\u0027ll highlight it."},{"Start":"03:04.430 ","End":"03:07.145","Text":"Now what about the minus 1?"},{"Start":"03:07.145 ","End":"03:10.610","Text":"Let\u0027s try substituting x equals minus 1."},{"Start":"03:10.610 ","End":"03:13.310","Text":"I start off with the square root of x minus 2,"},{"Start":"03:13.310 ","End":"03:15.845","Text":"which is minus 1 minus 2 and I say,"},{"Start":"03:15.845 ","End":"03:17.090","Text":"well, hold it there."},{"Start":"03:17.090 ","End":"03:18.530","Text":"I don\u0027t need to continue."},{"Start":"03:18.530 ","End":"03:26.600","Text":"Look, square root of a negative number violates the domain of definition,"},{"Start":"03:26.600 ","End":"03:29.645","Text":"or if you like it\u0027s an illegal value."},{"Start":"03:29.645 ","End":"03:36.120","Text":"I don\u0027t even need to continue and I can say that minus 1 is disqualified,"},{"Start":"03:36.120 ","End":"03:39.830","Text":"but it\u0027s not exactly in the same way as usual."},{"Start":"03:39.830 ","End":"03:43.925","Text":"It\u0027s not that we\u0027ve substituted in both sides and the equality fails."},{"Start":"03:43.925 ","End":"03:45.641","Text":"It\u0027s I can\u0027t even substitute,"},{"Start":"03:45.641 ","End":"03:49.220","Text":"that\u0027s another failure and I wanted to demonstrate that this can happen all,"},{"Start":"03:49.220 ","End":"03:55.835","Text":"and so there\u0027s no point in continuing the expansion because I can\u0027t compute this."},{"Start":"03:55.835 ","End":"04:01.695","Text":"Just clear the board and time for the next example which will probably be the last."},{"Start":"04:01.695 ","End":"04:07.695","Text":"Square root of 2x minus 3 plus"},{"Start":"04:07.695 ","End":"04:15.185","Text":"4 is equal to the square root of 8x plus 1."},{"Start":"04:15.185 ","End":"04:17.990","Text":"Now, it doesn\u0027t look very different,"},{"Start":"04:17.990 ","End":"04:21.560","Text":"but this equation is quite different because"},{"Start":"04:21.560 ","End":"04:25.925","Text":"the left-hand side is not the square root of something,"},{"Start":"04:25.925 ","End":"04:28.460","Text":"not even a constant times the square root of something,"},{"Start":"04:28.460 ","End":"04:30.140","Text":"I have a plus 4."},{"Start":"04:30.140 ","End":"04:32.030","Text":"Sure I could put the 4 on the other side."},{"Start":"04:32.030 ","End":"04:33.530","Text":"That would be the same thing."},{"Start":"04:33.530 ","End":"04:37.490","Text":"I could even put the 2 square roots on the same side and the 4 on another side,"},{"Start":"04:37.490 ","End":"04:39.740","Text":"then I\u0027d have a square root plus a square root."},{"Start":"04:39.740 ","End":"04:44.090","Text":"I still don\u0027t have a neat square root on each side,"},{"Start":"04:44.090 ","End":"04:48.065","Text":"so we\u0027re going to have to learn a new technique, very new."},{"Start":"04:48.065 ","End":"04:50.510","Text":"We\u0027re going to have to square both sides and then"},{"Start":"04:50.510 ","End":"04:53.150","Text":"later we\u0027re going to have to square something again."},{"Start":"04:53.150 ","End":"04:55.220","Text":"Let me show you, I\u0027ll demonstrate."},{"Start":"04:55.220 ","End":"04:59.540","Text":"We start off in this case by squaring both sides."},{"Start":"04:59.540 ","End":"05:01.850","Text":"Each side to the power of 2, if you like,"},{"Start":"05:01.850 ","End":"05:07.855","Text":"and I have to remind you of a law from algebra because people are forgetting it."},{"Start":"05:07.855 ","End":"05:11.645","Text":"That is that if I have a sum and I square it,"},{"Start":"05:11.645 ","End":"05:15.380","Text":"unlike a product I don\u0027t just square each 1 separately,"},{"Start":"05:15.380 ","End":"05:18.815","Text":"I do have an a^2 and I do have a b^2,"},{"Start":"05:18.815 ","End":"05:20.360","Text":"but there\u0027s a middle term,"},{"Start":"05:20.360 ","End":"05:25.490","Text":"a 2ab in the middle in this special binomial expansion or trinomial,"},{"Start":"05:25.490 ","End":"05:26.990","Text":"and I mustn\u0027t forget that."},{"Start":"05:26.990 ","End":"05:29.360","Text":"I might add that if it was a minus,"},{"Start":"05:29.360 ","End":"05:32.885","Text":"it would just mean that there\u0027s a minus in the middle term here."},{"Start":"05:32.885 ","End":"05:39.390","Text":"a^2 corresponds to this 1 squared which is 2x minus 3."},{"Start":"05:39.390 ","End":"05:42.694","Text":"Then I need a twice times the product of this and this."},{"Start":"05:42.694 ","End":"05:44.660","Text":"I\u0027ll take the 2 with the 4 first,"},{"Start":"05:44.660 ","End":"05:49.140","Text":"that\u0027s 8 and then the square root of 2x minus 3."},{"Start":"05:49.140 ","End":"05:50.640","Text":"Then the last 1 squared,"},{"Start":"05:50.640 ","End":"05:52.775","Text":"the b^2 is 16."},{"Start":"05:52.775 ","End":"05:56.000","Text":"On this side is just a straightforward square root and I square it,"},{"Start":"05:56.000 ","End":"05:59.660","Text":"so square root disappears and I have that."},{"Start":"05:59.660 ","End":"06:03.785","Text":"The thing is after I squared it I still have a square root."},{"Start":"06:03.785 ","End":"06:07.430","Text":"What we do here is we keep the square root on this side,"},{"Start":"06:07.430 ","End":"06:09.380","Text":"and the 8 doesn\u0027t hurt."},{"Start":"06:09.380 ","End":"06:13.595","Text":"Like I said, if you have a constant multiplier in front of a square root,"},{"Start":"06:13.595 ","End":"06:15.740","Text":"that\u0027s no problem at all."},{"Start":"06:15.740 ","End":"06:19.370","Text":"The rest of it I\u0027ll throw over to the right-hand side."},{"Start":"06:19.370 ","End":"06:24.000","Text":"Let\u0027s see. I have 8x and then I have minus 2x,"},{"Start":"06:24.000 ","End":"06:26.355","Text":"so I have 6x."},{"Start":"06:26.355 ","End":"06:29.315","Text":"Let\u0027s see, as far as numbers go I have a 1,"},{"Start":"06:29.315 ","End":"06:32.930","Text":"and then I have plus 3 makes it 4,"},{"Start":"06:32.930 ","End":"06:40.845","Text":"but then minus 16 makes it minus 12."},{"Start":"06:40.845 ","End":"06:44.060","Text":"I like to cancel wherever I can."},{"Start":"06:44.060 ","End":"06:47.060","Text":"I noticed that I can divide everything by 2."},{"Start":"06:47.060 ","End":"06:49.160","Text":"Let\u0027s just do that then the numbers are smaller,"},{"Start":"06:49.160 ","End":"06:51.004","Text":"less likely to make a mistake."},{"Start":"06:51.004 ","End":"06:57.655","Text":"4 times 2x minus 3 is just 3x and just 6."},{"Start":"06:57.655 ","End":"07:01.565","Text":"Now, I square both sides again,"},{"Start":"07:01.565 ","End":"07:03.890","Text":"and this time I will get rid of the square root."},{"Start":"07:03.890 ","End":"07:07.430","Text":"But remember, this is the formula for the sum."},{"Start":"07:07.430 ","End":"07:10.680","Text":"For a product it\u0027s still take each 1 separately."},{"Start":"07:10.680 ","End":"07:14.955","Text":"a times b^2 is still a^2 times b^2."},{"Start":"07:14.955 ","End":"07:22.970","Text":"In here, I get 4^2 is 16 and this thing squared is 2x minus 3 but in brackets."},{"Start":"07:22.970 ","End":"07:28.080","Text":"This equals this squared, (3x minus 6)^2."},{"Start":"07:28.280 ","End":"07:31.140","Text":"Expanding the left side,"},{"Start":"07:31.140 ","End":"07:38.300","Text":"I get 16 times 2 is 32x minus 16 times 3 is 48."},{"Start":"07:38.300 ","End":"07:41.840","Text":"Here, I\u0027ll use this formula with the minus,"},{"Start":"07:41.840 ","End":"07:47.345","Text":"so 3x all squared is 9x^2 squared, 3^2 times x^2,"},{"Start":"07:47.345 ","End":"07:49.970","Text":"and then minus twice this times this,"},{"Start":"07:49.970 ","End":"07:54.155","Text":"which comes out to be 36x and then plus 36."},{"Start":"07:54.155 ","End":"07:56.480","Text":"Let\u0027s bring everything to the right-hand side,"},{"Start":"07:56.480 ","End":"08:01.020","Text":"so I have 9x^2 and minus 36 minus"},{"Start":"08:01.020 ","End":"08:07.410","Text":"32 is minus 68x."},{"Start":"08:07.410 ","End":"08:09.020","Text":"Then if I bring this over,"},{"Start":"08:09.020 ","End":"08:16.355","Text":"it\u0027s 36 plus 48, which is 84."},{"Start":"08:16.355 ","End":"08:18.905","Text":"I make it, let\u0027s see."},{"Start":"08:18.905 ","End":"08:22.625","Text":"I don\u0027t think we can divide by anything."},{"Start":"08:22.625 ","End":"08:25.160","Text":"It\u0027s a quadratic equation."},{"Start":"08:25.160 ","End":"08:28.340","Text":"As usual, I\u0027m not going to solve quadratic equations."},{"Start":"08:28.340 ","End":"08:32.745","Text":"I\u0027ll just tell you the answer. There\u0027s 2 answers."},{"Start":"08:32.745 ","End":"08:34.725","Text":"One answer is 6,"},{"Start":"08:34.725 ","End":"08:43.030","Text":"and the other answer is 1 and 5/9 should be 14 over 9."},{"Start":"08:43.220 ","End":"08:45.500","Text":"We just have to check each 1."},{"Start":"08:45.500 ","End":"08:46.850","Text":"That\u0027s the most important part."},{"Start":"08:46.850 ","End":"08:51.770","Text":"Let\u0027s check the x equals 6 first. Do it over here."},{"Start":"08:51.770 ","End":"08:56.555","Text":"If x=6, and I\u0027m talking about the original equation,"},{"Start":"08:56.555 ","End":"09:03.770","Text":"then I get the square root of twice 6 minus 3 plus 4,"},{"Start":"09:03.770 ","End":"09:09.640","Text":"is this equal to the square root of 8 times 6 minus 1?"},{"Start":"09:09.640 ","End":"09:15.960","Text":"Let\u0027s see. Twice 6 minus 3 is 12 minus 3 is 9."},{"Start":"09:15.960 ","End":"09:20.070","Text":"I want to know if the square root of 9 plus 4,"},{"Start":"09:20.070 ","End":"09:22.635","Text":"this is a plus 1, I miss copied it."},{"Start":"09:22.635 ","End":"09:29.010","Text":"8 times 6 is 48 plus 1, so that\u0027s 49."},{"Start":"09:29.010 ","End":"09:31.830","Text":"This is reduced to asking,"},{"Start":"09:31.830 ","End":"09:36.360","Text":"is 3 plus 4 equal to 7?"},{"Start":"09:36.360 ","End":"09:39.490","Text":"I would say yes."},{"Start":"09:39.500 ","End":"09:42.915","Text":"This 1 has been validated."},{"Start":"09:42.915 ","End":"09:45.965","Text":"What about this 1?"},{"Start":"09:45.965 ","End":"09:47.360","Text":"Let\u0027s test it."},{"Start":"09:47.360 ","End":"09:51.470","Text":"Let\u0027s test what happens if we put x equals 1 and 5/9."},{"Start":"09:51.470 ","End":"09:57.695","Text":"We get the square root of twice 1 and 5/9,"},{"Start":"09:57.695 ","End":"10:00.325","Text":"minus 3 plus 4."},{"Start":"10:00.325 ","End":"10:07.460","Text":"Is this equal to the square root of 8 times 1 and 5/9?"},{"Start":"10:07.460 ","End":"10:08.750","Text":"Let me put brackets,"},{"Start":"10:08.750 ","End":"10:12.605","Text":"it looks confusing, plus 1, let\u0027s see."},{"Start":"10:12.605 ","End":"10:15.475","Text":"1 and 5/9 is 14 over 9."},{"Start":"10:15.475 ","End":"10:18.255","Text":"For double it, it\u0027s 28 over 9,"},{"Start":"10:18.255 ","End":"10:20.430","Text":"and 3 is 27 over 9,"},{"Start":"10:20.430 ","End":"10:24.525","Text":"so this comes out to square root of 1/9 plus 4."},{"Start":"10:24.525 ","End":"10:26.210","Text":"On this side, let\u0027s see."},{"Start":"10:26.210 ","End":"10:30.060","Text":"Again, it\u0027s 14 over 9."},{"Start":"10:30.060 ","End":"10:32.295","Text":"If I multiply that by 8,"},{"Start":"10:32.295 ","End":"10:36.975","Text":"14 times 8 is 112 over 9 plus 9 over 9,"},{"Start":"10:36.975 ","End":"10:40.765","Text":"so it\u0027s a 121 over 9."},{"Start":"10:40.765 ","End":"10:44.705","Text":"Take the square root of numerator and denominator separately,"},{"Start":"10:44.705 ","End":"10:52.380","Text":"and so this is 1 over 3 plus 4 and this is 11 over 3."},{"Start":"10:52.380 ","End":"10:56.930","Text":"I\u0027d have to say the answer to this question is no because"},{"Start":"10:56.930 ","End":"11:01.835","Text":"11 over 3 is less than 4 and this is more than 4 and it just isn\u0027t,"},{"Start":"11:01.835 ","End":"11:07.490","Text":"and so this 1 fails not because we can\u0027t substitute it,"},{"Start":"11:07.490 ","End":"11:09.245","Text":"not because of domain,"},{"Start":"11:09.245 ","End":"11:12.260","Text":"but just because the equality doesn\u0027t work."},{"Start":"11:12.260 ","End":"11:14.465","Text":"The answer is 6."},{"Start":"11:14.465 ","End":"11:19.456","Text":"There is plenty of solved examples so I could keep going,"},{"Start":"11:19.456 ","End":"11:25.320","Text":"and I think we\u0027ll just leave it at that and done with the tutorial."}],"ID":5030},{"Watched":false,"Name":"Exercise 1","Duration":"3m 9s","ChapterTopicVideoID":5031,"CourseChapterTopicPlaylistID":3873,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.995","Text":"In this exercise, we have a couple of equations with square roots in them."},{"Start":"00:04.995 ","End":"00:10.260","Text":"The first one is this and the usual procedure when you"},{"Start":"00:10.260 ","End":"00:15.330","Text":"have a single square root like this is to square both sides."},{"Start":"00:15.330 ","End":"00:18.960","Text":"I\u0027ll just make a note that I\u0027m going to square"},{"Start":"00:18.960 ","End":"00:24.000","Text":"both sides and we have to remember that when we square both sides,"},{"Start":"00:24.000 ","End":"00:26.790","Text":"we can get extra solutions,"},{"Start":"00:26.790 ","End":"00:29.690","Text":"sometimes called extraneous solutions creeping in,"},{"Start":"00:29.690 ","End":"00:32.559","Text":"so at the end we have to check our answers."},{"Start":"00:32.559 ","End":"00:34.550","Text":"If we just square it, here,"},{"Start":"00:34.550 ","End":"00:37.475","Text":"the square root sign just drops off."},{"Start":"00:37.475 ","End":"00:41.270","Text":"That\u0027s 4x-3 and if we square 5,"},{"Start":"00:41.270 ","End":"00:44.930","Text":"5 times 5 is 25."},{"Start":"00:44.930 ","End":"00:50.995","Text":"4x=28, x=7."},{"Start":"00:50.995 ","End":"00:56.300","Text":"But we have to check that our answer is correct."},{"Start":"00:56.300 ","End":"01:00.290","Text":"Let\u0027s see what happens if we put x=7 here."},{"Start":"01:00.290 ","End":"01:02.570","Text":"4 times 7 is 28,"},{"Start":"01:02.570 ","End":"01:08.039","Text":"28-3 is 25 and indeed,"},{"Start":"01:08.039 ","End":"01:12.840","Text":"the square root of 25 is 5."},{"Start":"01:12.840 ","End":"01:16.740","Text":"So x=7 can be highlighted."},{"Start":"01:16.740 ","End":"01:18.620","Text":"On to Part B."},{"Start":"01:18.620 ","End":"01:23.495","Text":"Part B, we also have the square root of something equals something."},{"Start":"01:23.495 ","End":"01:26.600","Text":"Again, we\u0027ll square both sides."},{"Start":"01:26.600 ","End":"01:29.480","Text":"Indicate that I\u0027ll square both sides."},{"Start":"01:29.480 ","End":"01:33.260","Text":"I get x+2=x^2."},{"Start":"01:33.260 ","End":"01:36.260","Text":"This time, we get a quadratic equation."},{"Start":"01:36.260 ","End":"01:38.900","Text":"Let\u0027s bring everything to the right-hand side,"},{"Start":"01:38.900 ","End":"01:46.270","Text":"but then put that on the left so we get x^2-x-2=0."},{"Start":"01:46.270 ","End":"01:48.680","Text":"Let\u0027s quickly use the formula,"},{"Start":"01:48.680 ","End":"01:53.090","Text":"x=-b plus or minus the square root"},{"Start":"01:53.090 ","End":"02:00.410","Text":"of b^2-4ac over 2a."},{"Start":"02:00.410 ","End":"02:03.965","Text":"Under the square root sign, we have 9,"},{"Start":"02:03.965 ","End":"02:12.165","Text":"so it\u0027s 1 plus or minus 3 over 2."},{"Start":"02:12.165 ","End":"02:14.070","Text":"If I take the plus,"},{"Start":"02:14.070 ","End":"02:16.070","Text":"it\u0027s 4/2 is 2."},{"Start":"02:16.070 ","End":"02:17.900","Text":"If I take the minus,"},{"Start":"02:17.900 ","End":"02:21.440","Text":"it\u0027s going to give me minus 1,"},{"Start":"02:21.440 ","End":"02:26.225","Text":"and so I have to check these candidates for solutions."},{"Start":"02:26.225 ","End":"02:28.415","Text":"Let\u0027s try putting them in."},{"Start":"02:28.415 ","End":"02:30.205","Text":"If x is 2,"},{"Start":"02:30.205 ","End":"02:32.250","Text":"2 plus 2 is 4,"},{"Start":"02:32.250 ","End":"02:34.080","Text":"square root of 4 is 2,"},{"Start":"02:34.080 ","End":"02:36.840","Text":"which is x, this is fine."},{"Start":"02:36.840 ","End":"02:39.770","Text":"With the other one, well,"},{"Start":"02:39.770 ","End":"02:42.890","Text":"already I can tell I have x on the right-hand side,"},{"Start":"02:42.890 ","End":"02:44.720","Text":"which is minus 1."},{"Start":"02:44.720 ","End":"02:47.240","Text":"The square root of anything is not going to be minus 1,"},{"Start":"02:47.240 ","End":"02:49.115","Text":"a square root is always positive."},{"Start":"02:49.115 ","End":"02:50.990","Text":"If we want to actually do it,"},{"Start":"02:50.990 ","End":"02:54.515","Text":"it\u0027s minus 1 plus 2 is 1,"},{"Start":"02:54.515 ","End":"02:56.510","Text":"but the square root of 1 is 1."},{"Start":"02:56.510 ","End":"02:59.875","Text":"The square root of 1 is not minus 1."},{"Start":"02:59.875 ","End":"03:04.025","Text":"This is ruled out, it just doesn\u0027t satisfy the equation."},{"Start":"03:04.025 ","End":"03:05.795","Text":"This is their only answer."},{"Start":"03:05.795 ","End":"03:08.820","Text":"That\u0027s the end of this exercise."}],"ID":5031},{"Watched":false,"Name":"Exercise 2","Duration":"5m 44s","ChapterTopicVideoID":5032,"CourseChapterTopicPlaylistID":3873,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.160","Text":"Here in this exercise,"},{"Start":"00:02.160 ","End":"00:05.385","Text":"we have a pair of equations with square roots."},{"Start":"00:05.385 ","End":"00:07.230","Text":"Will take the first 1."},{"Start":"00:07.230 ","End":"00:11.775","Text":"What I\u0027d like is the square root on its own here I have something else with it."},{"Start":"00:11.775 ","End":"00:17.100","Text":"I\u0027ll just get rid of this by throwing it to the other side and then we get"},{"Start":"00:17.100 ","End":"00:23.565","Text":"the square root of 3x plus 1=13 minus x."},{"Start":"00:23.565 ","End":"00:26.550","Text":"At this point, I square both sides."},{"Start":"00:26.550 ","End":"00:27.945","Text":"Just make a note of that."},{"Start":"00:27.945 ","End":"00:31.335","Text":"Square both sides."},{"Start":"00:31.335 ","End":"00:36.160","Text":"Then we get 3x plus 1 without the square root."},{"Start":"00:36.160 ","End":"00:41.255","Text":"Then I\u0027ll use the binomial expansion, the special product."},{"Start":"00:41.255 ","End":"00:43.740","Text":"Well, write it at the side,"},{"Start":"00:43.930 ","End":"00:49.385","Text":"a minus b^2 is a^2 minus 2ab plus b^2."},{"Start":"00:49.385 ","End":"00:58.885","Text":"Applying that here, 13^2 is 169 minus twice 13 is 26x plus x^2."},{"Start":"00:58.885 ","End":"01:01.010","Text":"Looks like I\u0027ve got a quadratic equation."},{"Start":"01:01.010 ","End":"01:03.350","Text":"Let\u0027s bring the 3x plus 1 over."},{"Start":"01:03.350 ","End":"01:07.670","Text":"Rearrange the order got x^2 minus"},{"Start":"01:07.670 ","End":"01:12.510","Text":"29x plus"},{"Start":"01:12.510 ","End":"01:18.360","Text":"168 is equal 0."},{"Start":"01:18.360 ","End":"01:20.660","Text":"A large number here."},{"Start":"01:20.660 ","End":"01:23.720","Text":"I don\u0027t want to spend too much time with quadratic equations."},{"Start":"01:23.720 ","End":"01:28.235","Text":"I did it on my own and found that there were 2 solutions."},{"Start":"01:28.235 ","End":"01:35.865","Text":"That x is either equal to 21 or x=8."},{"Start":"01:35.865 ","End":"01:37.845","Text":"Those are the 2 solutions."},{"Start":"01:37.845 ","End":"01:41.060","Text":"Let\u0027s see if each of these works."},{"Start":"01:41.060 ","End":"01:47.390","Text":"We substitute in the original equation and we\u0027ll take them 1 at a time."},{"Start":"01:47.390 ","End":"01:48.920","Text":"Let\u0027s take the 21."},{"Start":"01:48.920 ","End":"01:54.620","Text":"21 times 3 is 63 plus 1 is 64."},{"Start":"01:54.620 ","End":"01:57.500","Text":"Square root of 64 is 8,"},{"Start":"01:57.500 ","End":"02:02.675","Text":"8 plus 21 is not 13."},{"Start":"02:02.675 ","End":"02:04.115","Text":"Don\u0027t even have to say what is."},{"Start":"02:04.115 ","End":"02:07.580","Text":"Obviously 8 plus 21 is not 13."},{"Start":"02:07.580 ","End":"02:10.790","Text":"This one has been ruled out."},{"Start":"02:10.790 ","End":"02:12.475","Text":"Let\u0027s try the next,"},{"Start":"02:12.475 ","End":"02:14.730","Text":"I put in x=8."},{"Start":"02:14.730 ","End":"02:17.880","Text":"3 times 8 plus 1 is 25."},{"Start":"02:17.880 ","End":"02:19.470","Text":"Square root of 25 is 5,"},{"Start":"02:19.470 ","End":"02:21.435","Text":"5 plus 8 is 13."},{"Start":"02:21.435 ","End":"02:23.610","Text":"Yes, this one\u0027s okay."},{"Start":"02:23.610 ","End":"02:27.165","Text":"Our solution is just x=8."},{"Start":"02:27.165 ","End":"02:30.540","Text":"Onto Part B."},{"Start":"02:30.540 ","End":"02:33.040","Text":"In Part B,"},{"Start":"02:33.040 ","End":"02:37.340","Text":"I also want to get the square root isolated."},{"Start":"02:37.340 ","End":"02:42.485","Text":"I\u0027ll bring this to the left and the 2x to the right."},{"Start":"02:42.485 ","End":"02:47.450","Text":"Then we\u0027ll get 3 times the square root of x"},{"Start":"02:47.450 ","End":"02:53.310","Text":"minus 1 is equal to 16 minus 2x."},{"Start":"02:53.310 ","End":"02:57.755","Text":"The 3 here is okay, multiplies."},{"Start":"02:57.755 ","End":"03:06.825","Text":"Now we\u0027ll square both sides and we get each of them squared separately 3 times 3 is 9."},{"Start":"03:06.825 ","End":"03:09.170","Text":"The square root of x minus 1 when you square,"},{"Start":"03:09.170 ","End":"03:11.870","Text":"it just becomes x minus 1."},{"Start":"03:11.870 ","End":"03:16.835","Text":"On this side, we can use the formula,"},{"Start":"03:16.835 ","End":"03:23.585","Text":"a-b^2 is a^2 minus 2ab plus b^2."},{"Start":"03:23.585 ","End":"03:28.740","Text":"That gives us 16^2 is 256 minus"},{"Start":"03:28.740 ","End":"03:35.110","Text":"twice 2 times 16 is 64x plus 4x^2."},{"Start":"03:35.210 ","End":"03:38.654","Text":"Let\u0027s bring all this to the right hand side."},{"Start":"03:38.654 ","End":"03:42.765","Text":"I have 4x^2 from here."},{"Start":"03:42.765 ","End":"03:45.150","Text":"Then minus 64x,"},{"Start":"03:45.150 ","End":"03:50.760","Text":"minus 9x is minus, let\u0027s see,"},{"Start":"03:50.760 ","End":"03:56.895","Text":"64 and 9 is 73x,"},{"Start":"03:56.895 ","End":"04:00.810","Text":"then 256, this is minus 9,"},{"Start":"04:00.810 ","End":"04:02.475","Text":"so it\u0027s plus 9."},{"Start":"04:02.475 ","End":"04:12.095","Text":"256 plus 9 is 265 = 0."},{"Start":"04:12.095 ","End":"04:14.480","Text":"I went ahead and computed it for you."},{"Start":"04:14.480 ","End":"04:19.280","Text":"I didn\u0027t want to spend a lot of time just solving quadratic equations."},{"Start":"04:19.280 ","End":"04:20.735","Text":"That\u0027s not the point here."},{"Start":"04:20.735 ","End":"04:25.730","Text":"I got the answers, 5 and 13.25 For this quadratic."},{"Start":"04:25.730 ","End":"04:31.145","Text":"Now, we just have to check that these really do satisfy the equation."},{"Start":"04:31.145 ","End":"04:33.230","Text":"Equations with square roots."},{"Start":"04:33.230 ","End":"04:37.595","Text":"You get these external non-solution scraping in."},{"Start":"04:37.595 ","End":"04:40.600","Text":"Let\u0027s see. Let\u0027s take the 5 first."},{"Start":"04:40.600 ","End":"04:42.705","Text":"The left hand side is 10."},{"Start":"04:42.705 ","End":"04:44.610","Text":"Now let us see the right hand side."},{"Start":"04:44.610 ","End":"04:48.255","Text":"5 minus 1 is 4."},{"Start":"04:48.255 ","End":"04:50.490","Text":"Square root of 4 is 2,"},{"Start":"04:50.490 ","End":"04:53.730","Text":"2 times 3 is 6,"},{"Start":"04:53.730 ","End":"04:56.040","Text":"16 minus 6 is 10,"},{"Start":"04:56.040 ","End":"04:59.840","Text":"so yes, 5 is a good solution."},{"Start":"04:59.840 ","End":"05:02.155","Text":"Now what about 13 and 1/4?"},{"Start":"05:02.155 ","End":"05:04.224","Text":"On the left hand side,"},{"Start":"05:04.224 ","End":"05:07.395","Text":"we get 26 and 1/2."},{"Start":"05:07.395 ","End":"05:08.930","Text":"If I can remember that,"},{"Start":"05:08.930 ","End":"05:12.455","Text":"26 and 1/2, now what do I get on the right?"},{"Start":"05:12.455 ","End":"05:16.760","Text":"13 and 1/4 minus 1 is 12 and 1/4."},{"Start":"05:16.760 ","End":"05:21.975","Text":"The square root of 12 and 1/4 is 3 and 1/2,"},{"Start":"05:21.975 ","End":"05:25.155","Text":"3 times 3 and 1/2 is 10 and 1/2."},{"Start":"05:25.155 ","End":"05:33.015","Text":"16 minus 10 and 1/2 is not 26 and 1/2 if it was a plus it would be, but it\u0027s not."},{"Start":"05:33.015 ","End":"05:35.625","Text":"This one does not work."},{"Start":"05:35.625 ","End":"05:38.730","Text":"I cross it out, this is not a solution."},{"Start":"05:38.730 ","End":"05:43.960","Text":"x=5 is a solution to Part B and we\u0027re done."}],"ID":5032},{"Watched":false,"Name":"Exercise 3","Duration":"8m 14s","ChapterTopicVideoID":5033,"CourseChapterTopicPlaylistID":3873,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"Exercise, we have to solve these 2 equations,"},{"Start":"00:03.570 ","End":"00:05.520","Text":"each of them with square roots."},{"Start":"00:05.520 ","End":"00:08.505","Text":"In the first 1, I have a product of square roots."},{"Start":"00:08.505 ","End":"00:14.640","Text":"This is good, it\u0027s as if there was 1 square root because square root of a,"},{"Start":"00:14.640 ","End":"00:18.540","Text":"square root of b is the square root of ab,"},{"Start":"00:18.540 ","End":"00:22.830","Text":"so we really just have 1 square root here and here also."},{"Start":"00:22.830 ","End":"00:26.070","Text":"Why don\u0027t we just square both sides?"},{"Start":"00:26.070 ","End":"00:28.305","Text":"Remember that at this point,"},{"Start":"00:28.305 ","End":"00:30.840","Text":"all sorts of false solutions can creep in,"},{"Start":"00:30.840 ","End":"00:32.535","Text":"so at the end we need to check."},{"Start":"00:32.535 ","End":"00:37.440","Text":"If we do that, what I get here is x-1 times"},{"Start":"00:37.440 ","End":"00:44.430","Text":"2x-5 is in this principle or just directly this thing squared is this,"},{"Start":"00:44.430 ","End":"00:46.490","Text":"this thing squared is this."},{"Start":"00:46.490 ","End":"00:47.840","Text":"When we square a product,"},{"Start":"00:47.840 ","End":"00:55.355","Text":"we square each 1 and on the right-hand side we get 11-x^2."},{"Start":"00:55.355 ","End":"00:57.260","Text":"If we multiply out,"},{"Start":"00:57.260 ","End":"00:59.390","Text":"we get, let\u0027s see,"},{"Start":"00:59.390 ","End":"01:05.975","Text":"2x^2-2x-5x is this,"},{"Start":"01:05.975 ","End":"01:10.370","Text":"and then plus 5=11-x^2."},{"Start":"01:10.370 ","End":"01:13.040","Text":"Everything on the left gives us"},{"Start":"01:13.040 ","End":"01:22.620","Text":"3x^2-7x, 5-11 is -6=0."},{"Start":"01:22.620 ","End":"01:28.240","Text":"Let\u0027s quickly solve this with the formula -b+ or"},{"Start":"01:28.240 ","End":"01:33.375","Text":"- the square root of b^2-4ac."},{"Start":"01:33.375 ","End":"01:40.290","Text":"It\u0027s going to be plus 4 times 3 times 6 all over 6."},{"Start":"01:40.290 ","End":"01:42.420","Text":"Now this is 72,"},{"Start":"01:42.420 ","End":"01:45.630","Text":"72 and 49 is a 121,"},{"Start":"01:45.630 ","End":"01:52.005","Text":"so this is 7+ or -11 over 6."},{"Start":"01:52.005 ","End":"01:56.445","Text":"This gives us 18/ 6 is 3,"},{"Start":"01:56.445 ","End":"02:02.405","Text":"and -4/6 is -2/3."},{"Start":"02:02.405 ","End":"02:06.955","Text":"Two possible solutions for x."},{"Start":"02:06.955 ","End":"02:10.960","Text":"Let\u0027s see which of these works here."},{"Start":"02:10.960 ","End":"02:13.185","Text":"If I put in 3,"},{"Start":"02:13.185 ","End":"02:17.145","Text":"I\u0027ve got the square root of 2 times,"},{"Start":"02:17.145 ","End":"02:20.445","Text":"let\u0027s see, twice 3-5 is 1."},{"Start":"02:20.445 ","End":"02:24.840","Text":"We just got the square root of 2 on the left and on the right,"},{"Start":"02:24.840 ","End":"02:28.635","Text":"3^2 from 11 is 2 also square root of 2,"},{"Start":"02:28.635 ","End":"02:30.270","Text":"so 3 is okay."},{"Start":"02:30.270 ","End":"02:33.090","Text":"What about the other solution is -2/3?"},{"Start":"02:33.090 ","End":"02:39.000","Text":"Let\u0027s start, -2/3-1-1 and 2/3 square root,"},{"Start":"02:39.000 ","End":"02:43.220","Text":"can\u0027t take square root of a negative, we disqualified already."},{"Start":"02:43.220 ","End":"02:46.728","Text":"This is the only solution,"},{"Start":"02:46.728 ","End":"02:49.210","Text":"and onto part b,"},{"Start":"02:49.210 ","End":"02:51.540","Text":"see what we have here."},{"Start":"02:51.540 ","End":"02:54.380","Text":"Here we have square root here and here,"},{"Start":"02:54.380 ","End":"02:55.610","Text":"but we have a plus here."},{"Start":"02:55.610 ","End":"02:59.690","Text":"That\u0027s kind of a nuisance sum but it\u0027s doable."},{"Start":"02:59.690 ","End":"03:02.900","Text":"We just have to square this equation twice."},{"Start":"03:02.900 ","End":"03:07.725","Text":"Why don\u0027t we start with squaring it once,"},{"Start":"03:07.725 ","End":"03:09.665","Text":"so I\u0027ll square this,"},{"Start":"03:09.665 ","End":"03:14.035","Text":"indicate that I\u0027m going to square both sides."},{"Start":"03:14.035 ","End":"03:17.420","Text":"Using the special binomial product,"},{"Start":"03:17.420 ","End":"03:23.060","Text":"a+b^2 is a^2+2ab plus b^2."},{"Start":"03:23.060 ","End":"03:31.995","Text":"What I get here is this thing squared is 2x-1 plus twice this times this,"},{"Start":"03:31.995 ","End":"03:38.385","Text":"so twice the 3 gives me 6 square root of 2x-1."},{"Start":"03:38.385 ","End":"03:41.340","Text":"Then the last one squared is 9,"},{"Start":"03:41.340 ","End":"03:45.790","Text":"right-hand side squared is 7x-1."},{"Start":"03:45.790 ","End":"03:47.810","Text":"Now what I want to do,"},{"Start":"03:47.810 ","End":"03:50.435","Text":"there\u0027s only have 1 square root."},{"Start":"03:50.435 ","End":"03:55.760","Text":"I want to leave this thing on its own on the left and put everything else on the right."},{"Start":"03:55.760 ","End":"03:59.870","Text":"The 6 doesn\u0027t hurt if it stays so 6 square root of"},{"Start":"03:59.870 ","End":"04:08.550","Text":"2x-1 is equal to on the right I have 7x-2x is 5x."},{"Start":"04:08.550 ","End":"04:10.965","Text":"Then I have plus 1,"},{"Start":"04:10.965 ","End":"04:14.940","Text":"plus 1 is 2-9,"},{"Start":"04:14.940 ","End":"04:17.695","Text":"that makes it -7."},{"Start":"04:17.695 ","End":"04:21.725","Text":"Now I\u0027m going to square both sides again."},{"Start":"04:21.725 ","End":"04:26.465","Text":"This time I get the square of a product is product of the squares,"},{"Start":"04:26.465 ","End":"04:31.345","Text":"so it\u0027s 36 times 2x-1,"},{"Start":"04:31.345 ","End":"04:35.480","Text":"and here, using the similar formula there is a minus here,"},{"Start":"04:35.480 ","End":"04:36.950","Text":"it\u0027s a minus here,"},{"Start":"04:36.950 ","End":"04:44.000","Text":"so it\u0027s 5x all squared is 25x^2 minus twice 5 times"},{"Start":"04:44.000 ","End":"04:52.440","Text":"7 is -70x+49 and see,"},{"Start":"04:52.440 ","End":"05:02.870","Text":"expand this is 72x-36 equals the same."},{"Start":"05:02.870 ","End":"05:06.140","Text":"Now what I want to do is put this,"},{"Start":"05:06.140 ","End":"05:08.135","Text":"combine it with this."},{"Start":"05:08.135 ","End":"05:17.710","Text":"Let\u0027s see, 25x^2-70x minus 72x is"},{"Start":"05:17.710 ","End":"05:21.290","Text":"-142x and then"},{"Start":"05:21.290 ","End":"05:28.070","Text":"+49+36 is"},{"Start":"05:28.070 ","End":"05:32.870","Text":"85= 0."},{"Start":"05:32.870 ","End":"05:35.195","Text":"Quadratic equation."},{"Start":"05:35.195 ","End":"05:38.030","Text":"I\u0027m not going to spend time solving this,"},{"Start":"05:38.030 ","End":"05:42.320","Text":"I did it at aside separately and I brought you the solutions,"},{"Start":"05:42.320 ","End":"05:45.770","Text":"it\u0027s either 5 or 17/25."},{"Start":"05:45.770 ","End":"05:53.060","Text":"You should do this as an exercise on your own if you need the practice with quadratics."},{"Start":"05:53.060 ","End":"05:58.380","Text":"Then you write, we have to check these to see that they really do work."},{"Start":"05:58.380 ","End":"06:03.485","Text":"Let\u0027s try x equals 5 first and see what happens."},{"Start":"06:03.485 ","End":"06:04.985","Text":"If x is 5,"},{"Start":"06:04.985 ","End":"06:09.105","Text":"twice 5-1 is 9."},{"Start":"06:09.105 ","End":"06:13.920","Text":"Square root of 9 is 3, 3+3 is 6."},{"Start":"06:13.920 ","End":"06:16.215","Text":"Left-hand side is 6."},{"Start":"06:16.215 ","End":"06:18.585","Text":"Now let\u0027s check the right-hand side."},{"Start":"06:18.585 ","End":"06:22.270","Text":"7 times 5 is 35+1,"},{"Start":"06:22.270 ","End":"06:25.035","Text":"36, square root of that is 6."},{"Start":"06:25.035 ","End":"06:27.510","Text":"Yes, 6=6."},{"Start":"06:27.510 ","End":"06:29.525","Text":"This one is a good one."},{"Start":"06:29.525 ","End":"06:33.395","Text":"Now let\u0027s try the other, 17/25."},{"Start":"06:33.395 ","End":"06:37.730","Text":"Let\u0027s see if 17/25 twice"},{"Start":"06:37.730 ","End":"06:44.130","Text":"x is 34/25-1,"},{"Start":"06:44.130 ","End":"06:52.360","Text":"that comes out to be 34 minus 25/25, that\u0027s 9/25."},{"Start":"06:52.360 ","End":"06:55.490","Text":"If I take the square root of that,"},{"Start":"06:55.490 ","End":"07:00.205","Text":"then I\u0027ve got 3/5,"},{"Start":"07:00.205 ","End":"07:02.745","Text":"if I add 3,"},{"Start":"07:02.745 ","End":"07:07.920","Text":"the left-hand side is 3 and 3/5."},{"Start":"07:07.920 ","End":"07:10.980","Text":"Let\u0027s see, right-hand side,"},{"Start":"07:10.980 ","End":"07:15.900","Text":"7 times 17/25,"},{"Start":"07:15.900 ","End":"07:25.540","Text":"7 times 17 is 119/25 plus 1."},{"Start":"07:25.540 ","End":"07:28.205","Text":"I could just add 25 here."},{"Start":"07:28.205 ","End":"07:33.755","Text":"That would give me 144/25."},{"Start":"07:33.755 ","End":"07:36.890","Text":"If I take the square root of that,"},{"Start":"07:36.890 ","End":"07:44.230","Text":"that would give me 12/5."},{"Start":"07:44.230 ","End":"07:50.405","Text":"What did we get on the left-hand side? I wrote it here."},{"Start":"07:50.405 ","End":"07:54.230","Text":"This is the right-hand side,"},{"Start":"07:54.230 ","End":"07:56.360","Text":"which if I put it as a mixed number,"},{"Start":"07:56.360 ","End":"08:01.710","Text":"is 2 and 2/5 or certainly 2 and 2/5 is not the same as 3"},{"Start":"08:01.710 ","End":"08:09.005","Text":"and 3/5 and so it looks like this one is not a solution."},{"Start":"08:09.005 ","End":"08:14.650","Text":"Then only x=5 works. Yeah, that\u0027s it."}],"ID":5033},{"Watched":false,"Name":"Exercise 4","Duration":"11m 21s","ChapterTopicVideoID":5034,"CourseChapterTopicPlaylistID":3873,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"In this exercise, we have 2 equations,"},{"Start":"00:03.930 ","End":"00:07.365","Text":"each of them with 2 square roots."},{"Start":"00:07.365 ","End":"00:11.370","Text":"The technique will be to square both sides,"},{"Start":"00:11.370 ","End":"00:14.835","Text":"but we\u0027ll have to square something twice as you will see."},{"Start":"00:14.835 ","End":"00:21.165","Text":"What we will need in both of them is the following formula for the square of a binomial,"},{"Start":"00:21.165 ","End":"00:26.470","Text":"that (a+b)^2= a^2+2ab+b^2,"},{"Start":"00:27.350 ","End":"00:30.630","Text":"and a similar one if it\u0027s a minus here,"},{"Start":"00:30.630 ","End":"00:33.210","Text":"then it will be a minus here."},{"Start":"00:33.210 ","End":"00:35.445","Text":"Okay, in the first one,"},{"Start":"00:35.445 ","End":"00:37.155","Text":"after we square it,"},{"Start":"00:37.155 ","End":"00:40.230","Text":"we should only have one square root as you will see."},{"Start":"00:40.230 ","End":"00:46.830","Text":"If I use this, then a^2 will be just 9x-8."},{"Start":"00:46.830 ","End":"00:50.010","Text":"Then I have -2ab."},{"Start":"00:50.010 ","End":"00:53.175","Text":"2 times 3 is 6."},{"Start":"00:53.175 ","End":"00:58.675","Text":"Then it will be the square root of this times the square root of x+4."},{"Start":"00:58.675 ","End":"01:04.861","Text":"Then +b^2 gives me +9,"},{"Start":"01:04.861 ","End":"01:09.065","Text":"that\u0027s the 3^2 times x+4."},{"Start":"01:09.065 ","End":"01:13.255","Text":"On the right-hand side, -2^2 is 4."},{"Start":"01:13.255 ","End":"01:20.405","Text":"Now, let\u0027s leave the square root on the left and put the rest of it on the right."},{"Start":"01:20.405 ","End":"01:25.120","Text":"I have -6 square root"},{"Start":"01:25.120 ","End":"01:32.440","Text":"of 9x-8 square root of x+4 equals,"},{"Start":"01:32.440 ","End":"01:34.880","Text":"I had an idea it will be simpler if I put"},{"Start":"01:34.880 ","End":"01:37.910","Text":"the square root on the right and the 4 on the left,"},{"Start":"01:37.910 ","End":"01:39.590","Text":"and then switch sides."},{"Start":"01:39.590 ","End":"01:41.990","Text":"This comes out to be plus."},{"Start":"01:41.990 ","End":"01:45.630","Text":"Then what I have is the 9x-8+9(x+4)-4."},{"Start":"01:50.690 ","End":"01:53.210","Text":"The 4 on the left, this on the right,"},{"Start":"01:53.210 ","End":"01:54.890","Text":"and then I switch sides."},{"Start":"01:54.890 ","End":"01:58.400","Text":"I should square both sides."},{"Start":"01:58.400 ","End":"02:02.720","Text":"But, let\u0027s simplify this right-hand side first."},{"Start":"02:02.720 ","End":"02:05.030","Text":"What I have on the right is,"},{"Start":"02:05.030 ","End":"02:11.045","Text":"let\u0027s see the x\u0027s I have 9x+9x is 18x."},{"Start":"02:11.045 ","End":"02:12.665","Text":"As for the numbers,"},{"Start":"02:12.665 ","End":"02:14.555","Text":"what I have is,"},{"Start":"02:14.555 ","End":"02:18.640","Text":"let\u0027s see, +9 times 4 is 36,"},{"Start":"02:18.640 ","End":"02:21.865","Text":"and then -8-4 is -12."},{"Start":"02:21.865 ","End":"02:25.415","Text":"That leaves me with a +24,"},{"Start":"02:25.415 ","End":"02:27.825","Text":"and the same thing here."},{"Start":"02:27.825 ","End":"02:31.940","Text":"What I think we should do is 6 divides everything."},{"Start":"02:31.940 ","End":"02:36.538","Text":"We can simplify it by dividing everything by 6,"},{"Start":"02:36.538 ","End":"02:45.524","Text":"and we get square root of 9x-8 times the square root of x+4 equals,"},{"Start":"02:45.524 ","End":"02:50.190","Text":"dividing this by 6, 3x+4."},{"Start":"02:50.480 ","End":"02:54.830","Text":"This is what we got after squaring both sides and simplifying."},{"Start":"02:54.830 ","End":"02:56.995","Text":"Now we square again."},{"Start":"02:56.995 ","End":"03:00.425","Text":"The square of a product is the product of the squares."},{"Start":"03:00.425 ","End":"03:04.415","Text":"So we just get this thing squared, which is 9x-8."},{"Start":"03:04.415 ","End":"03:12.470","Text":"This is x+4 and here (3x+4)^2."},{"Start":"03:12.470 ","End":"03:19.415","Text":"Now we\u0027re going to use the special binomial expansion square of a binomial formula again."},{"Start":"03:19.415 ","End":"03:22.085","Text":"Just get a bit more space here."},{"Start":"03:22.085 ","End":"03:25.285","Text":"On the left we have,"},{"Start":"03:25.285 ","End":"03:28.110","Text":"let\u0027s just do it all in one go."},{"Start":"03:28.110 ","End":"03:33.510","Text":"9x times x is 9x^2."},{"Start":"03:33.510 ","End":"03:37.360","Text":"Then we have -8x plus 36x."},{"Start":"03:37.580 ","End":"03:42.015","Text":"That gives us plus 28x."},{"Start":"03:42.015 ","End":"03:45.720","Text":"Then -8 times 4 is -32."},{"Start":"03:45.720 ","End":"03:47.235","Text":"That\u0027s on this side."},{"Start":"03:47.235 ","End":"03:51.275","Text":"On this side we get, using this formula,"},{"Start":"03:51.275 ","End":"03:59.750","Text":"9x^2 twice 3 times 4 is 24x plus 16."},{"Start":"03:59.750 ","End":"04:07.135","Text":"We are somewhat in luck because the 9x^2 cancels and we don\u0027t have a quadratic."},{"Start":"04:07.135 ","End":"04:12.385","Text":"All we have left is 28 minus 24 is 4x,"},{"Start":"04:12.385 ","End":"04:18.530","Text":"and 32+16 is 48."},{"Start":"04:18.530 ","End":"04:25.730","Text":"If this is 48, then we have that x=12."},{"Start":"04:25.730 ","End":"04:27.950","Text":"Now, this is only a candidate,"},{"Start":"04:27.950 ","End":"04:31.819","Text":"we\u0027ve done to squarings and at each time you square both sides,"},{"Start":"04:31.819 ","End":"04:35.000","Text":"you can get all false solutions creeping in."},{"Start":"04:35.000 ","End":"04:40.670","Text":"The simplest thing to do is just to substitute in the original."},{"Start":"04:40.670 ","End":"04:44.015","Text":"I\u0027m going to take this x=12,"},{"Start":"04:44.015 ","End":"04:46.265","Text":"and substitute it in here."},{"Start":"04:46.265 ","End":"04:50.930","Text":"In fact, let\u0027s just substitute it in the left-hand side and see if we get -2,"},{"Start":"04:50.930 ","End":"04:52.715","Text":"which is the right-hand side."},{"Start":"04:52.715 ","End":"04:56.540","Text":"What we get is on the left-hand side on our continuing over here,"},{"Start":"04:56.540 ","End":"05:01.430","Text":"the square root of 9x-8,"},{"Start":"05:01.430 ","End":"05:05.230","Text":"see 9 times 12-8,"},{"Start":"05:05.230 ","End":"05:12.195","Text":"-3 times the square root of 12 plus 4."},{"Start":"05:12.195 ","End":"05:15.540","Text":"This is equal to, let\u0027s see,"},{"Start":"05:15.540 ","End":"05:21.585","Text":"9 times 12 is 108, -8 is 100."},{"Start":"05:21.585 ","End":"05:25.260","Text":"The square root of 16."},{"Start":"05:25.260 ","End":"05:30.636","Text":"This is 10-3 times 4,"},{"Start":"05:30.636 ","End":"05:34.600","Text":"and this is equal to 10-12."},{"Start":"05:37.340 ","End":"05:42.385","Text":"This is certainly equal to -2,"},{"Start":"05:42.385 ","End":"05:47.615","Text":"which is what is on the right-hand side."},{"Start":"05:47.615 ","End":"05:54.260","Text":"In other words, we started with left-hand side and ended up with right-hand side,"},{"Start":"05:54.260 ","End":"05:56.935","Text":"and so it satisfies."},{"Start":"05:56.935 ","End":"06:00.110","Text":"12 is the answer to part a,"},{"Start":"06:00.110 ","End":"06:03.635","Text":"and we can move on to part b."},{"Start":"06:03.635 ","End":"06:07.610","Text":"Again, we have two square roots."},{"Start":"06:07.610 ","End":"06:12.770","Text":"We raise both sides to the power of 2 and we square both sides,"},{"Start":"06:12.770 ","End":"06:14.540","Text":"I didn\u0027t even indicate that,"},{"Start":"06:14.540 ","End":"06:16.370","Text":"I should have written it also before."},{"Start":"06:16.370 ","End":"06:19.135","Text":"I\u0027ll just say it in words, we\u0027re squaring both sides."},{"Start":"06:19.135 ","End":"06:22.085","Text":"Using the same formula as we had above,"},{"Start":"06:22.085 ","End":"06:25.430","Text":"this thing squared is 2x-3,"},{"Start":"06:25.430 ","End":"06:30.110","Text":"then plus twice this times this, twice, let\u0027s see,"},{"Start":"06:30.110 ","End":"06:35.100","Text":"2x-3, 3-x in square roots."},{"Start":"06:35.100 ","End":"06:39.600","Text":"Then the last one squared is 3-x."},{"Start":"06:39.600 ","End":"06:43.090","Text":"This equals the right-hand side squared, which is 4."},{"Start":"06:43.090 ","End":"06:47.930","Text":"Now, this time I\u0027ll leave this on the left."},{"Start":"06:47.930 ","End":"06:56.320","Text":"Twice square root of 2x-3 square root of 3-x is equal to,"},{"Start":"06:56.320 ","End":"06:57.710","Text":"I\u0027ll put everything else on the right."},{"Start":"06:57.710 ","End":"07:00.575","Text":"Let\u0027s compute it. How many x\u0027s do we have?"},{"Start":"07:00.575 ","End":"07:05.300","Text":"On the left we have 2x-x is just x."},{"Start":"07:05.300 ","End":"07:08.610","Text":"On the right becomes -x."},{"Start":"07:08.610 ","End":"07:12.210","Text":"As for numbers, here we have -3+3,"},{"Start":"07:12.210 ","End":"07:15.000","Text":"so these two cancel each other out."},{"Start":"07:15.000 ","End":"07:19.430","Text":"We\u0027re just left with that same 4 that we had already."},{"Start":"07:19.430 ","End":"07:22.790","Text":"Now we square both sides again,"},{"Start":"07:22.790 ","End":"07:29.965","Text":"squaring this, we have 4(2x-3)(3-x)."},{"Start":"07:29.965 ","End":"07:33.195","Text":"I square each thing in the product."},{"Start":"07:33.195 ","End":"07:35.615","Text":"If I square the right-hand side,"},{"Start":"07:35.615 ","End":"07:38.675","Text":"again using the binomial expansion,"},{"Start":"07:38.675 ","End":"07:42.770","Text":"we get this thing squared is x^2 plus twice this,"},{"Start":"07:42.770 ","End":"07:44.150","Text":"times this, but it\u0027s a minus,"},{"Start":"07:44.150 ","End":"07:49.790","Text":"so we\u0027re going to have -8x and +16."},{"Start":"07:49.790 ","End":"07:53.510","Text":"Let\u0027s see, let\u0027s expand the left-hand side,"},{"Start":"07:53.510 ","End":"07:58.430","Text":"and I\u0027m going to just do it by collecting what I want."},{"Start":"07:58.430 ","End":"08:00.690","Text":"I want to collect the x^2 terms."},{"Start":"08:00.690 ","End":"08:02.700","Text":"From 2x times -x,"},{"Start":"08:02.700 ","End":"08:08.995","Text":"I\u0027ll get -2x^2 and multiply by 4, so -8x^2."},{"Start":"08:08.995 ","End":"08:11.675","Text":"Now, let\u0027s see where we can collect x\u0027s."},{"Start":"08:11.675 ","End":"08:15.580","Text":"x\u0027s we can get from -3 and -x."},{"Start":"08:15.580 ","End":"08:17.198","Text":"That gives me 3x,"},{"Start":"08:17.198 ","End":"08:19.920","Text":"and this with this gives me 6x."},{"Start":"08:19.920 ","End":"08:23.325","Text":"All together I have 9x from here,"},{"Start":"08:23.325 ","End":"08:26.925","Text":"but there\u0027s a 4, so it\u0027s 36x."},{"Start":"08:26.925 ","End":"08:28.500","Text":"Then just numbers,"},{"Start":"08:28.500 ","End":"08:31.545","Text":"-3 times 3 is -9."},{"Start":"08:31.545 ","End":"08:39.300","Text":"So it\u0027s minus 36 equals the same thing, x^2-8x+16."},{"Start":"08:39.300 ","End":"08:41.900","Text":"I could put it on either side."},{"Start":"08:41.900 ","End":"08:43.640","Text":"I prefer to move everything to the right,"},{"Start":"08:43.640 ","End":"08:46.160","Text":"but then put it on the left if you see what I mean,"},{"Start":"08:46.160 ","End":"08:49.145","Text":"because I want the x-squared coefficient to be positive."},{"Start":"08:49.145 ","End":"08:53.125","Text":"This to the right is going to be 9x^2."},{"Start":"08:53.125 ","End":"08:54.585","Text":"Moving this to the right,"},{"Start":"08:54.585 ","End":"09:01.545","Text":"I\u0027ve got -8, -36 is -44x."},{"Start":"09:01.545 ","End":"09:10.450","Text":"Then this on the right will give me 16+36 would be 52+52,"},{"Start":"09:12.010 ","End":"09:15.365","Text":"and that\u0027s equal to 0."},{"Start":"09:15.365 ","End":"09:18.020","Text":"This is a quadratic equation."},{"Start":"09:18.020 ","End":"09:19.954","Text":"I\u0027m not going to waste time."},{"Start":"09:19.954 ","End":"09:21.804","Text":"I\u0027ll just give you the answers."},{"Start":"09:21.804 ","End":"09:26.910","Text":"I get x is either 2 or 2, and 8/9."},{"Start":"09:26.910 ","End":"09:30.185","Text":"Now, each of these"},{"Start":"09:30.185 ","End":"09:34.760","Text":"is only a candidate for a solution because we raised things to the power of 2."},{"Start":"09:34.760 ","End":"09:37.370","Text":"We squared things, so we have to check in the original."},{"Start":"09:37.370 ","End":"09:39.650","Text":"Let\u0027s check each one of them."},{"Start":"09:39.650 ","End":"09:41.990","Text":"First of all, I\u0027m going to check,"},{"Start":"09:41.990 ","End":"09:43.595","Text":"let me do it over here."},{"Start":"09:43.595 ","End":"09:50.870","Text":"What I\u0027m doing now is I\u0027m going to check x=2 in the original on the left-hand side."},{"Start":"09:50.870 ","End":"09:56.420","Text":"I get that the left-hand side is equal to the square root of"},{"Start":"09:56.420 ","End":"10:02.840","Text":"2 times 2 -3 plus the square root of 3-2."},{"Start":"10:02.840 ","End":"10:04.250","Text":"Let\u0027s see what this equals."},{"Start":"10:04.250 ","End":"10:07.950","Text":"This is 4-3 is 1."},{"Start":"10:07.950 ","End":"10:11.070","Text":"It\u0027s the square root of 1 plus the square root of 1,"},{"Start":"10:11.070 ","End":"10:13.845","Text":"which is 1+1 which is 2."},{"Start":"10:13.845 ","End":"10:16.745","Text":"So this one is good. I can highlight it."},{"Start":"10:16.745 ","End":"10:18.365","Text":"Now let\u0027s try the second one."},{"Start":"10:18.365 ","End":"10:23.900","Text":"x equals 2 and 8/9 The left-hand side is"},{"Start":"10:23.900 ","End":"10:29.580","Text":"equal to square root of 2x-3."},{"Start":"10:29.580 ","End":"10:37.485","Text":"Let\u0027s see, twice this I make it 5 and 7/9 -3."},{"Start":"10:37.485 ","End":"10:44.130","Text":"Then square root of 3-x.3 minus this is just 1/9."},{"Start":"10:44.130 ","End":"10:51.090","Text":"This equals the square root of 2 and 7/9."},{"Start":"10:51.090 ","End":"10:53.490","Text":"2 and 7/9, you can check it,"},{"Start":"10:53.490 ","End":"10:58.100","Text":"is 25/9 plus the square root of 1/9,"},{"Start":"10:58.100 ","End":"11:04.270","Text":"take the top and bottom separately, this is 5/3."},{"Start":"11:04.270 ","End":"11:06.780","Text":"Here square root of 1 is 1."},{"Start":"11:06.780 ","End":"11:08.700","Text":"Square root of 9 again, is 3,"},{"Start":"11:08.700 ","End":"11:12.210","Text":"5/3 plus 1/3 is 6/3, which is 2."},{"Start":"11:12.210 ","End":"11:17.120","Text":"So yes, this thing turns out to also be a valid answer."},{"Start":"11:17.120 ","End":"11:21.150","Text":"We are done with part b, which means we\u0027re done."}],"ID":5034},{"Watched":false,"Name":"Exercise 5","Duration":"9m 4s","ChapterTopicVideoID":5035,"CourseChapterTopicPlaylistID":3873,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.825","Text":"In this exercise, we have a pair of equations with square roots."},{"Start":"00:03.825 ","End":"00:05.280","Text":"Each of them is similar,"},{"Start":"00:05.280 ","End":"00:08.355","Text":"square root plus square root equals square root."},{"Start":"00:08.355 ","End":"00:10.560","Text":"The way we\u0027re going to do this is we\u0027re going to square"},{"Start":"00:10.560 ","End":"00:13.410","Text":"both sides of the equation, and in both cases,"},{"Start":"00:13.410 ","End":"00:16.725","Text":"I\u0027ll write the formula we\u0027ll need is"},{"Start":"00:16.725 ","End":"00:22.440","Text":"a plus b squared equals a squared plus 2ab plus b squared,"},{"Start":"00:22.440 ","End":"00:24.675","Text":"you should really have this one memorized."},{"Start":"00:24.675 ","End":"00:26.670","Text":"After we do the squaring,"},{"Start":"00:26.670 ","End":"00:29.670","Text":"we\u0027ll only have one square root and then we\u0027ll have to square again,"},{"Start":"00:29.670 ","End":"00:31.950","Text":"the end of course we have to check our solution."},{"Start":"00:31.950 ","End":"00:34.110","Text":"Let\u0027s begin with part a."},{"Start":"00:34.110 ","End":"00:38.600","Text":"This one squared is x plus"},{"Start":"00:38.600 ","End":"00:45.965","Text":"3 plus twice this times this twice square root times square root x plus 3,"},{"Start":"00:45.965 ","End":"00:50.445","Text":"x minus 2, and then last one squared is just x minus 2,"},{"Start":"00:50.445 ","End":"00:54.145","Text":"is the right-hand side squared, 4x plus 1."},{"Start":"00:54.145 ","End":"00:56.990","Text":"Let\u0027s leave the square roots on this side,"},{"Start":"00:56.990 ","End":"00:58.745","Text":"I can leave the 2 with them."},{"Start":"00:58.745 ","End":"01:03.275","Text":"We get twice square root of x plus 3,"},{"Start":"01:03.275 ","End":"01:07.965","Text":"square root of x minus 2 equals,"},{"Start":"01:07.965 ","End":"01:10.485","Text":"and everything to the right-hand side."},{"Start":"01:10.485 ","End":"01:13.590","Text":"I have 4x, then minus x,"},{"Start":"01:13.590 ","End":"01:16.615","Text":"minus x leaves me with 2x."},{"Start":"01:16.615 ","End":"01:20.850","Text":"Then I have one from here and another 2 adds 3,"},{"Start":"01:20.850 ","End":"01:24.225","Text":"less 3 and that\u0027s it."},{"Start":"01:24.225 ","End":"01:27.810","Text":"Just 2x can cancel the 2,"},{"Start":"01:27.810 ","End":"01:30.275","Text":"that one goes and that one goes."},{"Start":"01:30.275 ","End":"01:33.080","Text":"Squaring again, each one of these is squared."},{"Start":"01:33.080 ","End":"01:35.020","Text":"The square root drops off,"},{"Start":"01:35.020 ","End":"01:37.410","Text":"this one times this,"},{"Start":"01:37.410 ","End":"01:40.695","Text":"and here we just get x squared. Let\u0027s see."},{"Start":"01:40.695 ","End":"01:41.880","Text":"Multiply this out,"},{"Start":"01:41.880 ","End":"01:50.550","Text":"I\u0027ve got x squared plus 3x minus 2x minus 6 equals x squared."},{"Start":"01:50.550 ","End":"01:53.040","Text":"Subtract x squared from both sides,"},{"Start":"01:53.040 ","End":"01:54.990","Text":"that leaves just 0 here."},{"Start":"01:54.990 ","End":"01:59.290","Text":"Now, we have 3x minus 2x is just x."},{"Start":"01:59.290 ","End":"02:02.040","Text":"On the right-hand side, x equals 6."},{"Start":"02:02.040 ","End":"02:05.355","Text":"That\u0027s our only suspect candidate,"},{"Start":"02:05.355 ","End":"02:07.605","Text":"we have just check in the original."},{"Start":"02:07.605 ","End":"02:11.880","Text":"Let\u0027s put x equals 6 in this original equation,"},{"Start":"02:11.880 ","End":"02:16.155","Text":"and let\u0027s see if the left-hand side equals the right-hand side."},{"Start":"02:16.155 ","End":"02:18.435","Text":"Let\u0027s write LHS,"},{"Start":"02:18.435 ","End":"02:21.400","Text":"acceptable abbreviation for left-hand side,"},{"Start":"02:21.400 ","End":"02:29.680","Text":"is the square root of 6 plus 3 plus the square root of 6 minus 2."},{"Start":"02:29.680 ","End":"02:35.010","Text":"This equals square root of 9 plus square root of 4,"},{"Start":"02:35.010 ","End":"02:41.185","Text":"this equals 3 plus 2, which equals 5."},{"Start":"02:41.185 ","End":"02:45.335","Text":"I just have to check the right-hand side."},{"Start":"02:45.335 ","End":"02:52.175","Text":"The right-hand side is the square root of 4x plus 1,"},{"Start":"02:52.175 ","End":"02:56.450","Text":"which is 4 times 6 plus 1,"},{"Start":"02:56.450 ","End":"03:00.045","Text":"which is square root of 24 plus 1 is 25,"},{"Start":"03:00.045 ","End":"03:01.275","Text":"which equals 5,"},{"Start":"03:01.275 ","End":"03:03.525","Text":"and yes, same thing."},{"Start":"03:03.525 ","End":"03:09.075","Text":"x equals 6 is the answer to part a."},{"Start":"03:09.075 ","End":"03:11.805","Text":"Let\u0027s get on to part b."},{"Start":"03:11.805 ","End":"03:14.735","Text":"Want to leave the formula here."},{"Start":"03:14.735 ","End":"03:18.020","Text":"Once again, we\u0027re going to square both sides."},{"Start":"03:18.020 ","End":"03:27.245","Text":"That was before this one squared is 2x minus 2 plus twice square root of 2x minus 2,"},{"Start":"03:27.245 ","End":"03:35.140","Text":"square root of 5x minus 4 plus last one squared is 5x minus 4."},{"Start":"03:35.140 ","End":"03:37.160","Text":"This equals right-hand side squared,"},{"Start":"03:37.160 ","End":"03:39.830","Text":"which is just 3x minus 2."},{"Start":"03:39.830 ","End":"03:41.960","Text":"On the left-hand side,"},{"Start":"03:41.960 ","End":"03:47.495","Text":"I leave the twice square root of 2x minus 2,"},{"Start":"03:47.495 ","End":"03:50.180","Text":"square root of 5x minus 4."},{"Start":"03:50.180 ","End":"03:52.970","Text":"The 2 doesn\u0027t bother me when I square it."},{"Start":"03:52.970 ","End":"03:54.380","Text":"On the right-hand side,"},{"Start":"03:54.380 ","End":"03:56.750","Text":"let\u0027s see what we have, x is first."},{"Start":"03:56.750 ","End":"04:02.395","Text":"We have 3x and then minus 5x, minus 2x,"},{"Start":"04:02.395 ","End":"04:06.960","Text":"altogether it\u0027s 3 minus 4x, I make it,"},{"Start":"04:06.960 ","End":"04:15.030","Text":"and numbers, we have a minus 2 and then we have plus 4, plus 2."},{"Start":"04:15.030 ","End":"04:18.550","Text":"I make that plus 4."},{"Start":"04:18.550 ","End":"04:20.840","Text":"I see that both sides are divisible by 2."},{"Start":"04:20.840 ","End":"04:24.260","Text":"Let\u0027s divide by 2 on both sides."},{"Start":"04:24.260 ","End":"04:29.440","Text":"Square root of 2x minus 2,"},{"Start":"04:29.440 ","End":"04:37.385","Text":"square root of 5x minus 4 equals minus 2x plus 2."},{"Start":"04:37.385 ","End":"04:41.340","Text":"Again, square both sides."},{"Start":"04:41.340 ","End":"04:44.945","Text":"I often don\u0027t bother writing as I really should."},{"Start":"04:44.945 ","End":"04:46.490","Text":"Like from here to here,"},{"Start":"04:46.490 ","End":"04:49.699","Text":"we square both sides."},{"Start":"04:49.699 ","End":"04:56.765","Text":"Here we divide by 2, here we square and then we get that 2x minus 2 not squared,"},{"Start":"04:56.765 ","End":"05:03.140","Text":"but it needs brackets and then 5x minus 4 equals this thing squared."},{"Start":"05:03.140 ","End":"05:05.660","Text":"We can just see the formula there."},{"Start":"05:05.660 ","End":"05:08.690","Text":"This is going to be 4x squared."},{"Start":"05:08.690 ","End":"05:11.765","Text":"The minus with a minus is plus and 2 with 2 is 4,"},{"Start":"05:11.765 ","End":"05:15.530","Text":"and then twice this times this is going to"},{"Start":"05:15.530 ","End":"05:16.920","Text":"be"},{"Start":"05:20.240 ","End":"05:28.200","Text":"minus 8x plus 4."},{"Start":"05:28.200 ","End":"05:30.230","Text":"I\u0027ll scroll down a little bit now, don\u0027t need the formula anymore. We have, let\u0027s see if"},{"Start":"05:30.230 ","End":"05:32.456","Text":"I multiply this out 2x"},{"Start":"05:32.456 ","End":"05:38.540","Text":"times 5x is 10x squared minus 10x,"},{"Start":"05:38.540 ","End":"05:45.870","Text":"this with this minus 8x plus 8 equals the same thing."},{"Start":"05:45.870 ","End":"05:49.739","Text":"I will copy it, shouldn\u0027t get lazy."},{"Start":"05:49.739 ","End":"05:51.360","Text":"Bring everything to the left-hand side."},{"Start":"05:51.360 ","End":"05:59.520","Text":"10x squared minus 4x squared is 6x squared minus 10 plus 8,"},{"Start":"05:59.520 ","End":"06:07.295","Text":"so it\u0027s just minus 10x, and then plus 8 minus 4 is plus 4 equals 0."},{"Start":"06:07.295 ","End":"06:09.935","Text":"Everything is divisible by 2."},{"Start":"06:09.935 ","End":"06:13.720","Text":"Again, I\u0027m going to divide both sides by 2 and we"},{"Start":"06:13.720 ","End":"06:22.190","Text":"get 3x squared minus 5x plus 2 equals 0."},{"Start":"06:22.640 ","End":"06:25.170","Text":"Usually, I just give you the answers,"},{"Start":"06:25.170 ","End":"06:27.540","Text":"let\u0027s solve the quadratic this time."},{"Start":"06:27.540 ","End":"06:32.620","Text":"x equals minus b plus or minus the square root of b"},{"Start":"06:32.620 ","End":"06:39.775","Text":"squared minus 4ac over 2a,"},{"Start":"06:39.775 ","End":"06:42.880","Text":"which is, let us see,"},{"Start":"06:42.880 ","End":"06:48.165","Text":"4 times 3 times 2 is 25 minus 24 is 1."},{"Start":"06:48.165 ","End":"06:54.180","Text":"It\u0027s 5 plus or minus 1 over 6,"},{"Start":"06:54.180 ","End":"06:57.180","Text":"which gives us 2 possibilities,"},{"Start":"06:57.180 ","End":"06:59.820","Text":"6 over 6 is 1,"},{"Start":"06:59.820 ","End":"07:02.600","Text":"and 4 over 6 is 2/3."},{"Start":"07:02.600 ","End":"07:08.380","Text":"We have to verify these solutions because we use the squaring operation twice even."},{"Start":"07:08.380 ","End":"07:13.045","Text":"I\u0027m going to take each one separately and plug it into this."},{"Start":"07:13.045 ","End":"07:15.340","Text":"Let\u0027s take x equals 1 first."},{"Start":"07:15.340 ","End":"07:20.350","Text":"I\u0027ll take the x equals 1 case and substitute here and see if it works."},{"Start":"07:20.350 ","End":"07:25.440","Text":"What we have is the square root of twice 1 minus"},{"Start":"07:25.440 ","End":"07:31.095","Text":"2 plus the square root of 5 times 1 minus 4."},{"Start":"07:31.095 ","End":"07:33.559","Text":"Both sides, but I\u0027ll put a question mark,"},{"Start":"07:33.559 ","End":"07:39.500","Text":"is it equal to the square root of 3 times 1 minus 2?"},{"Start":"07:39.500 ","End":"07:41.990","Text":"Well, I say yes,"},{"Start":"07:41.990 ","End":"07:45.335","Text":"because this is the square root of 0."},{"Start":"07:45.335 ","End":"07:47.440","Text":"2 times 1 minus 2 is 0,"},{"Start":"07:47.440 ","End":"07:49.860","Text":"under the square root here I have 1,"},{"Start":"07:49.860 ","End":"07:52.685","Text":"just 5 minus 4, and here I have 1."},{"Start":"07:52.685 ","End":"07:55.320","Text":"Basically, square root of 0 plus square root of 1,"},{"Start":"07:55.320 ","End":"07:56.780","Text":"is it equal to square root of 1?"},{"Start":"07:56.780 ","End":"07:59.200","Text":"Yes, because 0 plus 1 is 1."},{"Start":"07:59.200 ","End":"08:01.055","Text":"I say, \"Yes,"},{"Start":"08:01.055 ","End":"08:05.580","Text":"this is equal, which means that this one has been verified.\""},{"Start":"08:05.580 ","End":"08:09.050","Text":"We have to go for the other one, I\u0027ll do it over here."},{"Start":"08:09.050 ","End":"08:13.430","Text":"We\u0027re going to substitute x equals 2/3 in this equation."},{"Start":"08:13.430 ","End":"08:19.692","Text":"I want to know if the square root of twice 2/3 minus"},{"Start":"08:19.692 ","End":"08:27.925","Text":"2 plus the square root of 5 times 2/3 minus 4,"},{"Start":"08:27.925 ","End":"08:36.575","Text":"is this equal to the square root of 3 times 2/3 minus 2?"},{"Start":"08:36.575 ","End":"08:39.215","Text":"We have some fractions here."},{"Start":"08:39.215 ","End":"08:41.960","Text":"Let\u0027s see. 2 times 2/3 is 4/3,"},{"Start":"08:41.960 ","End":"08:44.974","Text":"which is 1 and 1/3 minus 2, oh, that\u0027s negative."},{"Start":"08:44.974 ","End":"08:48.395","Text":"I didn\u0027t need to have even bothered to write these because I didn\u0027t"},{"Start":"08:48.395 ","End":"08:51.860","Text":"know this one is square root of a negative,"},{"Start":"08:51.860 ","End":"08:56.300","Text":"not possible, so x equals 2/3 has been ruled out."},{"Start":"08:56.300 ","End":"09:00.560","Text":"Only x equals 1 is a solution."},{"Start":"09:00.560 ","End":"09:04.740","Text":"We are done with part b and with the exercise."}],"ID":5035}],"Thumbnail":null,"ID":3873},{"Name":"Equations with Absolute Value","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Absolute Value - Introduction","Duration":"6m 48s","ChapterTopicVideoID":5043,"CourseChapterTopicPlaylistID":3874,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.075","Text":"In this clip, we\u0027ll learn the concept of the absolute value of a number."},{"Start":"00:06.075 ","End":"00:11.475","Text":"I\u0027m going to bring in a picture to help me illustrate the concept."},{"Start":"00:11.475 ","End":"00:13.380","Text":"It\u0027s a picture of the number line."},{"Start":"00:13.380 ","End":"00:15.299","Text":"Of course, it goes on to infinity,"},{"Start":"00:15.299 ","End":"00:18.000","Text":"but I just did it from 10 - minus 10."},{"Start":"00:18.000 ","End":"00:21.255","Text":"The most special number is 0."},{"Start":"00:21.255 ","End":"00:23.625","Text":"This is going to help us with our definition."},{"Start":"00:23.625 ","End":"00:30.210","Text":"I wrote down the definition that the absolute value of a number is its distance from 0."},{"Start":"00:30.210 ","End":"00:32.490","Text":"I\u0027ll illustrate this using the number line."},{"Start":"00:32.490 ","End":"00:33.675","Text":"I\u0027ll give 3 examples,"},{"Start":"00:33.675 ","End":"00:36.210","Text":"I\u0027ll take a positive, a negative, and 0."},{"Start":"00:36.210 ","End":"00:37.890","Text":"For the positive,"},{"Start":"00:37.890 ","End":"00:40.270","Text":"I\u0027ll use the example of 3,"},{"Start":"00:40.270 ","End":"00:44.695","Text":"for the negative, I\u0027ll take the example minus 5,"},{"Start":"00:44.695 ","End":"00:47.709","Text":"and 0 is just 0."},{"Start":"00:47.709 ","End":"00:49.890","Text":"Now, the distance of 3 from 0,"},{"Start":"00:49.890 ","End":"00:53.912","Text":"I\u0027ll just put a little arc here to indicate that this is the distance,"},{"Start":"00:53.912 ","End":"00:55.238","Text":"we just count 1,"},{"Start":"00:55.238 ","End":"00:57.860","Text":"2, 3. Not a problem."},{"Start":"00:57.860 ","End":"01:00.685","Text":"The absolute value of 3 is 3."},{"Start":"01:00.685 ","End":"01:02.740","Text":"Now if I take minus 5,"},{"Start":"01:02.740 ","End":"01:04.670","Text":"the distance from it to 0, again,"},{"Start":"01:04.670 ","End":"01:08.945","Text":"I just put an arc here just to help me see what\u0027s going on."},{"Start":"01:08.945 ","End":"01:12.243","Text":"The distance from here to here is 1,"},{"Start":"01:12.243 ","End":"01:15.095","Text":"2, 3, 4, 5 units."},{"Start":"01:15.095 ","End":"01:18.865","Text":"The absolute value of minus 5 is 5."},{"Start":"01:18.865 ","End":"01:21.050","Text":"The distance from 0 - 0,"},{"Start":"01:21.050 ","End":"01:22.460","Text":"well, it\u0027s just 0."},{"Start":"01:22.460 ","End":"01:24.020","Text":"The distance of something to itself,"},{"Start":"01:24.020 ","End":"01:26.120","Text":"there is no distance, distance is 0."},{"Start":"01:26.120 ","End":"01:28.280","Text":"Now how do I write all this?"},{"Start":"01:28.280 ","End":"01:30.590","Text":"Take the example of 3."},{"Start":"01:30.590 ","End":"01:32.110","Text":"To say the absolute value,"},{"Start":"01:32.110 ","End":"01:35.465","Text":"you put it between 2 vertical bars like this."},{"Start":"01:35.465 ","End":"01:39.065","Text":"That\u0027s the mathematical notation for absolute value of a number."},{"Start":"01:39.065 ","End":"01:42.385","Text":"We say absolute value of 3 is 3."},{"Start":"01:42.385 ","End":"01:44.340","Text":"In the case of minus 5,"},{"Start":"01:44.340 ","End":"01:47.280","Text":"I put the bars around minus 5."},{"Start":"01:47.280 ","End":"01:52.885","Text":"I read this thing as saying the absolute value of minus 5 is 5."},{"Start":"01:52.885 ","End":"01:57.505","Text":"For 0, we decided that the absolute value of 0 is 0."},{"Start":"01:57.505 ","End":"01:59.825","Text":"Once again, to say what it means,"},{"Start":"01:59.825 ","End":"02:04.875","Text":"this means that the distance of 3 from 0 is 3."},{"Start":"02:04.875 ","End":"02:09.600","Text":"The distance of minus 5 from 0 is 5."},{"Start":"02:09.600 ","End":"02:12.720","Text":"The distance of 0 from 0 is just 0."},{"Start":"02:12.720 ","End":"02:14.960","Text":"It\u0027s pronounced absolute value."},{"Start":"02:14.960 ","End":"02:17.240","Text":"The absolute value and it\u0027s in bars."},{"Start":"02:17.240 ","End":"02:18.920","Text":"That\u0027s the introduction."},{"Start":"02:18.920 ","End":"02:22.010","Text":"The same notation is used for variables also."},{"Start":"02:22.010 ","End":"02:23.975","Text":"If I have a variable x,"},{"Start":"02:23.975 ","End":"02:29.810","Text":"the absolute value of x means the distance of x from 0 or from x - 0."},{"Start":"02:29.810 ","End":"02:31.415","Text":"That\u0027s how we write it."},{"Start":"02:31.415 ","End":"02:34.820","Text":"Notice that because distance is never negative,"},{"Start":"02:34.820 ","End":"02:38.075","Text":"this thing is always bigger or = 0."},{"Start":"02:38.075 ","End":"02:42.350","Text":"It\u0027s pretty much always positive with the 1 exception of 0,"},{"Start":"02:42.350 ","End":"02:44.555","Text":"where the distance can be 0."},{"Start":"02:44.555 ","End":"02:48.980","Text":"Now if I asked you what is the absolute value of,"},{"Start":"02:48.980 ","End":"02:51.680","Text":"say, minus 9,"},{"Start":"02:51.680 ","End":"02:55.790","Text":"you would tell me that it\u0027s from minus 9 - 0, it\u0027s 9."},{"Start":"02:55.790 ","End":"02:59.390","Text":"Absolute value of minus 9 is 9."},{"Start":"02:59.390 ","End":"03:03.140","Text":"We already had absolute value of minus 5 is 5."},{"Start":"03:03.140 ","End":"03:06.270","Text":"If I ask you what\u0027s the absolute value of minus 2,"},{"Start":"03:06.270 ","End":"03:09.000","Text":"then it\u0027s just distance of 2,"},{"Start":"03:09.000 ","End":"03:10.415","Text":"it\u0027s 2, and so on."},{"Start":"03:10.415 ","End":"03:13.760","Text":"You get the idea that if we take the absolute value of a negative number,"},{"Start":"03:13.760 ","End":"03:15.170","Text":"we just throw out the negative."},{"Start":"03:15.170 ","End":"03:16.880","Text":"Of course, for a positive,"},{"Start":"03:16.880 ","End":"03:19.325","Text":"like absolute value of 3 is 3."},{"Start":"03:19.325 ","End":"03:21.380","Text":"Distance of 3 from 0 is 3."},{"Start":"03:21.380 ","End":"03:23.825","Text":"If I asked you the absolute value of 6,"},{"Start":"03:23.825 ","End":"03:26.105","Text":"how far is it from 0 - 6?"},{"Start":"03:26.105 ","End":"03:27.560","Text":"It\u0027s just 6."},{"Start":"03:27.560 ","End":"03:30.185","Text":"How far is it from,"},{"Start":"03:30.185 ","End":"03:32.900","Text":"I don\u0027t know, from say, 1 - 0?"},{"Start":"03:32.900 ","End":"03:34.960","Text":"Absolute value of 1."},{"Start":"03:34.960 ","End":"03:38.075","Text":"1 unit, it\u0027s 1, and so on."},{"Start":"03:38.075 ","End":"03:40.820","Text":"For negatives, we throw out the negative,"},{"Start":"03:40.820 ","End":"03:42.583","Text":"for positives, it stays what it is,"},{"Start":"03:42.583 ","End":"03:44.960","Text":"and the absolute value of 0 is always 0."},{"Start":"03:44.960 ","End":"03:47.045","Text":"That\u0027s the only one that\u0027s not positive,"},{"Start":"03:47.045 ","End":"03:49.130","Text":"but it\u0027s never negative."},{"Start":"03:49.130 ","End":"03:53.735","Text":"In fact, this gives us a definition of absolute value of x."},{"Start":"03:53.735 ","End":"04:01.130","Text":"Without pictures, there\u0027s a way of defining it where we say the absolute value of x."},{"Start":"04:01.130 ","End":"04:05.750","Text":"There\u0027s a mathematical way of writing a definition with conditions."},{"Start":"04:05.750 ","End":"04:09.560","Text":"We say, if it\u0027s=x itself,"},{"Start":"04:09.560 ","End":"04:10.760","Text":"for the positive numbers,"},{"Start":"04:10.760 ","End":"04:12.650","Text":"the absolute value of x is x."},{"Start":"04:12.650 ","End":"04:14.930","Text":"Absolute value of 6 is 6, 3 is 3."},{"Start":"04:14.930 ","End":"04:16.625","Text":"We say it\u0027s x itself."},{"Start":"04:16.625 ","End":"04:22.250","Text":"Then usually a semicolon for the condition on the condition that x is positive,"},{"Start":"04:22.250 ","End":"04:24.665","Text":"which I write as x bigger than 0."},{"Start":"04:24.665 ","End":"04:29.075","Text":"To express this set of truths for negatives,"},{"Start":"04:29.075 ","End":"04:31.145","Text":"if x is negative,"},{"Start":"04:31.145 ","End":"04:33.800","Text":"the absolute value of x is, well,"},{"Start":"04:33.800 ","End":"04:36.710","Text":"how do I indicate that I throw out the minus?"},{"Start":"04:36.710 ","End":"04:39.515","Text":"The answer is just this, minus x."},{"Start":"04:39.515 ","End":"04:43.930","Text":"Because look, if x is minus 5,"},{"Start":"04:43.930 ","End":"04:46.450","Text":"minus x is plus 5."},{"Start":"04:46.450 ","End":"04:48.065","Text":"It works for negatives."},{"Start":"04:48.065 ","End":"04:49.224","Text":"If I use this definition,"},{"Start":"04:49.224 ","End":"04:52.895","Text":"I get absolute value of minus 5 is minus,"},{"Start":"04:52.895 ","End":"04:57.490","Text":"minus 5, that we usually write the brackets here, which is 5."},{"Start":"04:57.490 ","End":"04:59.368","Text":"It works for negatives,"},{"Start":"04:59.368 ","End":"05:00.980","Text":"well, works for negatives."},{"Start":"05:00.980 ","End":"05:04.710","Text":"For positives, absolute value of 3 says,"},{"Start":"05:04.710 ","End":"05:06.270","Text":"because 3 is bigger than 0,"},{"Start":"05:06.270 ","End":"05:07.675","Text":"it\u0027s just 3 itself."},{"Start":"05:07.675 ","End":"05:10.225","Text":"Absolute value of 0 is 0, less interesting."},{"Start":"05:10.225 ","End":"05:12.620","Text":"There\u0027s a third part of the definition."},{"Start":"05:12.620 ","End":"05:14.690","Text":"First, let me make this longer."},{"Start":"05:14.690 ","End":"05:16.465","Text":"Now here\u0027s room for the third part."},{"Start":"05:16.465 ","End":"05:19.010","Text":"If x is 0,"},{"Start":"05:19.010 ","End":"05:21.680","Text":"then the absolute value is 0."},{"Start":"05:21.680 ","End":"05:25.460","Text":"This formula tells me what to do in each case."},{"Start":"05:25.460 ","End":"05:26.810","Text":"It covers all the cases,"},{"Start":"05:26.810 ","End":"05:28.970","Text":"positive, negative, and 0."},{"Start":"05:28.970 ","End":"05:31.535","Text":"Positive, leave it as it is,"},{"Start":"05:31.535 ","End":"05:33.605","Text":"for negative, take minus x,"},{"Start":"05:33.605 ","End":"05:36.920","Text":"which actually makes the negative into a positive,"},{"Start":"05:36.920 ","End":"05:39.500","Text":"and for 0, it\u0027s just 0."},{"Start":"05:39.500 ","End":"05:44.540","Text":"This is a more formal mathematical type definition,"},{"Start":"05:44.540 ","End":"05:48.440","Text":"and the original one is just the distance from 0."},{"Start":"05:48.440 ","End":"05:52.530","Text":"Let\u0027s take a break here and continue in the next clip."}],"ID":5036},{"Watched":false,"Name":"Absolute Value - Properties","Duration":"6m 44s","ChapterTopicVideoID":5044,"CourseChapterTopicPlaylistID":3874,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.400","Text":"Continuing with absolute value,"},{"Start":"00:02.400 ","End":"00:04.230","Text":"let me erase the board,"},{"Start":"00:04.230 ","End":"00:08.100","Text":"but I\u0027ll keep this and this time we\u0027re going to talk about"},{"Start":"00:08.100 ","End":"00:12.194","Text":"the properties of the absolute value."},{"Start":"00:12.194 ","End":"00:13.860","Text":"There\u0027s quite a few of them,"},{"Start":"00:13.860 ","End":"00:15.915","Text":"and they\u0027re all fairly straightforward."},{"Start":"00:15.915 ","End":"00:22.560","Text":"The first property is if I take any number not equal to zero,"},{"Start":"00:22.560 ","End":"00:26.025","Text":"then its absolute value is always positive."},{"Start":"00:26.025 ","End":"00:28.110","Text":"There\u0027s only 1 exception to being positive,"},{"Start":"00:28.110 ","End":"00:29.730","Text":"and that\u0027s when A is 0,"},{"Start":"00:29.730 ","End":"00:31.575","Text":"like in this last row here."},{"Start":"00:31.575 ","End":"00:33.000","Text":"But if A is not 0,"},{"Start":"00:33.000 ","End":"00:35.040","Text":"the absolute value is a positive number."},{"Start":"00:35.040 ","End":"00:37.435","Text":"That\u0027s the first rule and straightforward."},{"Start":"00:37.435 ","End":"00:41.824","Text":"The second rule is that the absolute value of minus"},{"Start":"00:41.824 ","End":"00:47.630","Text":"a is the same as the absolute value of a and this holds for all a\u0027s whether positive,"},{"Start":"00:47.630 ","End":"00:49.160","Text":"negative, or zero."},{"Start":"00:49.160 ","End":"00:52.370","Text":"For example, if I take A is 3,"},{"Start":"00:52.370 ","End":"00:57.335","Text":"it says absolute value of minus 3 equals absolute value of 3 and this is true,"},{"Start":"00:57.335 ","End":"01:01.030","Text":"this is equal to 3 and this is equal to 3."},{"Start":"01:01.030 ","End":"01:04.370","Text":"If we use the original definition of the distance from 0,"},{"Start":"01:04.370 ","End":"01:06.830","Text":"both of these are located 3 units from 0,"},{"Start":"01:06.830 ","End":"01:09.560","Text":"or we could use this definition and in any event,"},{"Start":"01:09.560 ","End":"01:11.750","Text":"both of these will come out to be 3."},{"Start":"01:11.750 ","End":"01:14.300","Text":"If I take A as being negative,"},{"Start":"01:14.300 ","End":"01:15.740","Text":"say minus 5,"},{"Start":"01:15.740 ","End":"01:17.510","Text":"the absolute value of minus,"},{"Start":"01:17.510 ","End":"01:22.550","Text":"minus 5 is the same as the absolute value of minus 5."},{"Start":"01:22.550 ","End":"01:26.500","Text":"This is because this is the same as the absolute value of 5."},{"Start":"01:26.500 ","End":"01:31.790","Text":"Because minus, minus 5 is 5 and this is just like here on the reverse order,"},{"Start":"01:31.790 ","End":"01:34.580","Text":"a positive on the left and the negative on"},{"Start":"01:34.580 ","End":"01:39.200","Text":"the right and for zero it\u0027s certainly true because minus zero and zero are the same."},{"Start":"01:39.200 ","End":"01:41.480","Text":"This rule, seemingly trivial,"},{"Start":"01:41.480 ","End":"01:45.200","Text":"if you take a minus in front of a number or reverse its sign,"},{"Start":"01:45.200 ","End":"01:47.840","Text":"you\u0027re not going to change its distance from zero is just"},{"Start":"01:47.840 ","End":"01:51.185","Text":"going to move it back and forth from 1 side of the zero to the other."},{"Start":"01:51.185 ","End":"01:55.820","Text":"Rule number 3 talks about the absolute value of a product."},{"Start":"01:55.820 ","End":"01:59.945","Text":"If I multiply 2 numbers and take the absolute value,"},{"Start":"01:59.945 ","End":"02:05.809","Text":"it\u0027s the same as just taking the absolute value of each number and multiplying those."},{"Start":"02:05.809 ","End":"02:11.930","Text":"Basically what this is saying is if we multiply numbers and then throw out any minuses,"},{"Start":"02:11.930 ","End":"02:15.200","Text":"or if we just threw out the minuses first and then multiply,"},{"Start":"02:15.200 ","End":"02:17.015","Text":"it doesn\u0027t make any difference."},{"Start":"02:17.015 ","End":"02:21.625","Text":"If A is plus or minus 3 and b is plus or minus 5,"},{"Start":"02:21.625 ","End":"02:22.940","Text":"whichever way you do it,"},{"Start":"02:22.940 ","End":"02:26.600","Text":"you\u0027re going to get 3 times 5 is 15,"},{"Start":"02:26.600 ","End":"02:29.030","Text":"seemingly trivial, but it\u0027s a rule."},{"Start":"02:29.030 ","End":"02:31.415","Text":"Maybe I should call this 3 a,"},{"Start":"02:31.415 ","End":"02:35.450","Text":"because also same thing works with division."},{"Start":"02:35.450 ","End":"02:39.560","Text":"The absolute value of A over b is the same as"},{"Start":"02:39.560 ","End":"02:43.520","Text":"the absolute value of A over the absolute value of b."},{"Start":"02:43.520 ","End":"02:45.680","Text":"In other words, if I have a fraction and if I throw out"},{"Start":"02:45.680 ","End":"02:48.343","Text":"any minuses from the numerator or denominator,"},{"Start":"02:48.343 ","End":"02:52.475","Text":"it\u0027s the same as if I first divided and then throw out any minuses."},{"Start":"02:52.475 ","End":"02:58.790","Text":"Of course, this assumes that b is not equal to 0 because we can\u0027t divide by 0."},{"Start":"02:58.790 ","End":"03:03.005","Text":"The next rule is something that is going to be useful"},{"Start":"03:03.005 ","End":"03:07.470","Text":"for the section on equations with absolute value."},{"Start":"03:07.470 ","End":"03:10.520","Text":"Reminds us that it\u0027s for equations I\u0027ll use the letter x."},{"Start":"03:10.520 ","End":"03:11.960","Text":"Suppose I have an equation,"},{"Start":"03:11.960 ","End":"03:16.990","Text":"absolute value of x is equal to say, b."},{"Start":"03:16.990 ","End":"03:21.140","Text":"Then there are 2 solutions to this equation."},{"Start":"03:21.140 ","End":"03:28.790","Text":"Either x equals b or x equals minus b."},{"Start":"03:28.790 ","End":"03:31.745","Text":"But there\u0027s a very important restriction here,"},{"Start":"03:31.745 ","End":"03:34.610","Text":"b has to be bigger than 0."},{"Start":"03:34.610 ","End":"03:36.830","Text":"Actually, it could be bigger or equal to zero,"},{"Start":"03:36.830 ","End":"03:38.930","Text":"but it\u0027s less interesting when b is zero."},{"Start":"03:38.930 ","End":"03:41.645","Text":"I\u0027m going to just really think of it as positive."},{"Start":"03:41.645 ","End":"03:44.480","Text":"What it says is if I give you an equation,"},{"Start":"03:44.480 ","End":"03:51.085","Text":"suppose I asked you to solve absolute value of x is equal to 4."},{"Start":"03:51.085 ","End":"03:52.380","Text":"Then this what it says,"},{"Start":"03:52.380 ","End":"03:54.165","Text":"it\u0027s only 2 possibilities,"},{"Start":"03:54.165 ","End":"03:55.605","Text":"x could be 4,"},{"Start":"03:55.605 ","End":"03:58.815","Text":"or x could be minus 4."},{"Start":"03:58.815 ","End":"04:02.290","Text":"Those are the only 2 numbers whose absolute value is 4."},{"Start":"04:02.290 ","End":"04:05.465","Text":"Because which numbers have a distance of 4 from 0?"},{"Start":"04:05.465 ","End":"04:08.795","Text":"Either go 1 way or they go the other way on the number line."},{"Start":"04:08.795 ","End":"04:10.630","Text":"That\u0027s really all this is saying,"},{"Start":"04:10.630 ","End":"04:15.015","Text":"so absolute value, x equals 7."},{"Start":"04:15.015 ","End":"04:17.335","Text":"You say x is plus or minus 7."},{"Start":"04:17.335 ","End":"04:21.575","Text":"Just by the way, in case you\u0027re wondering what happens if b is less than zero,"},{"Start":"04:21.575 ","End":"04:23.165","Text":"then there\u0027s no solution."},{"Start":"04:23.165 ","End":"04:28.040","Text":"I can\u0027t have absolute value of x equals minus 3."},{"Start":"04:28.040 ","End":"04:30.500","Text":"We can\u0027t have absolute value of x being negative."},{"Start":"04:30.500 ","End":"04:32.420","Text":"That violates the First Law."},{"Start":"04:32.420 ","End":"04:34.250","Text":"The last rule that we need,"},{"Start":"04:34.250 ","End":"04:35.660","Text":"because there\u0027s plenty more rules,"},{"Start":"04:35.660 ","End":"04:39.580","Text":"but we don\u0027t need them is strange looking,"},{"Start":"04:39.580 ","End":"04:48.380","Text":"says that the square root of A squared is equal to the absolute value of A for all A."},{"Start":"04:48.380 ","End":"04:50.690","Text":"Now let\u0027s see why this makes sense because you"},{"Start":"04:50.690 ","End":"04:53.510","Text":"wondering if you square something and then take the square root,"},{"Start":"04:53.510 ","End":"04:55.640","Text":"don\u0027t I get back to the number itself?"},{"Start":"04:55.640 ","End":"04:57.905","Text":"Well, yes and no."},{"Start":"04:57.905 ","End":"05:00.470","Text":"Let me give an example with a positive,"},{"Start":"05:00.470 ","End":"05:02.989","Text":"if I take the square root of 3 squared,"},{"Start":"05:02.989 ","End":"05:04.480","Text":"so A is 3,"},{"Start":"05:04.480 ","End":"05:08.725","Text":"then it\u0027s the square root of 9 and I\u0027m back to 3."},{"Start":"05:08.725 ","End":"05:12.725","Text":"But if I take the square root of,"},{"Start":"05:12.725 ","End":"05:14.285","Text":"let\u0027s say A was negative,"},{"Start":"05:14.285 ","End":"05:16.880","Text":"say minus 5 squared."},{"Start":"05:16.880 ","End":"05:21.875","Text":"Then it\u0027s the square root of minus 5 times minus 5 is 25."},{"Start":"05:21.875 ","End":"05:25.545","Text":"The square root of 25 is 5."},{"Start":"05:25.545 ","End":"05:27.550","Text":"It\u0027s not minus 5."},{"Start":"05:27.550 ","End":"05:29.375","Text":"We didn\u0027t get back to where we started from."},{"Start":"05:29.375 ","End":"05:32.900","Text":"This is actually the absolute value of minus 5."},{"Start":"05:32.900 ","End":"05:38.690","Text":"For positives, you could drop the bars with the negatives we put a minus in,"},{"Start":"05:38.690 ","End":"05:43.010","Text":"but in either case, it\u0027s the absolute value and of course the zero, it also works."},{"Start":"05:43.010 ","End":"05:45.110","Text":"That\u0027s square root of zero is zero,"},{"Start":"05:45.110 ","End":"05:46.760","Text":"which is absolute value of zero."},{"Start":"05:46.760 ","End":"05:49.280","Text":"These are the rules that we\u0027re going to need and we\u0027re going to use them,"},{"Start":"05:49.280 ","End":"05:54.620","Text":"especially in the following section on equations with absolute value,"},{"Start":"05:54.620 ","End":"05:57.450","Text":"so for now, we\u0027re d1."}],"ID":5037},{"Watched":false,"Name":"Equations with Absolute Value","Duration":"17m 40s","ChapterTopicVideoID":5036,"CourseChapterTopicPlaylistID":3874,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.520","Text":"Continuing with the absolute value,"},{"Start":"00:02.520 ","End":"00:07.409","Text":"this clip will be about equations with absolute value,"},{"Start":"00:07.409 ","End":"00:10.185","Text":"and I\u0027ll just erase what I don\u0027t need."},{"Start":"00:10.185 ","End":"00:12.510","Text":"Just left the stuff I might need,"},{"Start":"00:12.510 ","End":"00:15.300","Text":"let\u0027s start with the first example of an equation."},{"Start":"00:15.300 ","End":"00:22.890","Text":"Absolute value of x+2=8."},{"Start":"00:22.890 ","End":"00:24.930","Text":"Now, it\u0027s similar to this form,"},{"Start":"00:24.930 ","End":"00:27.795","Text":"the absolute value of something equals something."},{"Start":"00:27.795 ","End":"00:31.649","Text":"What we say is that we have two possibilities,"},{"Start":"00:31.649 ","End":"00:36.870","Text":"x+2 could either equal 8 or -8 using this."},{"Start":"00:36.870 ","End":"00:38.850","Text":"I separate it into 2 equations,"},{"Start":"00:38.850 ","End":"00:42.130","Text":"x+2=8 and I\u0027ll solve this,"},{"Start":"00:42.130 ","End":"00:46.145","Text":"and then I\u0027ll solve x+2=-8 and see what we get."},{"Start":"00:46.145 ","End":"00:47.990","Text":"Well, in this case it\u0027s pretty clear,"},{"Start":"00:47.990 ","End":"00:57.060","Text":"we get x=6 and in this case we just get x=-10 and so these are the two solutions,"},{"Start":"00:57.060 ","End":"00:59.960","Text":"6 and -10 to the equation."},{"Start":"00:59.960 ","End":"01:01.850","Text":"Now let me take a variation,"},{"Start":"01:01.850 ","End":"01:07.895","Text":"how about absolute value of x+2=-3?"},{"Start":"01:07.895 ","End":"01:10.940","Text":"Don\u0027t go ahead and blindly apply this formula,"},{"Start":"01:10.940 ","End":"01:12.770","Text":"remember that when I wrote this formula,"},{"Start":"01:12.770 ","End":"01:16.160","Text":"I mentioned that if the b happens to be negative,"},{"Start":"01:16.160 ","End":"01:17.945","Text":"there is no solution."},{"Start":"01:17.945 ","End":"01:20.750","Text":"Absolute value of anything cannot be negative,"},{"Start":"01:20.750 ","End":"01:23.645","Text":"so this one has no solution and you\u0027re done,"},{"Start":"01:23.645 ","End":"01:25.719","Text":"so that\u0027s two equations solved."},{"Start":"01:25.719 ","End":"01:27.545","Text":"I mean, it\u0027s legitimate, no solution."},{"Start":"01:27.545 ","End":"01:30.250","Text":"Now let\u0027s take one slightly more involved,"},{"Start":"01:30.250 ","End":"01:32.490","Text":"let me just label our equations,"},{"Start":"01:32.490 ","End":"01:34.665","Text":"let\u0027s say this is number a,"},{"Start":"01:34.665 ","End":"01:43.935","Text":"this was number b and for number c we\u0027ll take absolute value of x^2-4=5."},{"Start":"01:43.935 ","End":"01:47.460","Text":"We\u0027re still going to use this rule Number 4 that we"},{"Start":"01:47.460 ","End":"01:50.865","Text":"had and say that we have two possibilities,"},{"Start":"01:50.865 ","End":"01:54.150","Text":"either x^2-4 is 5,"},{"Start":"01:54.150 ","End":"01:59.835","Text":"or x^2-4 is -5,"},{"Start":"01:59.835 ","End":"02:01.515","Text":"time to get rid of this."},{"Start":"02:01.515 ","End":"02:02.940","Text":"I start with this one,"},{"Start":"02:02.940 ","End":"02:05.730","Text":"bring the 4 over to the other side,"},{"Start":"02:05.730 ","End":"02:09.370","Text":"x^2=9 and when x^2=9,"},{"Start":"02:09.370 ","End":"02:12.830","Text":"we know that x is plus or minus the square root of 9,"},{"Start":"02:12.830 ","End":"02:17.350","Text":"x is plus or -3 and perhaps I\u0027d better write them separately,"},{"Start":"02:17.350 ","End":"02:20.190","Text":"I write that as 3 or -3,"},{"Start":"02:20.190 ","End":"02:21.314","Text":"so that\u0027s two solutions."},{"Start":"02:21.314 ","End":"02:22.470","Text":"But we might get some more,"},{"Start":"02:22.470 ","End":"02:27.450","Text":"let\u0027s look at the other possibility x^2-4=-5,"},{"Start":"02:27.450 ","End":"02:32.685","Text":"bring the minus 4 over it becomes plus 4, so x^2=-1."},{"Start":"02:32.685 ","End":"02:34.680","Text":"A square can never be negative,"},{"Start":"02:34.680 ","End":"02:37.030","Text":"so from here we have no solution,"},{"Start":"02:37.030 ","End":"02:39.170","Text":"I don\u0027t mean that we have no solution in general,"},{"Start":"02:39.170 ","End":"02:40.370","Text":"we\u0027ve already found two of them,"},{"Start":"02:40.370 ","End":"02:43.625","Text":"but this branch, there\u0027s two branches, this is one, this is the other."},{"Start":"02:43.625 ","End":"02:45.160","Text":"From here we get no solutions,"},{"Start":"02:45.160 ","End":"02:51.750","Text":"so what we\u0027re left with is the original 3 and -3."},{"Start":"02:51.750 ","End":"02:53.910","Text":"Next example d, well,"},{"Start":"02:53.910 ","End":"02:55.620","Text":"I\u0027m going to erase this first,"},{"Start":"02:55.620 ","End":"03:00.395","Text":"example d will be 3 times absolute value of"},{"Start":"03:00.395 ","End":"03:09.230","Text":"2x+1 minus absolute value of 2x-1=2."},{"Start":"03:09.230 ","End":"03:10.850","Text":"Before I solve it, let me just show you"},{"Start":"03:10.850 ","End":"03:16.375","Text":"a simplified form of this alternate definition of absolute value of x."},{"Start":"03:16.375 ","End":"03:21.715","Text":"We said that the absolute value is the thing itself if it\u0027s positive or it\u0027s negative,"},{"Start":"03:21.715 ","End":"03:25.075","Text":"minus it if it\u0027s negative and for zero at zero now,"},{"Start":"03:25.075 ","End":"03:29.440","Text":"zero actually works with both of them because if x is 0,"},{"Start":"03:29.440 ","End":"03:33.220","Text":"the absolute value of zero is both zero and minus zero."},{"Start":"03:33.220 ","End":"03:36.130","Text":"I can throw in this case with either one of these,"},{"Start":"03:36.130 ","End":"03:39.910","Text":"I\u0027ll throw it in with the top one and write that"},{"Start":"03:39.910 ","End":"03:44.410","Text":"this is true if x is bigger or equal to 0 and now I don\u0027t need this line,"},{"Start":"03:44.410 ","End":"03:46.795","Text":"so I can just erase it."},{"Start":"03:46.795 ","End":"03:51.745","Text":"I\u0027ll just tidy up a bit and I\u0027ll just put two cases and this over here,"},{"Start":"03:51.745 ","End":"03:55.065","Text":"that\u0027s better, I\u0027m going to need this in this exercise."},{"Start":"03:55.065 ","End":"03:57.860","Text":"Now this exercise is quite involved and"},{"Start":"03:57.860 ","End":"04:00.710","Text":"it\u0027s fairly typical and if we follow this then most of"},{"Start":"04:00.710 ","End":"04:03.470","Text":"it\u0027ll easily follow the exercises"},{"Start":"04:03.470 ","End":"04:07.009","Text":"after this tutorial and there\u0027ll be plenty of exercises."},{"Start":"04:07.009 ","End":"04:09.215","Text":"Let\u0027s see one solved in detail."},{"Start":"04:09.215 ","End":"04:13.040","Text":"The idea when you have two absolute values is to write"},{"Start":"04:13.040 ","End":"04:17.075","Text":"each absolute value first in terms of this kind of definition,"},{"Start":"04:17.075 ","End":"04:19.760","Text":"equal a piecewise defined which says in"},{"Start":"04:19.760 ","End":"04:23.600","Text":"one place it\u0027s defined like this and in another place it\u0027s defined like that."},{"Start":"04:23.600 ","End":"04:28.845","Text":"Absolute value of 2x+1,"},{"Start":"04:28.845 ","End":"04:32.015","Text":"make these bold or it looks like a 1,"},{"Start":"04:32.015 ","End":"04:35.935","Text":"this is equal to either 2x+1,"},{"Start":"04:35.935 ","End":"04:44.565","Text":"provided that 2x+1 is bigger or equal to 0 as here and it\u0027s equal to -(2x+1),"},{"Start":"04:44.565 ","End":"04:47.615","Text":"if it happens to be negative."},{"Start":"04:47.615 ","End":"04:49.615","Text":"I\u0027m going to rewrite this,"},{"Start":"04:49.615 ","End":"04:52.785","Text":"this part will be the same 2x+1,"},{"Start":"04:52.785 ","End":"04:55.870","Text":"we\u0027ll expand the brackets -2x-1."},{"Start":"04:55.870 ","End":"04:57.590","Text":"I want to rewrite the inequality."},{"Start":"04:57.590 ","End":"04:59.165","Text":"So I\u0027ll do the side,"},{"Start":"04:59.165 ","End":"05:02.195","Text":"2x+1 bigger or equal to 0,"},{"Start":"05:02.195 ","End":"05:04.535","Text":"I want to know what it is in terms of x,"},{"Start":"05:04.535 ","End":"05:11.100","Text":"so I can subtract 1 from both sides and then we get 2x bigger or equal to minus 1,"},{"Start":"05:11.100 ","End":"05:14.600","Text":"you are allowed to add or subtract the numbers from both sides of an inequality."},{"Start":"05:14.600 ","End":"05:18.325","Text":"You\u0027re also allowed to divide by a positive number,"},{"Start":"05:18.325 ","End":"05:19.759","Text":"and 2 is positive,"},{"Start":"05:19.759 ","End":"05:21.380","Text":"so I divide by 2,"},{"Start":"05:21.380 ","End":"05:24.110","Text":"I get x bigger or equal to -1/2,"},{"Start":"05:24.110 ","End":"05:25.975","Text":"which is minus 1/2."},{"Start":"05:25.975 ","End":"05:29.960","Text":"I\u0027m just rewriting the condition as a condition on x,"},{"Start":"05:29.960 ","End":"05:33.830","Text":"that x should be bigger or equal to minus 1/2 and"},{"Start":"05:33.830 ","End":"05:37.990","Text":"the other one comes out to be x less than minus 1/2,"},{"Start":"05:37.990 ","End":"05:40.190","Text":"that\u0027s for this absolute value."},{"Start":"05:40.190 ","End":"05:42.095","Text":"Now let\u0027s do the other absolute value,"},{"Start":"05:42.095 ","End":"05:46.085","Text":"absolute value of 2x-1, same idea."},{"Start":"05:46.085 ","End":"05:53.010","Text":"It\u0027s 2x-1, provided that 2x-1 is bigger or equal to 0 as here and"},{"Start":"05:53.010 ","End":"06:00.005","Text":"in the case where 2x-1 is negative, then it\u0027s -(2x-1)."},{"Start":"06:00.005 ","End":"06:04.025","Text":"Once again, I rewrite it the same 2x-1,"},{"Start":"06:04.025 ","End":"06:09.170","Text":"this, I\u0027ll just remove the brackets and write it as -2x+1."},{"Start":"06:09.170 ","End":"06:12.680","Text":"Now, this inequality, similar to here,"},{"Start":"06:12.680 ","End":"06:14.240","Text":"I bring the 1 over to the other side,"},{"Start":"06:14.240 ","End":"06:15.410","Text":"divide by 2,"},{"Start":"06:15.410 ","End":"06:19.970","Text":"I get x bigger or equal to 1/2 won\u0027t split it out this"},{"Start":"06:19.970 ","End":"06:25.025","Text":"time and this one gives us that x is less than 1/2."},{"Start":"06:25.025 ","End":"06:29.125","Text":"Now, what I do next is I look and see,"},{"Start":"06:29.125 ","End":"06:31.850","Text":"I don\u0027t even need this anymore, so there are two,"},{"Start":"06:31.850 ","End":"06:35.405","Text":"let\u0027s call them interesting points for this equation,"},{"Start":"06:35.405 ","End":"06:40.280","Text":"one of them is minus 1/2 and also x equals plus 1/2,"},{"Start":"06:40.280 ","End":"06:41.450","Text":"so what we do is we put"},{"Start":"06:41.450 ","End":"06:45.840","Text":"these two special values on the number line and I have a picture,"},{"Start":"06:45.840 ","End":"06:47.810","Text":"here is this diagram."},{"Start":"06:47.810 ","End":"06:49.670","Text":"This is the number line,"},{"Start":"06:49.670 ","End":"06:55.250","Text":"here are the two special values that we found minus 1/2 and"},{"Start":"06:55.250 ","End":"07:01.650","Text":"1/2 and the green represents the second one,"},{"Start":"07:01.650 ","End":"07:09.755","Text":"the solids dot means that the 1/2 is on this bit and on the less than 1/2,"},{"Start":"07:09.755 ","End":"07:13.595","Text":"not including the 1/2, not very important."},{"Start":"07:13.595 ","End":"07:17.945","Text":"Here the same thing for minus 1/2 that\u0027s the top equation is the red one,"},{"Start":"07:17.945 ","End":"07:21.770","Text":"and it means that this part is x bigger or equal to minus"},{"Start":"07:21.770 ","End":"07:26.050","Text":"1/2 and this part for x less than minus 1/2."},{"Start":"07:26.050 ","End":"07:31.540","Text":"There are actually three pieces to the graph."},{"Start":"07:31.540 ","End":"07:34.705","Text":"There\u0027s the first bit up to minus 1/2,"},{"Start":"07:34.705 ","End":"07:36.370","Text":"then there\u0027s the bit in-between the two,"},{"Start":"07:36.370 ","End":"07:38.255","Text":"and then the bit from here onwards."},{"Start":"07:38.255 ","End":"07:40.190","Text":"In each of the three bits,"},{"Start":"07:40.190 ","End":"07:44.880","Text":"we\u0027ll take either the top row or the bottom row from the definition."},{"Start":"07:44.880 ","End":"07:46.925","Text":"I\u0027ll show you what I mean."},{"Start":"07:46.925 ","End":"07:49.480","Text":"Let\u0027s just get a bit more space here."},{"Start":"07:49.480 ","End":"07:51.335","Text":"What I\u0027m saying is,"},{"Start":"07:51.335 ","End":"07:52.870","Text":"going from left to right,"},{"Start":"07:52.870 ","End":"07:57.320","Text":"the first bit is x less than minus 1/2."},{"Start":"07:57.320 ","End":"08:00.025","Text":"When x is less than minus 1/2,"},{"Start":"08:00.025 ","End":"08:05.080","Text":"then we are in the bottom part of this definition,"},{"Start":"08:05.080 ","End":"08:09.095","Text":"so we know that this absolute value is what\u0027s written here and not here."},{"Start":"08:09.095 ","End":"08:11.450","Text":"Now, if x is less than minus 1/2,"},{"Start":"08:11.450 ","End":"08:15.310","Text":"algebraically, we can see that it\u0027s also less than 1/2,"},{"Start":"08:15.310 ","End":"08:16.750","Text":"or from the picture,"},{"Start":"08:16.750 ","End":"08:22.075","Text":"this red bit on the left is included in this green bit on the left."},{"Start":"08:22.075 ","End":"08:26.360","Text":"In both cases, we\u0027re going to be reading off the bottom,"},{"Start":"08:26.360 ","End":"08:33.760","Text":"and so what we get for the equation is that we get 3."},{"Start":"08:33.760 ","End":"08:36.180","Text":"Now, the absolute value of 2x+1,"},{"Start":"08:36.180 ","End":"08:39.365","Text":"we said we\u0027re reading it off the bottom row,"},{"Start":"08:39.365 ","End":"08:44.530","Text":"so that\u0027s -2x-1 minus,"},{"Start":"08:44.530 ","End":"08:46.285","Text":"and then this bit,"},{"Start":"08:46.285 ","End":"08:48.300","Text":"absolute value of 2x-1,"},{"Start":"08:48.300 ","End":"08:52.390","Text":"is also read off the bottom row,"},{"Start":"08:52.390 ","End":"08:56.020","Text":"it\u0027s this bit here, -2x+1."},{"Start":"08:56.020 ","End":"08:58.885","Text":"This is what I get on the left,"},{"Start":"08:58.885 ","End":"09:01.910","Text":"equals 2. You know what?"},{"Start":"09:01.910 ","End":"09:05.060","Text":"I\u0027ll write them all out and do the simplifications later because I"},{"Start":"09:05.060 ","End":"09:09.430","Text":"don\u0027t want to lose momentum on this bit of breaking up the line into three bits."},{"Start":"09:09.430 ","End":"09:12.390","Text":"Now, the middle bit begins when"},{"Start":"09:12.390 ","End":"09:21.065","Text":"x=-1/2 and continues up to 1/2 but not including the 1/2,"},{"Start":"09:21.065 ","End":"09:23.240","Text":"because here, we\u0027re not including it."},{"Start":"09:23.240 ","End":"09:25.520","Text":"When this is true,"},{"Start":"09:25.520 ","End":"09:28.020","Text":"then we\u0027re bigger or equal to minus 1/2,"},{"Start":"09:28.020 ","End":"09:31.040","Text":"so this absolute value is coming off the top row,"},{"Start":"09:31.040 ","End":"09:32.735","Text":"but we\u0027re still less than 1/2,"},{"Start":"09:32.735 ","End":"09:35.160","Text":"so this one is coming from the bottom row."},{"Start":"09:35.160 ","End":"09:38.045","Text":"What we get is 3 times,"},{"Start":"09:38.045 ","End":"09:42.245","Text":"from the top row, 2x+1 minus,"},{"Start":"09:42.245 ","End":"09:43.505","Text":"now, the other one,"},{"Start":"09:43.505 ","End":"09:47.155","Text":"we\u0027re still reading off the bottom row because we\u0027re still less than 1/2,"},{"Start":"09:47.155 ","End":"09:49.740","Text":"so that\u0027s minus 2x+1."},{"Start":"09:49.740 ","End":"09:52.680","Text":"That\u0027s the same as the row above,"},{"Start":"09:52.680 ","End":"09:54.954","Text":"and that\u0027s equal to 2."},{"Start":"09:54.954 ","End":"10:00.315","Text":"The third case is when x is bigger or equal to 1/2,"},{"Start":"10:00.315 ","End":"10:03.780","Text":"wherein this top green bit were"},{"Start":"10:03.780 ","End":"10:07.405","Text":"bigger or equal to 1/2 so we\u0027re reading this off the top row,"},{"Start":"10:07.405 ","End":"10:11.040","Text":"but we\u0027re automatically also bigger or equal to minus 1/2,"},{"Start":"10:11.040 ","End":"10:13.830","Text":"because you can see this green bit is included in this red bit,"},{"Start":"10:13.830 ","End":"10:19.295","Text":"what we get is 3 times absolute value of 2x+1,"},{"Start":"10:19.295 ","End":"10:25.800","Text":"because we\u0027re bigger or equal to minus 1/2 is 2x+1, just like above."},{"Start":"10:25.800 ","End":"10:33.640","Text":"This time, the 2x-1 from here is also from the top row,"},{"Start":"10:33.640 ","End":"10:37.415","Text":"so it\u0027s just 2x-1,"},{"Start":"10:37.415 ","End":"10:41.375","Text":"and this is equal to 2, just as before."},{"Start":"10:41.375 ","End":"10:44.630","Text":"Perhaps, I also could have emphasized the middle bit."},{"Start":"10:44.630 ","End":"10:48.785","Text":"This bit, I could\u0027ve marked as being from here,"},{"Start":"10:48.785 ","End":"10:56.060","Text":"including the minus 1/2 and up to but not including the 1/2."},{"Start":"10:56.060 ","End":"11:00.650","Text":"Perhaps I could\u0027ve added 1/3 just to emphasize the middle bit."},{"Start":"11:00.650 ","End":"11:02.915","Text":"Here, we have this part."},{"Start":"11:02.915 ","End":"11:08.080","Text":"Here, we have the part that was labeled in the diagram like this."},{"Start":"11:08.080 ","End":"11:10.930","Text":"In the middle, the thing that\u0027s colored in"},{"Start":"11:10.930 ","End":"11:14.810","Text":"blue with the solid circle here and the hollow circle here."},{"Start":"11:14.810 ","End":"11:17.030","Text":"Now, it\u0027s time to actually solve these."},{"Start":"11:17.030 ","End":"11:18.800","Text":"There\u0027s three parts to solve."},{"Start":"11:18.800 ","End":"11:20.840","Text":"Let\u0027s just scroll down a bit."},{"Start":"11:20.840 ","End":"11:23.035","Text":"Let\u0027s take the first one."},{"Start":"11:23.035 ","End":"11:24.310","Text":"Perhaps, I\u0027ll label them."},{"Start":"11:24.310 ","End":"11:25.855","Text":"I\u0027ll call this one A,"},{"Start":"11:25.855 ","End":"11:29.660","Text":"this one B, this one C. We\u0027ll go over to a different color."},{"Start":"11:29.660 ","End":"11:31.640","Text":"Let\u0027s solve A first."},{"Start":"11:31.640 ","End":"11:33.735","Text":"What we have is,"},{"Start":"11:33.735 ","End":"11:35.855","Text":"let\u0027s take the equation part,"},{"Start":"11:35.855 ","End":"11:37.210","Text":"and then the condition part,"},{"Start":"11:37.210 ","End":"11:38.510","Text":"we\u0027ll put in later."},{"Start":"11:38.510 ","End":"11:46.458","Text":"I have, just copy it, 3(-2x-1)-(-2x+1)=2."},{"Start":"11:46.458 ","End":"11:48.805","Text":"Open up the brackets."},{"Start":"11:48.805 ","End":"11:57.600","Text":"-6x-3+2x-1=2,"},{"Start":"11:57.600 ","End":"12:03.325","Text":"x\u0027s on the left, -6x+2x is -4x."},{"Start":"12:03.325 ","End":"12:05.575","Text":"Numbers on the right,"},{"Start":"12:05.575 ","End":"12:14.030","Text":"2+1+3 is 2+4 is 6."},{"Start":"12:14.030 ","End":"12:22.975","Text":"Divide both sides by -4 and we have that x is equal to -6/4,"},{"Start":"12:22.975 ","End":"12:25.480","Text":"which is -3/2,"},{"Start":"12:25.480 ","End":"12:28.115","Text":"or if you like, -1/2."},{"Start":"12:28.115 ","End":"12:30.145","Text":"Now, when I get to this,"},{"Start":"12:30.145 ","End":"12:34.535","Text":"I have to make sure that it\u0027s within range because A only"},{"Start":"12:34.535 ","End":"12:39.045","Text":"applies to x\u0027s that are less than -1/2."},{"Start":"12:39.045 ","End":"12:45.160","Text":"This thing -1/2 is certainly less than -1/2. That\u0027s okay."},{"Start":"12:45.160 ","End":"12:46.985","Text":"This is a good solution,"},{"Start":"12:46.985 ","End":"12:49.555","Text":"and so it deserves to be highlighted."},{"Start":"12:49.555 ","End":"12:51.740","Text":"Now, let\u0027s go to part B."},{"Start":"12:51.740 ","End":"12:54.829","Text":"In part B, I\u0027ll just copy it first."},{"Start":"12:54.829 ","End":"12:56.570","Text":"That\u0027s this bit."},{"Start":"12:56.570 ","End":"13:05.885","Text":"It\u0027s 3 times 2x+1-(-2x+1)=2."},{"Start":"13:05.885 ","End":"13:07.880","Text":"Open the brackets."},{"Start":"13:07.880 ","End":"13:14.825","Text":"6x+3+2x-1=2,"},{"Start":"13:14.825 ","End":"13:17.435","Text":"x\u0027s to the left numbers to the right,"},{"Start":"13:17.435 ","End":"13:18.865","Text":"6 and 2 is,"},{"Start":"13:18.865 ","End":"13:21.880","Text":"we have 8x equals,"},{"Start":"13:21.880 ","End":"13:27.160","Text":"let\u0027s see, 2+1-3 is 0,"},{"Start":"13:27.160 ","End":"13:30.245","Text":"so x is 0."},{"Start":"13:30.245 ","End":"13:31.650","Text":"Then when we get the answer,"},{"Start":"13:31.650 ","End":"13:33.500","Text":"we have to check the range."},{"Start":"13:33.500 ","End":"13:37.130","Text":"Is zero between -1/2 and 1/2?"},{"Start":"13:37.130 ","End":"13:42.830","Text":"Yes, certainly, zero is between -1/2 and 1/2,"},{"Start":"13:42.830 ","End":"13:45.009","Text":"so zero is a good solution."},{"Start":"13:45.009 ","End":"13:47.545","Text":"If it wasn\u0027t, we just wouldn\u0027t include it."},{"Start":"13:47.545 ","End":"13:49.475","Text":"But it is in range."},{"Start":"13:49.475 ","End":"13:52.680","Text":"Let\u0027s see what happens in Part C. First of all,"},{"Start":"13:52.680 ","End":"13:55.675","Text":"I\u0027ll just copy the equation part."},{"Start":"13:55.675 ","End":"14:06.124","Text":"We have 3 times 2x+1-2x-1=2."},{"Start":"14:06.124 ","End":"14:08.465","Text":"We\u0027ll need to scroll here."},{"Start":"14:08.465 ","End":"14:10.445","Text":"Opening the brackets."},{"Start":"14:10.445 ","End":"14:20.260","Text":"6x+3-2x+1=2, x\u0027s to the left,"},{"Start":"14:20.260 ","End":"14:23.720","Text":"4x, numbers to the right,"},{"Start":"14:23.720 ","End":"14:29.840","Text":"2 and here, -3-1 is -2."},{"Start":"14:29.840 ","End":"14:34.060","Text":"X is equal to -2/4 which is -1/2."},{"Start":"14:34.060 ","End":"14:36.920","Text":"The question is, is -1/2 in the range?"},{"Start":"14:36.920 ","End":"14:38.830","Text":"Let\u0027s see, what was that range?"},{"Start":"14:38.830 ","End":"14:43.185","Text":"We need x bigger or equal to 1/2."},{"Start":"14:43.185 ","End":"14:47.060","Text":"This is definitely not in this range."},{"Start":"14:47.060 ","End":"14:53.500","Text":"This one has to be excluded because these two we were lucky with,"},{"Start":"14:53.500 ","End":"14:56.720","Text":"just happened to be that this was within the range,"},{"Start":"14:56.720 ","End":"14:58.385","Text":"this was within the range."},{"Start":"14:58.385 ","End":"15:00.300","Text":"This is not."},{"Start":"15:00.300 ","End":"15:07.715","Text":"Our final answers are just -1 1/2 and 0."},{"Start":"15:07.715 ","End":"15:12.920","Text":"Those are the two answers, and we\u0027re done."}],"ID":5038},{"Watched":false,"Name":"Exercise 1","Duration":"3m 55s","ChapterTopicVideoID":5037,"CourseChapterTopicPlaylistID":3874,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"In this exercise, we\u0027re given a pair of"},{"Start":"00:03.030 ","End":"00:06.285","Text":"equations to solve each of them with absolute value."},{"Start":"00:06.285 ","End":"00:08.115","Text":"Let\u0027s start with the first,"},{"Start":"00:08.115 ","End":"00:11.310","Text":"absolute value of x minus 4 is 6."},{"Start":"00:11.310 ","End":"00:17.145","Text":"This fits into the pattern of absolute value of a=b."},{"Start":"00:17.145 ","End":"00:18.630","Text":"When have this pattern,"},{"Start":"00:18.630 ","End":"00:21.450","Text":"then there\u0027s 2 possibilities,"},{"Start":"00:21.450 ","End":"00:28.695","Text":"we either have that a=b or that a equals minus b."},{"Start":"00:28.695 ","End":"00:31.770","Text":"This only applies when this b,"},{"Start":"00:31.770 ","End":"00:35.985","Text":"is bigger or equal to 0 because if I have a negative b here,"},{"Start":"00:35.985 ","End":"00:37.665","Text":"I stop right away and say,"},{"Start":"00:37.665 ","End":"00:42.365","Text":"\"No solution,\" because I know absolute value of anything is not negative."},{"Start":"00:42.365 ","End":"00:43.865","Text":"Now in our case,"},{"Start":"00:43.865 ","End":"00:45.840","Text":"6 is not negative,"},{"Start":"00:45.840 ","End":"00:49.450","Text":"it\u0027s positive, and so we have 2 possibilities."},{"Start":"00:49.450 ","End":"00:58.425","Text":"We either have x minus 4 equals 6 or x minus 4 is minus 6,"},{"Start":"00:58.425 ","End":"01:01.290","Text":"and we can work on each one separately,"},{"Start":"01:01.290 ","End":"01:07.605","Text":"x minus 4 equals 6 gives me x=10,"},{"Start":"01:07.605 ","End":"01:11.160","Text":"and x minus 4 equals minus 6,"},{"Start":"01:11.160 ","End":"01:13.124","Text":"if I bring the 4 to the other side,"},{"Start":"01:13.124 ","End":"01:14.970","Text":"minus 6 plus 4,"},{"Start":"01:14.970 ","End":"01:18.075","Text":"or x is minus 2,"},{"Start":"01:18.075 ","End":"01:21.510","Text":"just put this at the side there, it\u0027s in the way."},{"Start":"01:21.510 ","End":"01:24.645","Text":"Those are the 2 solutions,"},{"Start":"01:24.645 ","End":"01:27.720","Text":"10 or minus 2."},{"Start":"01:27.720 ","End":"01:29.640","Text":"Let\u0027s get onto the next one b."},{"Start":"01:29.640 ","End":"01:33.890","Text":"Same idea, absolute value of"},{"Start":"01:33.890 ","End":"01:40.025","Text":"something is equal to something and that something is non-negative."},{"Start":"01:40.025 ","End":"01:46.530","Text":"It\u0027s important, I might even just emphasize it that this is bigger or equal to 0,"},{"Start":"01:46.530 ","End":"01:49.320","Text":"suppose I should stress that here also,"},{"Start":"01:49.320 ","End":"01:52.565","Text":"otherwise, we always say we can\u0027t continue no solution."},{"Start":"01:52.565 ","End":"01:54.635","Text":"Using the same pattern,"},{"Start":"01:54.635 ","End":"01:58.780","Text":"we either have the 2x plus 5 is 11,"},{"Start":"01:58.780 ","End":"02:06.290","Text":"or 2x plus 5 is equal to minus 11."},{"Start":"02:06.290 ","End":"02:07.880","Text":"We\u0027ve learned this in the theory,"},{"Start":"02:07.880 ","End":"02:09.485","Text":"but just to explain it again,"},{"Start":"02:09.485 ","End":"02:11.930","Text":"the only way the absolute value of something can be"},{"Start":"02:11.930 ","End":"02:15.170","Text":"11 is if it\u0027s 11 itself or if it\u0027s minus 11,"},{"Start":"02:15.170 ","End":"02:17.840","Text":"in which case the absolute value also comes out to be 11,"},{"Start":"02:17.840 ","End":"02:19.970","Text":"that\u0027s why we have the 2 possibilities."},{"Start":"02:19.970 ","End":"02:22.980","Text":"In this case 2x equals,"},{"Start":"02:22.980 ","End":"02:24.735","Text":"11 minus 5 is 6,"},{"Start":"02:24.735 ","End":"02:26.535","Text":"and so x equals 3,"},{"Start":"02:26.535 ","End":"02:27.870","Text":"and in this case,"},{"Start":"02:27.870 ","End":"02:30.680","Text":"2x is minus 11 minus 5,"},{"Start":"02:30.680 ","End":"02:32.330","Text":"which is minus 16,"},{"Start":"02:32.330 ","End":"02:36.515","Text":"which gives us x equals minus 8,"},{"Start":"02:36.515 ","End":"02:41.450","Text":"and these here are the 2 solutions for x."},{"Start":"02:41.450 ","End":"02:43.500","Text":"Now of course as in all equations,"},{"Start":"02:43.500 ","End":"02:45.125","Text":"when we get to the end,"},{"Start":"02:45.125 ","End":"02:48.980","Text":"we sometimes should check and substitute in the original equation."},{"Start":"02:48.980 ","End":"02:50.554","Text":"Let\u0027s try, for example,"},{"Start":"02:50.554 ","End":"02:53.840","Text":"putting this x=3,"},{"Start":"02:53.840 ","End":"02:57.635","Text":"let\u0027s substitute it in this equation and see what we get."},{"Start":"02:57.635 ","End":"03:03.365","Text":"Well, the left-hand side is absolute value of 2x plus 5,"},{"Start":"03:03.365 ","End":"03:08.180","Text":"is absolute value of twice 3 plus 5,"},{"Start":"03:08.180 ","End":"03:13.805","Text":"which is absolute value of 6 plus 5 is 11,"},{"Start":"03:13.805 ","End":"03:15.995","Text":"which equals, indeed,"},{"Start":"03:15.995 ","End":"03:17.930","Text":"the right-hand side 11."},{"Start":"03:17.930 ","End":"03:22.549","Text":"If you start from the left-hand side and end up in the right-hand side,"},{"Start":"03:22.549 ","End":"03:24.900","Text":"then your equality is verified,"},{"Start":"03:24.900 ","End":"03:26.225","Text":"so this is a solution,"},{"Start":"03:26.225 ","End":"03:27.910","Text":"similarly minus 8,"},{"Start":"03:27.910 ","End":"03:30.920","Text":"and you could go back to the previous one and check each one."},{"Start":"03:30.920 ","End":"03:34.010","Text":"For example, if I put here x=-2,"},{"Start":"03:34.010 ","End":"03:36.740","Text":"I\u0027d get absolute value of minus 2 minus 4,"},{"Start":"03:36.740 ","End":"03:39.770","Text":"which is absolute value of minus 6, which is 6,"},{"Start":"03:39.770 ","End":"03:43.130","Text":"and then this would be verified and depending on how much time you"},{"Start":"03:43.130 ","End":"03:46.970","Text":"have and how many doubts you have that you\u0027ve done it correctly,"},{"Start":"03:46.970 ","End":"03:49.985","Text":"then you should verify by substitution."},{"Start":"03:49.985 ","End":"03:52.280","Text":"Very well. If we\u0027ve done part b,"},{"Start":"03:52.280 ","End":"03:55.590","Text":"then that\u0027s the end of this exercise."}],"ID":5039},{"Watched":false,"Name":"Exercise 2","Duration":"10m 7s","ChapterTopicVideoID":5038,"CourseChapterTopicPlaylistID":3874,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:13.070 ","End":"00:19.875","Text":"In this exercise, we have a pair of equations to solve involving absolute values."},{"Start":"00:19.875 ","End":"00:21.855","Text":"Let\u0027s start with the first one."},{"Start":"00:21.855 ","End":"00:24.000","Text":"The standard form that we know,"},{"Start":"00:24.000 ","End":"00:29.300","Text":"I mean the basic case is of the form absolute value of a=b,"},{"Start":"00:29.300 ","End":"00:33.215","Text":"where its not necessarily numbers but could be expressions."},{"Start":"00:33.215 ","End":"00:34.790","Text":"Now this is not in that form,"},{"Start":"00:34.790 ","End":"00:39.050","Text":"but I could bring it to that form if I just take the x to the other side,"},{"Start":"00:39.050 ","End":"00:46.085","Text":"and then I would get absolute value of 4x-3=2-x."},{"Start":"00:46.085 ","End":"00:49.220","Text":"Now, generally, when we get to this form,"},{"Start":"00:49.220 ","End":"00:53.450","Text":"we split it up into two cases a is plus or minus b."},{"Start":"00:53.450 ","End":"00:57.590","Text":"We take a=b or a=-b."},{"Start":"00:57.590 ","End":"01:00.815","Text":"Now, if it was just a number on the right,"},{"Start":"01:00.815 ","End":"01:06.530","Text":"that would be easier and then we just go ahead and solve and get our solutions."},{"Start":"01:06.530 ","End":"01:09.230","Text":"The thing is when it\u0027s not a constant on the right,"},{"Start":"01:09.230 ","End":"01:11.780","Text":"we don\u0027t know if this is positive or negative."},{"Start":"01:11.780 ","End":"01:13.040","Text":"If b was a number,"},{"Start":"01:13.040 ","End":"01:15.110","Text":"I\u0027d say if b is negative,"},{"Start":"01:15.110 ","End":"01:16.355","Text":"there is no solution."},{"Start":"01:16.355 ","End":"01:18.520","Text":"If b is positive, these are the solutions."},{"Start":"01:18.520 ","End":"01:19.670","Text":"When it\u0027s a variable,"},{"Start":"01:19.670 ","End":"01:22.850","Text":"what we do is we proceed as usual, but at the end,"},{"Start":"01:22.850 ","End":"01:27.860","Text":"we have to check our answer to see if it fits because of the absolute value,"},{"Start":"01:27.860 ","End":"01:33.305","Text":"things might not work out quite right so let\u0027s just solve it by saying,"},{"Start":"01:33.305 ","End":"01:35.450","Text":"we split up into two cases,"},{"Start":"01:35.450 ","End":"01:41.130","Text":"we either take 4x-3=2-x,"},{"Start":"01:41.130 ","End":"01:42.390","Text":"or the other case,"},{"Start":"01:42.390 ","End":"01:46.585","Text":"we have 4x-3 is minus of this,"},{"Start":"01:46.585 ","End":"01:53.025","Text":"I\u0027ll just reverse the order because minus of a subtraction is just the reverse,"},{"Start":"01:53.025 ","End":"01:56.705","Text":"and we\u0027ll solve then we\u0027ll get to the end and then we\u0027ll check our solutions."},{"Start":"01:56.705 ","End":"02:00.215","Text":"So here we get bringing the x\u0027s to the left,"},{"Start":"02:00.215 ","End":"02:05.970","Text":"5x and the numbers to the right equals 5 and so x=1,"},{"Start":"02:05.970 ","End":"02:10.700","Text":"but I\u0027m not highlighting it yet because I need to check it and if I do this one,"},{"Start":"02:10.700 ","End":"02:13.760","Text":"bring x to the left, it\u0027s 3x."},{"Start":"02:13.760 ","End":"02:15.710","Text":"Here I bring the 3 to the right,"},{"Start":"02:15.710 ","End":"02:17.465","Text":"-2+3 is 1,"},{"Start":"02:17.465 ","End":"02:20.065","Text":"and so x is 1/3."},{"Start":"02:20.065 ","End":"02:23.100","Text":"At this point, I need to check."},{"Start":"02:23.100 ","End":"02:26.460","Text":"I\u0027m going to check each one separately, first, the 1."},{"Start":"02:26.460 ","End":"02:29.860","Text":"The thing is I don\u0027t have to actually go all the way back to the beginning"},{"Start":"02:29.860 ","End":"02:33.830","Text":"it\u0027s enough to go back to the point before I did the split,"},{"Start":"02:33.830 ","End":"02:36.570","Text":"so I\u0027m going to take this part here."},{"Start":"02:36.570 ","End":"02:42.687","Text":"Let\u0027s copy it again. Absolute value of 4x-3=2-x."},{"Start":"02:42.687 ","End":"02:46.865","Text":"First of all, I\u0027ll try x=1,"},{"Start":"02:46.865 ","End":"02:55.740","Text":"and then what I get is absolute value of 4 times 1-3=?"},{"Start":"02:55.740 ","End":"03:00.360","Text":"That\u0027s what I\u0027m checking, 2-1."},{"Start":"03:00.360 ","End":"03:02.895","Text":"Is this true? Well, let\u0027s see."},{"Start":"03:02.895 ","End":"03:08.415","Text":"It says absolute value of 4-3, which is 1."},{"Start":"03:08.415 ","End":"03:13.890","Text":"Basically, this says absolute value of 1 equals 1 and this is true."},{"Start":"03:13.890 ","End":"03:18.210","Text":"Yes, so we\u0027ve confirmed x=1."},{"Start":"03:18.210 ","End":"03:22.560","Text":"Now let\u0027s try the other one, x=1/3."},{"Start":"03:22.560 ","End":"03:25.440","Text":"We get in here absolute value of"},{"Start":"03:25.440 ","End":"03:34.020","Text":"4 times 1/3-3=2-x is a 1/3?"},{"Start":"03:34.020 ","End":"03:36.788","Text":"Because this is what I\u0027m checking."},{"Start":"03:36.788 ","End":"03:38.340","Text":"So absolute value."},{"Start":"03:38.340 ","End":"03:42.570","Text":"Now, 4 times a 1/3 is 1 1/3,"},{"Start":"03:42.570 ","End":"03:52.560","Text":"1 1/3 minus 3 is -1 and 2/3 so it\u0027s -1 and 2/3 in absolute value."},{"Start":"03:52.560 ","End":"03:55.500","Text":"Distinguish the bars from the 1."},{"Start":"03:55.500 ","End":"03:59.135","Text":"These are bars from the absolute value equals?"},{"Start":"03:59.135 ","End":"04:03.790","Text":"2-1/3 is 1 and 2/3."},{"Start":"04:03.790 ","End":"04:06.170","Text":"Now, I have to say the answer to this is yes"},{"Start":"04:06.170 ","End":"04:09.290","Text":"because what is written in here is simply the negative of this,"},{"Start":"04:09.290 ","End":"04:10.939","Text":"but then absolute value,"},{"Start":"04:10.939 ","End":"04:13.040","Text":"it becomes the same thing so,"},{"Start":"04:13.040 ","End":"04:18.220","Text":"yes, we\u0027ve confirmed this one too and so we have two answers."},{"Start":"04:18.220 ","End":"04:19.820","Text":"This is the way to do it,"},{"Start":"04:19.820 ","End":"04:21.620","Text":"but I\u0027d like to mention a shortcut."},{"Start":"04:21.620 ","End":"04:23.585","Text":"Feel free to use or not."},{"Start":"04:23.585 ","End":"04:28.040","Text":"I mentioned that the problem with splitting when b is"},{"Start":"04:28.040 ","End":"04:33.125","Text":"not a number but an expression is that we don\u0027t know if it\u0027s positive or negative."},{"Start":"04:33.125 ","End":"04:37.490","Text":"An actual fact, it would be enough to check each of these two solutions"},{"Start":"04:37.490 ","End":"04:41.885","Text":"in the right-hand side and if it\u0027s positive in both cases,"},{"Start":"04:41.885 ","End":"04:43.590","Text":"then we know we\u0027re okay."},{"Start":"04:43.590 ","End":"04:47.100","Text":"If I put in x=1, 2-1 is positive."},{"Start":"04:47.100 ","End":"04:51.810","Text":"Yes. If I put an x=1/3, 2-1/3 is positive."},{"Start":"04:51.810 ","End":"04:57.920","Text":"Yes. As long as this thing comes up positive in both cases or non-negative,"},{"Start":"04:57.920 ","End":"05:01.910","Text":"should I say because absolute value it could be also zero, then we\u0027re okay."},{"Start":"05:01.910 ","End":"05:06.200","Text":"I didn\u0027t write anything down that\u0027s the shortcut this is the full way of doing it."},{"Start":"05:06.200 ","End":"05:08.115","Text":"You can check by substitution."},{"Start":"05:08.115 ","End":"05:10.670","Text":"The only thing you can save as you don\u0027t have to go all"},{"Start":"05:10.670 ","End":"05:13.920","Text":"the way back to the beginning you can go back to before the split."},{"Start":"05:13.920 ","End":"05:21.030","Text":"Let\u0027s go to part b and scroll down a bit maybe keep a bit of part a first."},{"Start":"05:21.030 ","End":"05:22.900","Text":"In part b,"},{"Start":"05:22.900 ","End":"05:28.195","Text":"we also pretty much have absolute value of something equals something."},{"Start":"05:28.195 ","End":"05:34.940","Text":"What I need to do is divide by 2 and get this as x+3 over 2 and"},{"Start":"05:34.940 ","End":"05:37.910","Text":"then I do have the case absolute value of something"},{"Start":"05:37.910 ","End":"05:41.750","Text":"equals something and then we do the split."},{"Start":"05:41.750 ","End":"05:44.390","Text":"I\u0027ll just get a bit more space here now,"},{"Start":"05:44.390 ","End":"05:52.815","Text":"so we say that there are two possibilities either x-1 equals this thing,"},{"Start":"05:52.815 ","End":"06:00.395","Text":"as is or the other possibility that x-1 is minus this thing,"},{"Start":"06:00.395 ","End":"06:06.290","Text":"minus of x+3 over 2 because this expression is not"},{"Start":"06:06.290 ","End":"06:09.330","Text":"a constant number then at the end we\u0027re going to have to"},{"Start":"06:09.330 ","End":"06:13.140","Text":"check our solution so let\u0027s just continue meanwhile to solve."},{"Start":"06:13.140 ","End":"06:15.560","Text":"Let\u0027s see, clearing of fractions,"},{"Start":"06:15.560 ","End":"06:21.765","Text":"multiply both sides by 2 twice x-1=x+3,"},{"Start":"06:21.765 ","End":"06:25.430","Text":"let\u0027s bring everything to one side and the other in our heads."},{"Start":"06:25.430 ","End":"06:29.770","Text":"On the left, I have 2x and -x is just x."},{"Start":"06:29.770 ","End":"06:34.060","Text":"On the right, I have 3 and from here I have -2."},{"Start":"06:34.060 ","End":"06:37.845","Text":"That becomes a plus 2, so it\u0027s 5."},{"Start":"06:37.845 ","End":"06:42.780","Text":"Here we could\u0027ve drawn another line 2x-2=x+3."},{"Start":"06:42.780 ","End":"06:45.380","Text":"Anyway, x is 5 and we have to check this,"},{"Start":"06:45.380 ","End":"06:47.360","Text":"so I\u0027m not highlighting it yet."},{"Start":"06:47.360 ","End":"06:53.550","Text":"The other one, also clear fractions I get twice x-1,"},{"Start":"06:53.550 ","End":"06:56.265","Text":"this time -x,"},{"Start":"06:56.265 ","End":"07:00.735","Text":"and then -3 because the whole x-3 on the left,"},{"Start":"07:00.735 ","End":"07:07.560","Text":"2x+x is 3x and on the right I have -2,"},{"Start":"07:07.560 ","End":"07:09.735","Text":"which comes over as plus 2."},{"Start":"07:09.735 ","End":"07:14.775","Text":"Plus 2-3 is -1 and so"},{"Start":"07:14.775 ","End":"07:21.645","Text":"x=-1/3 so these are the two candidates."},{"Start":"07:21.645 ","End":"07:23.625","Text":"I have to verify these,"},{"Start":"07:23.625 ","End":"07:28.130","Text":"I can verify them either in the original equation or in"},{"Start":"07:28.130 ","End":"07:32.630","Text":"this equation as long as it\u0027s before the split because of the fraction,"},{"Start":"07:32.630 ","End":"07:35.390","Text":"I think I\u0027d rather go for the original equation."},{"Start":"07:35.390 ","End":"07:38.690","Text":"Let me take the original equation and I\u0027ll"},{"Start":"07:38.690 ","End":"07:43.520","Text":"substitute x=5 once and see what I get and then"},{"Start":"07:43.520 ","End":"07:52.940","Text":"afterwards I\u0027ll substitute x=-1/3 and see if what we get x=5 in the original."},{"Start":"07:52.940 ","End":"07:57.290","Text":"I get twice absolute value of 5-1."},{"Start":"07:57.290 ","End":"07:59.705","Text":"Is this equal to?"},{"Start":"07:59.705 ","End":"08:02.540","Text":"25+3."},{"Start":"08:02.540 ","End":"08:04.760","Text":"Well, let\u0027s carry on with this one."},{"Start":"08:04.760 ","End":"08:12.270","Text":"This says that 2 times absolute value of 4 equals?"},{"Start":"08:12.270 ","End":"08:17.325","Text":"5 plus 3 absolute value of 4 is 4,"},{"Start":"08:17.325 ","End":"08:19.905","Text":"2 times 4 equals 8?"},{"Start":"08:19.905 ","End":"08:31.020","Text":"I would say yes and so we verified the 5 and I\u0027ll highlight it and now on to the -1/3."},{"Start":"08:31.020 ","End":"08:34.370","Text":"Looking at the equation, the original one,"},{"Start":"08:34.370 ","End":"08:38.585","Text":"I am going to see if twice absolute value of"},{"Start":"08:38.585 ","End":"08:48.600","Text":"-1/3-1 is equal to -1/3 and then plus 3."},{"Start":"08:48.600 ","End":"08:54.450","Text":"Let\u0027s see, -1/3-1 is -1"},{"Start":"08:54.450 ","End":"09:01.040","Text":"1/3 so this is twice and absolute value of -1 1/3."},{"Start":"09:01.040 ","End":"09:05.075","Text":"I have to distinguish the absolute value from a 1, anyway."},{"Start":"09:05.075 ","End":"09:06.545","Text":"Now, this is equal to,"},{"Start":"09:06.545 ","End":"09:09.050","Text":"I\u0027m just continue with the left-hand side for a moment."},{"Start":"09:09.050 ","End":"09:15.515","Text":"This is absolute value is 1 1/3 and twice 1 1/3 is 2, and 2/3."},{"Start":"09:15.515 ","End":"09:21.300","Text":"On the other hand, -1/3+3 is"},{"Start":"09:21.300 ","End":"09:27.240","Text":"also 2 and 2/3 and so we get equality."},{"Start":"09:27.240 ","End":"09:33.995","Text":"The answer is yes and so this is also a solution and I\u0027ll highlight that."},{"Start":"09:33.995 ","End":"09:35.750","Text":"Now again, I\u0027ll just mention"},{"Start":"09:35.750 ","End":"09:41.060","Text":"my shortcut method is that if you have absolute value of something equals something,"},{"Start":"09:41.060 ","End":"09:44.600","Text":"if this right-hand side is positive,"},{"Start":"09:44.600 ","End":"09:50.750","Text":"then it confirms that this is a solution so if I put 5 in here,"},{"Start":"09:50.750 ","End":"09:55.025","Text":"I see it\u0027s positive if I put -1/3 it still comes out positive,"},{"Start":"09:55.025 ","End":"09:58.280","Text":"so both of these are valid solutions,"},{"Start":"09:58.280 ","End":"10:03.020","Text":"but this is the more correct way of doing it."},{"Start":"10:03.020 ","End":"10:05.030","Text":"Anyway, we\u0027re done with part b,"},{"Start":"10:05.030 ","End":"10:08.100","Text":"which means that we\u0027re done with this exercise."}],"ID":5040},{"Watched":false,"Name":"Exercise 3 part a","Duration":"11m 30s","ChapterTopicVideoID":5039,"CourseChapterTopicPlaylistID":3874,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.709","Text":"In this exercise,"},{"Start":"00:01.709 ","End":"00:04.470","Text":"there are 2 equations with absolute value,"},{"Start":"00:04.470 ","End":"00:06.345","Text":"and notice that in each of them,"},{"Start":"00:06.345 ","End":"00:08.910","Text":"there\u0027s more than 1 absolute value."},{"Start":"00:08.910 ","End":"00:12.990","Text":"Here, demonstrate how we handle this situation."},{"Start":"00:12.990 ","End":"00:15.015","Text":"Start with the first one."},{"Start":"00:15.015 ","End":"00:19.320","Text":"But I\u0027d like to remind you at the side here of what is the definition of"},{"Start":"00:19.320 ","End":"00:23.850","Text":"absolute value of some number, say a."},{"Start":"00:23.850 ","End":"00:25.980","Text":"It\u0027s equal to a itself,"},{"Start":"00:25.980 ","End":"00:34.665","Text":"provided that a is positive and it\u0027s equal to minus a if a is negative."},{"Start":"00:34.665 ","End":"00:38.400","Text":"Think of it. If a is say minus 6,"},{"Start":"00:38.400 ","End":"00:42.870","Text":"then absolute value of a is 6 which is minus minus 6."},{"Start":"00:42.870 ","End":"00:45.170","Text":"Absolute value of 0 is 0."},{"Start":"00:45.170 ","End":"00:46.310","Text":"We could stick it in either one."},{"Start":"00:46.310 ","End":"00:49.070","Text":"We usually stick it here by saying that if a is"},{"Start":"00:49.070 ","End":"00:52.845","Text":"bigger or equal to 0 it\u0027s also equal to a."},{"Start":"00:52.845 ","End":"00:55.755","Text":"It makes sense for a equals 0 also."},{"Start":"00:55.755 ","End":"01:01.595","Text":"Now using this, I\u0027m going to rewrite each of these absolute values like this."},{"Start":"01:01.595 ","End":"01:07.470","Text":"The absolute value of 2 minus x from here is equal to,"},{"Start":"01:07.470 ","End":"01:09.365","Text":"again, 2 cases."},{"Start":"01:09.365 ","End":"01:12.153","Text":"It\u0027s equal to the thing itself,"},{"Start":"01:12.153 ","End":"01:13.560","Text":"2 minus x,"},{"Start":"01:13.560 ","End":"01:18.630","Text":"in the case where 2 minus x is bigger or equal to 0,"},{"Start":"01:18.630 ","End":"01:20.385","Text":"and it\u0027s minus this,"},{"Start":"01:20.385 ","End":"01:25.480","Text":"remember to negate a difference we reverse the order."},{"Start":"01:25.480 ","End":"01:31.110","Text":"So it\u0027s x minus 2 if 2 minus x is negative."},{"Start":"01:31.110 ","End":"01:35.990","Text":"Now what we\u0027re going to do is reinterpret these conditions in terms of just"},{"Start":"01:35.990 ","End":"01:40.980","Text":"x. I\u0027m going to rewrite this as 2 minus x."},{"Start":"01:40.980 ","End":"01:42.615","Text":"This part\u0027s the same,"},{"Start":"01:42.615 ","End":"01:44.310","Text":"and x minus 2."},{"Start":"01:44.310 ","End":"01:47.820","Text":"But instead of saying 2 minus x bigger or equal to 0,"},{"Start":"01:47.820 ","End":"01:49.650","Text":"if I bring the other x to the other side,"},{"Start":"01:49.650 ","End":"01:51.330","Text":"it\u0027s x less than or equal to 2."},{"Start":"01:51.330 ","End":"01:53.520","Text":"2 bigger or equal to x, or in other words,"},{"Start":"01:53.520 ","End":"01:55.950","Text":"x less than or equal to 2."},{"Start":"01:55.950 ","End":"01:58.170","Text":"Here, 2 minus x is negative,"},{"Start":"01:58.170 ","End":"02:01.170","Text":"means that x is bigger than 2."},{"Start":"02:01.170 ","End":"02:08.000","Text":"That\u0027s how this absolute value is defined separately for this range and for this range,"},{"Start":"02:08.000 ","End":"02:12.525","Text":"and similarly, we\u0027ll get 2 ranges for the 2x plus 3."},{"Start":"02:12.525 ","End":"02:16.795","Text":"This is equal to 2x plus 3 itself,"},{"Start":"02:16.795 ","End":"02:23.120","Text":"provided that 2x plus 3 is bigger or equal to 0 and negative of this,"},{"Start":"02:23.120 ","End":"02:29.695","Text":"or write it as minus 2x minus 3 if 2x plus 3 is negative."},{"Start":"02:29.695 ","End":"02:36.110","Text":"Once again, I just rewrite this to make the condition in terms of x."},{"Start":"02:36.110 ","End":"02:38.410","Text":"It\u0027s 2x plus 3,"},{"Start":"02:38.410 ","End":"02:41.165","Text":"if this is bigger or equal to 0 and mentally we could figure,"},{"Start":"02:41.165 ","End":"02:44.150","Text":"that means 2x is bigger or equal to minus 3."},{"Start":"02:44.150 ","End":"02:53.235","Text":"X is bigger or equal to minus 1.5, whichever."},{"Start":"02:53.235 ","End":"02:56.600","Text":"This is minus 2x minus 3,"},{"Start":"02:56.600 ","End":"03:00.080","Text":"provided it\u0027s always just the opposite of this. It\u0027d be bigger."},{"Start":"03:00.080 ","End":"03:01.280","Text":"So the bigger or equal to,"},{"Start":"03:01.280 ","End":"03:04.660","Text":"it will be less than minus 1.5."},{"Start":"03:04.660 ","End":"03:05.900","Text":"Now on the face of it,"},{"Start":"03:05.900 ","End":"03:07.965","Text":"it looks like there are 4 different ranges."},{"Start":"03:07.965 ","End":"03:11.008","Text":"There is this and this and this and this."},{"Start":"03:11.008 ","End":"03:13.770","Text":"But actually each one of these and with each one of"},{"Start":"03:13.770 ","End":"03:16.605","Text":"these gives 2 times 2 is 4 combinations,"},{"Start":"03:16.605 ","End":"03:18.905","Text":"but actually there\u0027s only 3 combinations."},{"Start":"03:18.905 ","End":"03:22.145","Text":"What we do is we take the numbers that appear here."},{"Start":"03:22.145 ","End":"03:25.565","Text":"From here we see that something happens when x is 2."},{"Start":"03:25.565 ","End":"03:30.155","Text":"From here we see something happens when x is minus 1.5."},{"Start":"03:30.155 ","End":"03:34.520","Text":"What I do is take these numbers and put them on the number line."},{"Start":"03:34.520 ","End":"03:39.365","Text":"I draw a number line and label it x,"},{"Start":"03:39.365 ","End":"03:43.025","Text":"and then I put these 2 values on the number line."},{"Start":"03:43.025 ","End":"03:45.215","Text":"It doesn\u0027t have to be to scale or anything,"},{"Start":"03:45.215 ","End":"03:49.490","Text":"the important thing is that 2 should be to the right of minus 1.5."},{"Start":"03:49.490 ","End":"03:53.585","Text":"That\u0027s the main thing is just 2 numbers or more, get them in order."},{"Start":"03:53.585 ","End":"04:01.325","Text":"Let\u0027s just say that I\u0027ll place the 2 here and the minus 1.5 here."},{"Start":"04:01.325 ","End":"04:05.765","Text":"This breaks the line up into 3 pieces."},{"Start":"04:05.765 ","End":"04:13.670","Text":"This piece here is where x is less than minus 1.5."},{"Start":"04:13.670 ","End":"04:18.020","Text":"To decide whether I include or don\u0027t include the minus 1.5 I look at"},{"Start":"04:18.020 ","End":"04:22.970","Text":"where the part that contains minus 1.5 and I see it\u0027s less than."},{"Start":"04:22.970 ","End":"04:27.130","Text":"This bit is where x is bigger or equal to"},{"Start":"04:27.130 ","End":"04:31.370","Text":"minus 1.5 or minus 1.5 is less than or equal to x,"},{"Start":"04:31.370 ","End":"04:33.080","Text":"and then I go up to,"},{"Start":"04:33.080 ","End":"04:38.260","Text":"now look at the 2, and I see that the less than or equal to,"},{"Start":"04:38.260 ","End":"04:43.875","Text":"the equals is on this side so this goes up to and includes the 2,"},{"Start":"04:43.875 ","End":"04:46.770","Text":"and then from 2 onwards,"},{"Start":"04:46.770 ","End":"04:49.245","Text":"I get the x bigger than 2,"},{"Start":"04:49.245 ","End":"04:51.995","Text":"so I get 3 ranges."},{"Start":"04:51.995 ","End":"04:53.480","Text":"Then each of these,"},{"Start":"04:53.480 ","End":"04:57.500","Text":"I actually know what is the absolute value,"},{"Start":"04:57.500 ","End":"04:58.820","Text":"because for example,"},{"Start":"04:58.820 ","End":"05:00.490","Text":"if x is bigger than 2,"},{"Start":"05:00.490 ","End":"05:07.010","Text":"then automatically it\u0027s also bigger or equal to minus 1.5."},{"Start":"05:07.010 ","End":"05:10.330","Text":"If x is less than minus 1.5,"},{"Start":"05:10.330 ","End":"05:13.790","Text":"it\u0027s automatically less than or equal to 2,"},{"Start":"05:13.790 ","End":"05:17.330","Text":"so you don\u0027t get all the combinations of one of these and one of these,"},{"Start":"05:17.330 ","End":"05:19.895","Text":"you just get these 3 intervals as follows."},{"Start":"05:19.895 ","End":"05:25.685","Text":"Now let\u0027s compute the equation what it is in each of the 3 intervals."},{"Start":"05:25.685 ","End":"05:29.230","Text":"Just scroll down a bit. Let\u0027s see now."},{"Start":"05:29.230 ","End":"05:36.805","Text":"In the interval where x is less than minus 1.5,"},{"Start":"05:36.805 ","End":"05:38.405","Text":"what do I get?"},{"Start":"05:38.405 ","End":"05:41.330","Text":"Just go down a bit again so I can see this equation."},{"Start":"05:41.330 ","End":"05:44.090","Text":"This is the equation I want. You know what?"},{"Start":"05:44.090 ","End":"05:45.950","Text":"Just while I\u0027m with this color,"},{"Start":"05:45.950 ","End":"05:51.420","Text":"I\u0027ll just write the other intervals minus 1.5 less than or equal to x,"},{"Start":"05:51.420 ","End":"05:53.065","Text":"less than or equal to 2."},{"Start":"05:53.065 ","End":"05:58.175","Text":"What happens here and what happens when x is bigger than 2?"},{"Start":"05:58.175 ","End":"06:00.535","Text":"There are 3 cases."},{"Start":"06:00.535 ","End":"06:02.735","Text":"Now in this case,"},{"Start":"06:02.735 ","End":"06:04.430","Text":"looking all the time with this equation,"},{"Start":"06:04.430 ","End":"06:08.945","Text":"I think I\u0027ll highlight it just so I\u0027ll see it better when referring to it."},{"Start":"06:08.945 ","End":"06:11.970","Text":"We get 3 times."},{"Start":"06:11.970 ","End":"06:18.185","Text":"Now the absolute value of 2 minus x on this interval where we\u0027re less than minus 1/2,"},{"Start":"06:18.185 ","End":"06:21.280","Text":"we\u0027re also less than or equal to 2."},{"Start":"06:21.280 ","End":"06:23.890","Text":"It\u0027s 2 minus x itself,"},{"Start":"06:23.890 ","End":"06:27.595","Text":"so just regular brackets for 2 minus x,"},{"Start":"06:27.595 ","End":"06:32.990","Text":"and then minus, and then absolute value of 2x plus 3."},{"Start":"06:32.990 ","End":"06:40.195","Text":"Absolute value of 2x plus 3 on this part is minus 2x minus 3,"},{"Start":"06:40.195 ","End":"06:44.040","Text":"and then equals minus 2."},{"Start":"06:44.040 ","End":"06:48.040","Text":"Simplify this and let\u0027s see what we get."},{"Start":"06:48.040 ","End":"06:49.745","Text":"X\u0027s on the left."},{"Start":"06:49.745 ","End":"06:52.730","Text":"We didn\u0027t do this alone in our heads. Let\u0027s see."},{"Start":"06:52.730 ","End":"07:01.620","Text":"Minus 3x plus 2x is minus x on the left,"},{"Start":"07:01.620 ","End":"07:03.270","Text":"numbers on the right."},{"Start":"07:03.270 ","End":"07:09.420","Text":"Let\u0027s see. On the left I have 6 plus 3 is 9."},{"Start":"07:09.420 ","End":"07:13.835","Text":"If I throw the 9 over to the right-hand side,"},{"Start":"07:13.835 ","End":"07:16.970","Text":"I get minus 11,"},{"Start":"07:16.970 ","End":"07:19.400","Text":"which means if minus x is minus 11,"},{"Start":"07:19.400 ","End":"07:22.255","Text":"it means that x equals 11."},{"Start":"07:22.255 ","End":"07:25.700","Text":"Now I have to rule this out right away"},{"Start":"07:25.700 ","End":"07:31.805","Text":"because I am in this interval x less than minus 1.5,"},{"Start":"07:31.805 ","End":"07:36.560","Text":"and I couldn\u0027t possibly get an 11 in this interval so I"},{"Start":"07:36.560 ","End":"07:41.180","Text":"cross this out and you write some words like not in the range,"},{"Start":"07:41.180 ","End":"07:43.955","Text":"not in the interval or anyway."},{"Start":"07:43.955 ","End":"07:47.420","Text":"Now let\u0027s go on to the next interval,"},{"Start":"07:47.420 ","End":"07:49.685","Text":"and hope maybe we\u0027ll find something there."},{"Start":"07:49.685 ","End":"07:55.445","Text":"Once again, I\u0027m looking at this original equation 3 is the 3,"},{"Start":"07:55.445 ","End":"07:59.095","Text":"and let\u0027s see what is the absolute value of 2 minus x."},{"Start":"07:59.095 ","End":"08:04.050","Text":"Absolute value of 2 minus x. I\u0027m still less than or equal"},{"Start":"08:04.050 ","End":"08:09.870","Text":"to 2 so this is still 2 minus x minus."},{"Start":"08:09.870 ","End":"08:13.180","Text":"Now absolute value of 2x plus 3."},{"Start":"08:13.180 ","End":"08:21.305","Text":"This is defined in this interval now bigger or equal to 1.5 as 2x plus 3."},{"Start":"08:21.305 ","End":"08:28.870","Text":"This time it\u0027s 2x plus 3 with pluses in the brackets and still minus 2."},{"Start":"08:28.870 ","End":"08:31.280","Text":"Let\u0027s see what this gives us. Let\u0027s see."},{"Start":"08:31.280 ","End":"08:33.185","Text":"It\u0027s collect x\u0027s on the left."},{"Start":"08:33.185 ","End":"08:43.140","Text":"On the left I have minus 3x minus 2x is minus 5x."},{"Start":"08:43.140 ","End":"08:45.780","Text":"Let\u0027s see. Numbers on the right, but first of all,"},{"Start":"08:45.780 ","End":"08:49.910","Text":"on the left I have 6 minus 3 is 3."},{"Start":"08:49.910 ","End":"08:52.190","Text":"If I bring it to the right,"},{"Start":"08:52.190 ","End":"08:56.175","Text":"it\u0027s minus 2 minus 3 is minus 5,"},{"Start":"08:56.175 ","End":"09:02.940","Text":"which gives me that x equals minus 5 over minus 5 is 1."},{"Start":"09:02.940 ","End":"09:07.130","Text":"Now is 1 inside this interval? Yes, it is."},{"Start":"09:07.130 ","End":"09:09.260","Text":"Somewhere here, I don\u0027t know where 1 is,"},{"Start":"09:09.260 ","End":"09:12.770","Text":"but it\u0027s obviously between minus a 1/2 and 2,"},{"Start":"09:12.770 ","End":"09:14.890","Text":"so this one is okay."},{"Start":"09:14.890 ","End":"09:16.815","Text":"Let me highlight this."},{"Start":"09:16.815 ","End":"09:18.530","Text":"But with absolute values,"},{"Start":"09:18.530 ","End":"09:21.870","Text":"it\u0027s always best to check in the original equation."},{"Start":"09:21.870 ","End":"09:23.363","Text":"Let\u0027s do that."},{"Start":"09:23.363 ","End":"09:25.790","Text":"But let\u0027s just do it mentally. Remember x is 1."},{"Start":"09:25.790 ","End":"09:28.625","Text":"2 minus 1 is 1."},{"Start":"09:28.625 ","End":"09:30.845","Text":"Absolute value of 1 is 1."},{"Start":"09:30.845 ","End":"09:32.983","Text":"This part is 3."},{"Start":"09:32.983 ","End":"09:36.125","Text":"Now let\u0027s see what this part is inside the absolute value."},{"Start":"09:36.125 ","End":"09:39.860","Text":"2 times 1 plus 3 is 5,"},{"Start":"09:39.860 ","End":"09:42.430","Text":"absolute value of 5 is 5."},{"Start":"09:42.430 ","End":"09:46.060","Text":"Now we\u0027re up to 3 minus 5,"},{"Start":"09:46.060 ","End":"09:49.530","Text":"and 3 minus 5 really is minus 2,"},{"Start":"09:49.530 ","End":"09:52.020","Text":"so we even extra-checked it."},{"Start":"09:52.020 ","End":"09:54.920","Text":"Now let\u0027s go to the third interval,"},{"Start":"09:54.920 ","End":"09:57.950","Text":"this part here where x is bigger than 2."},{"Start":"09:57.950 ","End":"10:00.440","Text":"What does our equation become?"},{"Start":"10:00.440 ","End":"10:03.720","Text":"Let\u0027s see now. 3,"},{"Start":"10:03.720 ","End":"10:07.380","Text":"and then absolute value of 2 minus x."},{"Start":"10:07.380 ","End":"10:16.530","Text":"Absolute value of 2 minus x in the range where x is bigger than 2 is actually x minus 2."},{"Start":"10:16.530 ","End":"10:20.190","Text":"Here I put x minus 2 minus."},{"Start":"10:20.190 ","End":"10:24.595","Text":"Now let\u0027s see. What is the absolute value of 2x plus 3."},{"Start":"10:24.595 ","End":"10:26.890","Text":"Well, when I\u0027m bigger than 2,"},{"Start":"10:26.890 ","End":"10:32.180","Text":"I\u0027m certainly bigger or equal to minus 1.5,"},{"Start":"10:32.180 ","End":"10:35.795","Text":"and so this is the top line is what I read off."},{"Start":"10:35.795 ","End":"10:37.955","Text":"It\u0027s 2x plus 3,"},{"Start":"10:37.955 ","End":"10:41.930","Text":"and still same right-hand side minus 2."},{"Start":"10:41.930 ","End":"10:42.965","Text":"What does this give us?"},{"Start":"10:42.965 ","End":"10:44.900","Text":"Again, we\u0027re going to collect x\u0027s on the left."},{"Start":"10:44.900 ","End":"10:50.910","Text":"Here I have 3x minus 2x is x. Let\u0027s see."},{"Start":"10:50.910 ","End":"10:56.220","Text":"Numbers. 3 times minus 2 is minus 6,"},{"Start":"10:56.220 ","End":"10:59.415","Text":"and then we have minus 3 is minus 9,"},{"Start":"10:59.415 ","End":"11:02.570","Text":"but it goes over to the other side as plus 9,"},{"Start":"11:02.570 ","End":"11:05.075","Text":"so that gives us 7."},{"Start":"11:05.075 ","End":"11:12.553","Text":"Now 7 certainly is in the range x bigger than 2 so this gives us another solution,"},{"Start":"11:12.553 ","End":"11:14.450","Text":"and I\u0027ll leave it to you."},{"Start":"11:14.450 ","End":"11:17.510","Text":"It\u0027s optional to check that really,"},{"Start":"11:17.510 ","End":"11:20.650","Text":"this satisfies this equation."},{"Start":"11:20.650 ","End":"11:24.030","Text":"That does it for part a."},{"Start":"11:24.030 ","End":"11:27.555","Text":"2 solutions, x equals 1 and x equals 7,"},{"Start":"11:27.555 ","End":"11:30.850","Text":"and now on to part b."}],"ID":5041},{"Watched":false,"Name":"Exercise 3 part b","Duration":"2m 23s","ChapterTopicVideoID":5040,"CourseChapterTopicPlaylistID":3874,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:10.530","Text":"Part B looks like it could be done similarly to Part A because it has 2 absolute values."},{"Start":"00:10.530 ","End":"00:15.030","Text":"We could do it by breaking up the range and finding out where"},{"Start":"00:15.030 ","End":"00:20.094","Text":"this changes definition and where this changes definition. That\u0027s possible."},{"Start":"00:20.094 ","End":"00:23.775","Text":"But I noted on the right-hand side there is a 0,"},{"Start":"00:23.775 ","End":"00:30.145","Text":"so I want to exploit this and just bring this over to the other side,"},{"Start":"00:30.145 ","End":"00:32.970","Text":"and then we\u0027ll get x-1,"},{"Start":"00:32.970 ","End":"00:34.170","Text":"in absolute value,"},{"Start":"00:34.170 ","End":"00:38.055","Text":"equals 3x-1, in absolute value."},{"Start":"00:38.055 ","End":"00:44.090","Text":"This is of the format absolute value of a equals absolute value of b,"},{"Start":"00:44.090 ","End":"00:53.650","Text":"and this also has a solution that a=b or a=-b."},{"Start":"00:53.650 ","End":"00:56.300","Text":"Because if 2 numbers have the same absolute value,"},{"Start":"00:56.300 ","End":"00:59.240","Text":"they have the same magnitude if you ignore the sign,"},{"Start":"00:59.240 ","End":"01:02.410","Text":"and they could be the same sign or opposite signs."},{"Start":"01:02.410 ","End":"01:05.250","Text":"We get, let\u0027s see,"},{"Start":"01:05.250 ","End":"01:13.095","Text":"either x-1=3x-1 or x-1"},{"Start":"01:13.095 ","End":"01:18.165","Text":"is minus this, which is -3x+1."},{"Start":"01:18.165 ","End":"01:24.975","Text":"In this case, we get -2x=0,"},{"Start":"01:24.975 ","End":"01:26.640","Text":"this cancels with this,"},{"Start":"01:26.640 ","End":"01:29.700","Text":"and so x is 0."},{"Start":"01:29.700 ","End":"01:33.365","Text":"The other possibility, let\u0027s see,"},{"Start":"01:33.365 ","End":"01:36.080","Text":"x+3x is 4x,"},{"Start":"01:36.080 ","End":"01:43.320","Text":"1+1 is 2, x=1/2."},{"Start":"01:43.320 ","End":"01:45.575","Text":"Just to be sure,"},{"Start":"01:45.575 ","End":"01:47.930","Text":"we\u0027ll substitute in the original,"},{"Start":"01:47.930 ","End":"01:49.010","Text":"just do it mentally,"},{"Start":"01:49.010 ","End":"01:51.155","Text":"if x is 0,"},{"Start":"01:51.155 ","End":"01:54.890","Text":"we\u0027ve got absolute value of -1,"},{"Start":"01:54.890 ","End":"01:57.940","Text":"less absolute value of -1,"},{"Start":"01:57.940 ","End":"02:00.265","Text":"which is certainly 0."},{"Start":"02:00.265 ","End":"02:05.990","Text":"If x is 1/2, and we have absolute value of -1/2,"},{"Start":"02:05.990 ","End":"02:10.550","Text":"less absolute value of 1/2-1 is 1/2."},{"Start":"02:10.550 ","End":"02:13.775","Text":"Again, we get 1/2-1/2 is 0."},{"Start":"02:13.775 ","End":"02:16.470","Text":"Both of these are okay,"},{"Start":"02:16.470 ","End":"02:20.160","Text":"and these are the 2 solutions of part b,"},{"Start":"02:20.160 ","End":"02:23.050","Text":"and so we\u0027re done with that."}],"ID":5042},{"Watched":false,"Name":"Exercise 4","Duration":"17m 29s","ChapterTopicVideoID":5041,"CourseChapterTopicPlaylistID":3874,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"In this exercise, we have a pair of equations with"},{"Start":"00:03.690 ","End":"00:07.980","Text":"absolute values and they each have 2 absolute values."},{"Start":"00:07.980 ","End":"00:13.980","Text":"In both cases we\u0027re going to use the technique of breaking the range of the number line."},{"Start":"00:13.980 ","End":"00:16.245","Text":"Let\u0027s start with the first 1,"},{"Start":"00:16.245 ","End":"00:22.470","Text":"and let\u0027s look at each of the absolute value bits and see how to define it."},{"Start":"00:22.470 ","End":"00:28.620","Text":"Start with the absolute value of x+3, from the definition,"},{"Start":"00:28.620 ","End":"00:32.145","Text":"this is equal to x+3 itself,"},{"Start":"00:32.145 ","End":"00:35.910","Text":"provided that x+3 is bigger or equal to 0,"},{"Start":"00:35.910 ","End":"00:38.175","Text":"and it\u0027s equal to minus this."},{"Start":"00:38.175 ","End":"00:41.715","Text":"I\u0027ll just put it as a -(x+3)."},{"Start":"00:41.715 ","End":"00:45.180","Text":"If x+3 is negative,"},{"Start":"00:45.180 ","End":"00:48.080","Text":"then the absolute value is minus it,"},{"Start":"00:48.080 ","End":"00:50.420","Text":"minus the negative is the positive."},{"Start":"00:50.420 ","End":"00:52.040","Text":"What I\u0027d like to do though,"},{"Start":"00:52.040 ","End":"00:58.185","Text":"is just change the conditions on just x bigger or less than or whatever."},{"Start":"00:58.185 ","End":"01:03.720","Text":"This part\u0027s the same x+3,"},{"Start":"01:03.720 ","End":"01:06.650","Text":"but the condition x+3 bigger or equal to 0,"},{"Start":"01:06.650 ","End":"01:10.115","Text":"I\u0027ll write it as x bigger or equal to -3."},{"Start":"01:10.115 ","End":"01:11.480","Text":"Let\u0027s bring the 3 over."},{"Start":"01:11.480 ","End":"01:20.305","Text":"The other 1 will be minus x+3, provided that x\u003c-3."},{"Start":"01:20.305 ","End":"01:28.520","Text":"Now, the other 1, absolute value of x-1 is equal to either x-1 itself,"},{"Start":"01:28.520 ","End":"01:32.675","Text":"provided that this is non-negative or bigger or equal to 0."},{"Start":"01:32.675 ","End":"01:34.070","Text":"The opposite of it,"},{"Start":"01:34.070 ","End":"01:37.345","Text":"the negative of it -x-1,"},{"Start":"01:37.345 ","End":"01:40.240","Text":"if it happens to be less than 0."},{"Start":"01:40.240 ","End":"01:41.810","Text":"Again, we\u0027ll rewrite this,"},{"Start":"01:41.810 ","End":"01:44.225","Text":"the condition on x is clearer."},{"Start":"01:44.225 ","End":"01:49.804","Text":"It\u0027s x-1, provided that x is bigger or equal to 1"},{"Start":"01:49.804 ","End":"01:59.130","Text":"and -x-1 in the case that x is less than 1."},{"Start":"01:59.130 ","End":"02:03.200","Text":"We have 2 numbers that we\u0027re going to put on the number line."},{"Start":"02:03.200 ","End":"02:06.320","Text":"We\u0027re going to put -3 there and we\u0027re going to put 1 there."},{"Start":"02:06.320 ","End":"02:10.895","Text":"These are the 2 and they\u0027re going to break up our number line into 3 sections."},{"Start":"02:10.895 ","End":"02:16.790","Text":"Here provided a quick sketch of a number line with the 2 numbers on it,"},{"Start":"02:16.790 ","End":"02:19.340","Text":"it\u0027s important that they should be in the right order,"},{"Start":"02:19.340 ","End":"02:22.880","Text":"-3 is on the left and 1 is on the right."},{"Start":"02:22.880 ","End":"02:26.810","Text":"Now we describe each of the 3 intervals."},{"Start":"02:26.810 ","End":"02:30.555","Text":"This 1 would be x."},{"Start":"02:30.555 ","End":"02:35.630","Text":"First of all, write \u003c-3 and then see if it doesn\u0027t include, it doesn\u0027t include."},{"Start":"02:35.630 ","End":"02:38.510","Text":"Well, if here it doesn\u0027t include the -3,"},{"Start":"02:38.510 ","End":"02:40.250","Text":"then here also not."},{"Start":"02:40.250 ","End":"02:44.155","Text":"On the other side, the -3 would be included."},{"Start":"02:44.155 ","End":"02:46.515","Text":"It goes up to 1,"},{"Start":"02:46.515 ","End":"02:48.810","Text":"but the 1 here is not included."},{"Start":"02:48.810 ","End":"02:50.300","Text":"This is how it is."},{"Start":"02:50.300 ","End":"02:54.410","Text":"That\u0027s an extent revolve from 3 inclusive to 1 non-inclusive."},{"Start":"02:54.410 ","End":"02:56.750","Text":"Then here we would just have the 1,"},{"Start":"02:56.750 ","End":"03:00.320","Text":"would have to be included x bigger or equal to 1."},{"Start":"03:00.320 ","End":"03:03.035","Text":"Now for each of these 3 ranges,"},{"Start":"03:03.035 ","End":"03:05.840","Text":"the absolute value expressions are going to be either"},{"Start":"03:05.840 ","End":"03:09.305","Text":"the upper or the lower definition, that\u0027s column."},{"Start":"03:09.305 ","End":"03:12.760","Text":"Just make it a little bit more room."},{"Start":"03:12.760 ","End":"03:16.760","Text":"Let\u0027s look what happens in each of the 3 ranges."},{"Start":"03:16.760 ","End":"03:20.390","Text":"The first 1 is x\u003c-3,"},{"Start":"03:20.390 ","End":"03:23.615","Text":"let\u0027s just write them all now and then we\u0027ll see."},{"Start":"03:23.615 ","End":"03:25.240","Text":"Well, I\u0027ve got the black ink,"},{"Start":"03:25.240 ","End":"03:27.375","Text":"-3 less than or equal to x\u003c1."},{"Start":"03:27.375 ","End":"03:34.965","Text":"X bigger or equal to 1 colon, colon."},{"Start":"03:34.965 ","End":"03:37.855","Text":"When we\u0027re \u003c-3,"},{"Start":"03:37.855 ","End":"03:45.110","Text":"then this absolute value of x+3 would be from the bottom row."},{"Start":"03:45.110 ","End":"03:52.050","Text":"For \u003c-3, we\u0027re certainly less than 1 and so this is also from the bottom row."},{"Start":"03:52.050 ","End":"03:53.510","Text":"I\u0027m taking from here and here."},{"Start":"03:53.510 ","End":"03:56.540","Text":"What I mean is I\u0027m going to look this equation."},{"Start":"03:56.540 ","End":"04:02.305","Text":"I\u0027ll highlight it and I\u0027m going to transcribe it without the absolute values."},{"Start":"04:02.305 ","End":"04:07.625","Text":"We start out with x+3 has nothing to do with absolute values plus."},{"Start":"04:07.625 ","End":"04:09.950","Text":"Then I have the absolute value of x+3."},{"Start":"04:09.950 ","End":"04:13.955","Text":"We already decided that we\u0027re taking it off the bottom row."},{"Start":"04:13.955 ","End":"04:23.570","Text":"It\u0027s not plus, it\u0027s actually minus, then (x+3)-(x+3)=2."},{"Start":"04:23.570 ","End":"04:25.205","Text":"Now the x-1,"},{"Start":"04:25.205 ","End":"04:28.910","Text":"we also said that it\u0027s from the bottom row."},{"Start":"04:28.910 ","End":"04:32.030","Text":"It\u0027s -2(x-1) this."},{"Start":"04:32.030 ","End":"04:36.430","Text":"I\u0027ll put the minus here, the x-1 here."},{"Start":"04:36.430 ","End":"04:39.140","Text":"This is the equation we get for this range,"},{"Start":"04:39.140 ","End":"04:41.240","Text":"that\u0027s continue to develop it."},{"Start":"04:41.240 ","End":"04:44.105","Text":"All the x\u0027s on the left."},{"Start":"04:44.105 ","End":"04:53.165","Text":"Let\u0027s see, we have x-x just cancels and then we have -2x so it\u0027s +2x on the left."},{"Start":"04:53.165 ","End":"04:54.650","Text":"What else do we have?"},{"Start":"04:54.650 ","End":"05:02.564","Text":"3-3 cancels, and we\u0027re left with -2 times -1 is +2 on the right,"},{"Start":"05:02.564 ","End":"05:05.705","Text":"which gives us that x=1."},{"Start":"05:05.705 ","End":"05:11.340","Text":"Now we have to check is x=1 in this range that we\u0027re looking for? The answer is, no."},{"Start":"05:11.340 ","End":"05:16.910","Text":"1 is not \u003c-3 so this 1 is ruled out."},{"Start":"05:16.910 ","End":"05:21.050","Text":"We write some reason not in the range of something."},{"Start":"05:21.050 ","End":"05:24.980","Text":"Now let\u0027s go to the next and maybe we\u0027ll find the solution here."},{"Start":"05:24.980 ","End":"05:30.064","Text":"Once again, I interpret this equation without the absolute values,"},{"Start":"05:30.064 ","End":"05:32.975","Text":"starts out the same x+3."},{"Start":"05:32.975 ","End":"05:36.250","Text":"Now the absolute value of x+3,"},{"Start":"05:36.250 ","End":"05:39.925","Text":"I\u0027m still less than 1,"},{"Start":"05:39.925 ","End":"05:44.917","Text":"which means that the other absolute value is from here but"},{"Start":"05:44.917 ","End":"05:52.175","Text":"the x+3 now changes because I\u0027m now bigger or equal to -3."},{"Start":"05:52.175 ","End":"05:55.195","Text":"It\u0027s plus x+3."},{"Start":"05:55.195 ","End":"05:59.425","Text":"This absolute value is +x+3."},{"Start":"05:59.425 ","End":"06:06.925","Text":"As I said, the other 1 stays as -x-1 because I\u0027m still below 1."},{"Start":"06:06.925 ","End":"06:09.365","Text":"This 1 changes sign, this 1 doesn\u0027t."},{"Start":"06:09.365 ","End":"06:17.205","Text":"This gives us let\u0027s see on the left 2x on the right -2 comes over, so that\u0027s 4x."},{"Start":"06:17.205 ","End":"06:24.045","Text":"On the left, we have 6 on the right +2 to -6,"},{"Start":"06:24.045 ","End":"06:32.450","Text":"-4, which means that x is -2 and -2 is in this range."},{"Start":"06:32.450 ","End":"06:35.225","Text":"I don\u0027t know exactly somewhere here anyway."},{"Start":"06:35.225 ","End":"06:40.130","Text":"-2 is between -3 and 1 so this one\u0027s okay."},{"Start":"06:40.130 ","End":"06:47.915","Text":"This 1 gets so highlight and let\u0027s just to be safe mentally check the -2 fits the bill."},{"Start":"06:47.915 ","End":"06:49.640","Text":"If it satisfies the equation,"},{"Start":"06:49.640 ","End":"06:55.700","Text":"so to speak, -2+3=1."},{"Start":"06:55.700 ","End":"06:57.570","Text":"Again, -2+3 is 1."},{"Start":"06:57.570 ","End":"06:59.580","Text":"Absolute value of 1 is 1."},{"Start":"06:59.580 ","End":"07:02.295","Text":"On the left we have 1+1=2."},{"Start":"07:02.295 ","End":"07:08.840","Text":"On the other hand, -2-1=-3."},{"Start":"07:08.840 ","End":"07:10.460","Text":"Absolute value is 3,"},{"Start":"07:10.460 ","End":"07:13.595","Text":"2 times 3 is 6."},{"Start":"07:13.595 ","End":"07:18.070","Text":"No, it does not satisfy the equation."},{"Start":"07:18.070 ","End":"07:22.835","Text":"There we were looking good for a moment because it was in range,"},{"Start":"07:22.835 ","End":"07:27.890","Text":"but still gets eliminated because it doesn\u0027t satisfy the equation."},{"Start":"07:27.890 ","End":"07:31.280","Text":"Maybe we\u0027ll still get lucky with the last 1."},{"Start":"07:31.280 ","End":"07:35.465","Text":"We started off with x+3 because that\u0027s not an absolute value."},{"Start":"07:35.465 ","End":"07:37.400","Text":"Now this 1,"},{"Start":"07:37.400 ","End":"07:44.600","Text":"we take from the top line because it\u0027s still bigger than -3."},{"Start":"07:44.600 ","End":"07:46.415","Text":"We\u0027re a lot bigger already."},{"Start":"07:46.415 ","End":"07:52.740","Text":"This is +x+3= -2, or rather 2."},{"Start":"07:52.740 ","End":"07:58.585","Text":"This time, since where bigger or equal to 1,"},{"Start":"07:58.585 ","End":"08:07.445","Text":"then we take the definition of this absolute value from here without the x-1."},{"Start":"08:07.445 ","End":"08:09.560","Text":"This gives us, let\u0027s see."},{"Start":"08:09.560 ","End":"08:15.394","Text":"The left-hand side is 2x and on the right-hand side,"},{"Start":"08:15.394 ","End":"08:17.675","Text":"we also have 2x."},{"Start":"08:17.675 ","End":"08:24.235","Text":"That means the x\u0027s and 3+3 is 6."},{"Start":"08:24.235 ","End":"08:26.640","Text":"Here we have -2."},{"Start":"08:26.640 ","End":"08:29.935","Text":"We get 8=0,"},{"Start":"08:29.935 ","End":"08:33.875","Text":"which is certainly not a solution."},{"Start":"08:33.875 ","End":"08:38.180","Text":"Which means that this equation basically has no solutions at all regardless of"},{"Start":"08:38.180 ","End":"08:42.860","Text":"what range or an haven\u0027t even got to the range part because the x\u0027s canceled out."},{"Start":"08:42.860 ","End":"08:47.015","Text":"Or we could leave it as 6=-2 rather,"},{"Start":"08:47.015 ","End":"08:50.485","Text":"whatever way you do it has no solution."},{"Start":"08:50.485 ","End":"08:52.670","Text":"I\u0027ll just write that."},{"Start":"08:52.670 ","End":"08:54.199","Text":"When you get to a contradiction,"},{"Start":"08:54.199 ","End":"08:56.900","Text":"it means that this equation has no solution."},{"Start":"08:56.900 ","End":"09:00.680","Text":"We didn\u0027t find anything anywhere for different reasons."},{"Start":"09:00.680 ","End":"09:02.675","Text":"This was outside the range."},{"Start":"09:02.675 ","End":"09:04.580","Text":"This was in the range,"},{"Start":"09:04.580 ","End":"09:07.445","Text":"but the original equation wasn\u0027t satisfied."},{"Start":"09:07.445 ","End":"09:13.075","Text":"Here we couldn\u0027t even get a candidate because this part had no solution."},{"Start":"09:13.075 ","End":"09:17.900","Text":"There we are altogether it\u0027s summarizes as no solution,"},{"Start":"09:17.900 ","End":"09:20.420","Text":"and so on to part B."},{"Start":"09:20.420 ","End":"09:26.165","Text":"Let\u0027s see, we have better luck there finding a solution. That\u0027s what we want to do."},{"Start":"09:26.165 ","End":"09:30.080","Text":"Again, we have 2 absolute values, but in fact,"},{"Start":"09:30.080 ","End":"09:34.685","Text":"the extra absolute value that surround both of them here and here"},{"Start":"09:34.685 ","End":"09:39.860","Text":"is redundant because you can\u0027t get something negative inside it."},{"Start":"09:39.860 ","End":"09:42.530","Text":"This has to be 0 or positive,"},{"Start":"09:42.530 ","End":"09:45.040","Text":"this 1 has to be 0 or positive and when you add them,"},{"Start":"09:45.040 ","End":"09:46.990","Text":"it\u0027s still 0 or positive."},{"Start":"09:46.990 ","End":"09:54.855","Text":"We can really just write it as the absolute value of x-3 plus absolute value of 2x-1=4."},{"Start":"09:54.855 ","End":"09:59.615","Text":"Just dispense with those to trick you or make it more difficult."},{"Start":"09:59.615 ","End":"10:04.735","Text":"As before we write down the definitions."},{"Start":"10:04.735 ","End":"10:08.470","Text":"Now, I can show you a shortcut."},{"Start":"10:08.470 ","End":"10:12.340","Text":"We don\u0027t have to go through the whole definitions thing."},{"Start":"10:12.340 ","End":"10:16.735","Text":"We can actually just see where each of these is 0."},{"Start":"10:16.735 ","End":"10:18.730","Text":"We continue as before,"},{"Start":"10:18.730 ","End":"10:22.300","Text":"we write down the definition of each of the absolute values."},{"Start":"10:22.300 ","End":"10:27.595","Text":"So Let\u0027s see, x minus 3 is either x minus 3 itself or"},{"Start":"10:27.595 ","End":"10:34.134","Text":"minus x minus 3 depending on whether x minus 3 is bigger or equal to 0,"},{"Start":"10:34.134 ","End":"10:38.230","Text":"or x minus 3 is less than 0."},{"Start":"10:38.230 ","End":"10:42.970","Text":"I\u0027m just going to rewrite this as more conveniently as x"},{"Start":"10:42.970 ","End":"10:48.160","Text":"minus 3 if x is bigger or equal to 3,"},{"Start":"10:48.160 ","End":"10:54.355","Text":"and minus of x minus 3 if x is less than 3."},{"Start":"10:54.355 ","End":"10:56.215","Text":"And a similar thing here,"},{"Start":"10:56.215 ","End":"11:00.865","Text":"absolute value of 2x minus 1 is equal to,"},{"Start":"11:00.865 ","End":"11:07.945","Text":"it\u0027s either 2x minus 1 if 2x minus 1 is bigger or equal to 0,"},{"Start":"11:07.945 ","End":"11:11.530","Text":"and minus 2x minus 1,"},{"Start":"11:11.530 ","End":"11:16.405","Text":"in the case that 2x minus 1 is less than 0."},{"Start":"11:16.405 ","End":"11:24.805","Text":"Again, I rewrite this as 2x minus 1 minus 2x minus 1."},{"Start":"11:24.805 ","End":"11:29.170","Text":"This condition comes out to x bigger or equal to a half,"},{"Start":"11:29.170 ","End":"11:30.835","Text":"I think you can see that."},{"Start":"11:30.835 ","End":"11:32.920","Text":"Bring the 1 over and divide by 2."},{"Start":"11:32.920 ","End":"11:35.500","Text":"You\u0027re allowed to divide by positive number."},{"Start":"11:35.500 ","End":"11:40.135","Text":"In the other case we get x less than 1/2."},{"Start":"11:40.135 ","End":"11:46.960","Text":"The 2 numbers that go on the number line would be 3 and 1/2."},{"Start":"11:46.960 ","End":"11:48.760","Text":"This one\u0027s on the left,"},{"Start":"11:48.760 ","End":"11:50.380","Text":"this one\u0027s on the right."},{"Start":"11:50.380 ","End":"11:52.900","Text":"If you don\u0027t feel like doing a drawing,"},{"Start":"11:52.900 ","End":"11:57.790","Text":"I did the drawing a couple of times in the exercises we can do without a drawing."},{"Start":"11:57.790 ","End":"11:59.140","Text":"We can just say,"},{"Start":"11:59.140 ","End":"12:04.195","Text":"we have to break up the x interval into 3 ranges."},{"Start":"12:04.195 ","End":"12:06.565","Text":"I\u0027m looking at the leftmost 1,"},{"Start":"12:06.565 ","End":"12:08.125","Text":"which is the half."},{"Start":"12:08.125 ","End":"12:10.060","Text":"I see that there,"},{"Start":"12:10.060 ","End":"12:12.775","Text":"what\u0027s left of it doesn\u0027t include the half."},{"Start":"12:12.775 ","End":"12:14.350","Text":"I have 1 range,"},{"Start":"12:14.350 ","End":"12:18.324","Text":"which is x less than a half."},{"Start":"12:18.324 ","End":"12:20.680","Text":"The other range is between the 2 numbers,"},{"Start":"12:20.680 ","End":"12:22.585","Text":"between a 1/2 and 3."},{"Start":"12:22.585 ","End":"12:27.415","Text":"So it\u0027s a half. I\u0027ll just put the x and I\u0027ll put the 3 and let\u0027s see."},{"Start":"12:27.415 ","End":"12:29.050","Text":"Here it was less than,"},{"Start":"12:29.050 ","End":"12:31.315","Text":"so here it has to be bigger or equal to,"},{"Start":"12:31.315 ","End":"12:33.625","Text":"that\u0027s this bigger or equal to here."},{"Start":"12:33.625 ","End":"12:38.380","Text":"As for the 3 on the left side of 3 doesn\u0027t include,"},{"Start":"12:38.380 ","End":"12:40.120","Text":"so this is how it goes."},{"Start":"12:40.120 ","End":"12:45.930","Text":"Then on the rightmost range it\u0027s x bigger or equal to 3."},{"Start":"12:45.930 ","End":"12:48.570","Text":"Less than a half strictly,"},{"Start":"12:48.570 ","End":"12:52.080","Text":"or from a half inclusive to 3, not inclusive."},{"Start":"12:52.080 ","End":"12:54.035","Text":"Then from 3 onwards."},{"Start":"12:54.035 ","End":"12:55.630","Text":"Now in each of these,"},{"Start":"12:55.630 ","End":"12:59.590","Text":"I can write what the equation is."},{"Start":"12:59.590 ","End":"13:05.485","Text":"I\u0027ll use this equation without the extra absolute values."},{"Start":"13:05.485 ","End":"13:09.355","Text":"Let\u0027s see. I\u0027ll interpret it without absolute value."},{"Start":"13:09.355 ","End":"13:11.530","Text":"When we\u0027re less than a half,"},{"Start":"13:11.530 ","End":"13:16.270","Text":"then we read from the bottom line for this 1."},{"Start":"13:16.270 ","End":"13:19.750","Text":"For less than a half is certainly less than 3 also."},{"Start":"13:19.750 ","End":"13:22.330","Text":"So it\u0027s the bottom line for this 1."},{"Start":"13:22.330 ","End":"13:31.870","Text":"The equation comes out to be absolute value of x minus 3 is minus x minus 3,"},{"Start":"13:31.870 ","End":"13:34.750","Text":"and absolute value of 2x minus 1,"},{"Start":"13:34.750 ","End":"13:38.320","Text":"can we set the bottom line is minus 2x minus 1."},{"Start":"13:38.320 ","End":"13:44.695","Text":"So I\u0027ll just change this plus to a minus and then we have a 2x minus 1,"},{"Start":"13:44.695 ","End":"13:47.350","Text":"and it\u0027s equal to 4."},{"Start":"13:47.350 ","End":"13:49.945","Text":"Now what does this equation giving us?"},{"Start":"13:49.945 ","End":"13:57.865","Text":"X is on the left minus x minus 2x is minus 3x."},{"Start":"13:57.865 ","End":"14:01.885","Text":"Then we have numbers on the right."},{"Start":"14:01.885 ","End":"14:07.975","Text":"But on the left we have 3 plus 1 is 4 on the right."},{"Start":"14:07.975 ","End":"14:13.195","Text":"This is equal to 0. This gives us x equals 0."},{"Start":"14:13.195 ","End":"14:16.584","Text":"Now is x equals 0 in the range x less than a half?"},{"Start":"14:16.584 ","End":"14:21.145","Text":"Yes. I\u0027ll temporarily highlighted it."},{"Start":"14:21.145 ","End":"14:22.825","Text":"It looks promising."},{"Start":"14:22.825 ","End":"14:28.135","Text":"I just have to do the final check of putting it in the original equation."},{"Start":"14:28.135 ","End":"14:30.250","Text":"Let\u0027s see if x is 0,"},{"Start":"14:30.250 ","End":"14:34.560","Text":"absolute value of 0 minus 3 is 3."},{"Start":"14:34.560 ","End":"14:38.055","Text":"Absolute value of twice 0 minus 1 is 1,"},{"Start":"14:38.055 ","End":"14:39.960","Text":"3 plus 1 is 4."},{"Start":"14:39.960 ","End":"14:42.165","Text":"Yes, this 1."},{"Start":"14:42.165 ","End":"14:43.710","Text":"Give it a check-mark."},{"Start":"14:43.710 ","End":"14:46.925","Text":"Yes, it\u0027s been definitely checked in the original."},{"Start":"14:46.925 ","End":"14:49.975","Text":"The next 1. In this range,"},{"Start":"14:49.975 ","End":"14:53.515","Text":"x is bigger or equal to a half, which is this."},{"Start":"14:53.515 ","End":"14:54.895","Text":"It\u0027s the top row,"},{"Start":"14:54.895 ","End":"14:58.000","Text":"but it\u0027s still less than 3 is the bottom row for this."},{"Start":"14:58.000 ","End":"15:01.285","Text":"So these are the 2 definitions we take."},{"Start":"15:01.285 ","End":"15:09.565","Text":"Our equation becomes x minus 3 from here is minus x minus 3, same as above."},{"Start":"15:09.565 ","End":"15:12.175","Text":"But this 1 now is from the top."},{"Start":"15:12.175 ","End":"15:15.130","Text":"It\u0027s plus 2x minus 1,"},{"Start":"15:15.130 ","End":"15:16.765","Text":"and it\u0027s still 4."},{"Start":"15:16.765 ","End":"15:18.490","Text":"What does that give us?"},{"Start":"15:18.490 ","End":"15:23.125","Text":"Minus x plus 2x is x."},{"Start":"15:23.125 ","End":"15:28.675","Text":"The numbers, let\u0027s see on the left I have 3 minus 1 is 2."},{"Start":"15:28.675 ","End":"15:31.990","Text":"The 2 then can come over as minus 2."},{"Start":"15:31.990 ","End":"15:34.240","Text":"I get x equals 2,"},{"Start":"15:34.240 ","End":"15:36.280","Text":"and is 2 in this range?"},{"Start":"15:36.280 ","End":"15:38.080","Text":"Certainly is."},{"Start":"15:38.080 ","End":"15:45.340","Text":"I\u0027ll highlight it, but still need the final okay by putting in the original equation."},{"Start":"15:45.340 ","End":"15:51.145","Text":"2 minus 3 is minus 1 and the absolute value is 1."},{"Start":"15:51.145 ","End":"15:54.580","Text":"This 1 is twice,"},{"Start":"15:54.580 ","End":"15:56.650","Text":"2 minus 1 is 3,"},{"Start":"15:56.650 ","End":"16:00.205","Text":"and the absolute value is 3.1 plus 3 is 4."},{"Start":"16:00.205 ","End":"16:02.275","Text":"Yes, indeed it is."},{"Start":"16:02.275 ","End":"16:04.915","Text":"This 1 is okay too."},{"Start":"16:04.915 ","End":"16:08.500","Text":"Now let\u0027s check what we have in this range."},{"Start":"16:08.500 ","End":"16:11.755","Text":"When x is bigger or equal to 3, we\u0027re here,"},{"Start":"16:11.755 ","End":"16:16.359","Text":"but we\u0027re also definitely bigger or equal to a half, so we\u0027re also here."},{"Start":"16:16.359 ","End":"16:19.720","Text":"So we\u0027re reading off the top line in each of them."},{"Start":"16:19.720 ","End":"16:23.965","Text":"We get plus x minus 3,"},{"Start":"16:23.965 ","End":"16:26.320","Text":"I mean just this x minus 3."},{"Start":"16:26.320 ","End":"16:33.190","Text":"And then plus this 2x minus 1 equals 4."},{"Start":"16:33.190 ","End":"16:34.600","Text":"What does that give us?"},{"Start":"16:34.600 ","End":"16:42.100","Text":"X plus 2x is 3x and on the left I have minus 3 minus 1,"},{"Start":"16:42.100 ","End":"16:43.960","Text":"which is minus 4,"},{"Start":"16:43.960 ","End":"16:46.480","Text":"but it comes over as plus 4,"},{"Start":"16:46.480 ","End":"16:48.880","Text":"which is a total of 8."},{"Start":"16:48.880 ","End":"16:55.450","Text":"Which means that x is equal to 8/3,"},{"Start":"16:55.450 ","End":"16:58.435","Text":"I have somehow gotten crooked here."},{"Start":"16:58.435 ","End":"17:02.605","Text":"Never mind. That is equal to,"},{"Start":"17:02.605 ","End":"17:04.990","Text":"3 goes into 8 twice,"},{"Start":"17:04.990 ","End":"17:08.395","Text":"and remainder 2 and 2/3."},{"Start":"17:08.395 ","End":"17:11.530","Text":"That is definitely less than 3."},{"Start":"17:11.530 ","End":"17:16.315","Text":"So it is not in the range x bigger or equal to 3."},{"Start":"17:16.315 ","End":"17:19.465","Text":"This 1 needs to be disqualified."},{"Start":"17:19.465 ","End":"17:23.725","Text":"Line through it as not being in the range."},{"Start":"17:23.725 ","End":"17:25.795","Text":"We have 2 answers,"},{"Start":"17:25.795 ","End":"17:29.690","Text":"0 or 2, and that\u0027s it."}],"ID":5043},{"Watched":false,"Name":"Exercise 5","Duration":"11m 16s","ChapterTopicVideoID":5042,"CourseChapterTopicPlaylistID":3874,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"We have here another pair of equations with absolute values."},{"Start":"00:04.890 ","End":"00:07.650","Text":"In each case we have 2 absolute values,"},{"Start":"00:07.650 ","End":"00:10.245","Text":"so we know how to handle these."},{"Start":"00:10.245 ","End":"00:14.250","Text":"We break up the number line into 3 intervals."},{"Start":"00:14.250 ","End":"00:16.125","Text":"Let\u0027s start with the first."},{"Start":"00:16.125 ","End":"00:21.855","Text":"We define each of the absolute values in terms of the definition."},{"Start":"00:21.855 ","End":"00:28.170","Text":"Like absolute value of 5 plus x is either 5 plus x itself,"},{"Start":"00:28.170 ","End":"00:30.915","Text":"or minus 5 plus x,"},{"Start":"00:30.915 ","End":"00:38.075","Text":"depending on whether 5 plus x is bigger or equal to 0 or whether it\u0027s less than 0."},{"Start":"00:38.075 ","End":"00:44.520","Text":"Similarly, the absolute value of 3x minus 2 is either"},{"Start":"00:44.520 ","End":"00:51.195","Text":"3x minus 2 or minus 3x minus 2."},{"Start":"00:51.195 ","End":"00:54.720","Text":"Depending on whether 3x minus 2 is bigger,"},{"Start":"00:54.720 ","End":"00:59.735","Text":"or equal to 0, or 3x minus 2 is less than 0."},{"Start":"00:59.735 ","End":"01:04.520","Text":"Let\u0027s just rewrite this in a slightly more convenient form."},{"Start":"01:04.520 ","End":"01:10.145","Text":"The absolute value of 5 plus x is equal to"},{"Start":"01:10.145 ","End":"01:16.380","Text":"5 plus x if x is bigger or equal to minus 5."},{"Start":"01:16.380 ","End":"01:19.744","Text":"I bring the minus 5 over to the other side,"},{"Start":"01:19.744 ","End":"01:28.065","Text":"and it\u0027ll be minus 5 plus x if x is less than minus 5."},{"Start":"01:28.065 ","End":"01:33.555","Text":"This one we can write as 3x minus 2."},{"Start":"01:33.555 ","End":"01:35.190","Text":"Let\u0027s see what this becomes."},{"Start":"01:35.190 ","End":"01:37.650","Text":"3x bigger, or equal to 2,"},{"Start":"01:37.650 ","End":"01:40.590","Text":"x bigger or equal to 2/3."},{"Start":"01:40.590 ","End":"01:45.065","Text":"Just bring the 2 over and you\u0027re allowed to divide by a positive number,"},{"Start":"01:45.065 ","End":"01:49.975","Text":"and it\u0027s minus 3x minus 2,"},{"Start":"01:49.975 ","End":"01:51.350","Text":"that would be in the other case,"},{"Start":"01:51.350 ","End":"01:54.095","Text":"x less than 2/3."},{"Start":"01:54.095 ","End":"02:00.280","Text":"The interesting numbers, minus 5 and 2/3."},{"Start":"02:00.280 ","End":"02:03.830","Text":"These are the ones I would put on the number line."},{"Start":"02:03.830 ","End":"02:06.245","Text":"I don\u0027t actually have to draw the sketch,"},{"Start":"02:06.245 ","End":"02:10.670","Text":"I will this time a line with an arrow,"},{"Start":"02:10.670 ","End":"02:12.815","Text":"and the label x."},{"Start":"02:12.815 ","End":"02:14.960","Text":"Then these 2 values,"},{"Start":"02:14.960 ","End":"02:20.045","Text":"it doesn\u0027t matter the scale just as long as you get them in the right order."},{"Start":"02:20.045 ","End":"02:24.445","Text":"Certainly, minus 5 is less than 2/3,"},{"Start":"02:24.445 ","End":"02:25.860","Text":"so something like this,"},{"Start":"02:25.860 ","End":"02:28.009","Text":"and then the range has become,"},{"Start":"02:28.009 ","End":"02:36.545","Text":"this one is to the left of minus 5 so it\u0027s this one x less than minus 5."},{"Start":"02:36.545 ","End":"02:42.510","Text":"Here we have minus 5 is less than or equal to x,"},{"Start":"02:42.510 ","End":"02:44.745","Text":"and then when we get to the 2/3,"},{"Start":"02:44.745 ","End":"02:46.755","Text":"less than 2/3,"},{"Start":"02:46.755 ","End":"02:51.960","Text":"and then bigger or equal to 2/3."},{"Start":"02:51.960 ","End":"02:53.675","Text":"Each of these ranges,"},{"Start":"02:53.675 ","End":"02:58.190","Text":"we have to decide which definition of the absolute value to take the top one,"},{"Start":"02:58.190 ","End":"02:59.435","Text":"or the bottom one."},{"Start":"02:59.435 ","End":"03:01.985","Text":"Let\u0027s start with the first range."},{"Start":"03:01.985 ","End":"03:08.955","Text":"The one that is x less than minus 5. We\u0027ll do all of them."},{"Start":"03:08.955 ","End":"03:14.745","Text":"After that, we\u0027ll do minus 5 less than or equal to x less than 2/3,"},{"Start":"03:14.745 ","End":"03:17.670","Text":"and after that, we\u0027ll do x bigger,"},{"Start":"03:17.670 ","End":"03:19.645","Text":"or equal to 2/3."},{"Start":"03:19.645 ","End":"03:21.645","Text":"Just label them well first,"},{"Start":"03:21.645 ","End":"03:23.535","Text":"let\u0027s see what we have."},{"Start":"03:23.535 ","End":"03:26.960","Text":"We\u0027re going to interpret this equation without"},{"Start":"03:26.960 ","End":"03:30.680","Text":"the absolute value in each of these 3 cases."},{"Start":"03:30.680 ","End":"03:34.130","Text":"Now if I\u0027m less than minus 5 then I\u0027m certainly"},{"Start":"03:34.130 ","End":"03:40.325","Text":"less than 2/3 so both this inequality holds,"},{"Start":"03:40.325 ","End":"03:46.195","Text":"and this inequality so I\u0027m taking them both from the bottom line,"},{"Start":"03:46.195 ","End":"03:50.337","Text":"and so this equation becomes 5 plus x,"},{"Start":"03:50.337 ","End":"03:56.735","Text":"absolute value is minus 5 plus x."},{"Start":"03:56.735 ","End":"04:02.150","Text":"Then minus 1 equals absolute value of 3x minus 2."},{"Start":"04:02.150 ","End":"04:04.115","Text":"We also said it\u0027s the bottom one,"},{"Start":"04:04.115 ","End":"04:08.505","Text":"minus 3x minus 2,"},{"Start":"04:08.505 ","End":"04:10.640","Text":"and let\u0027s see what that gives us."},{"Start":"04:10.640 ","End":"04:17.595","Text":"How many x do we have on the left minus x and then plus"},{"Start":"04:17.595 ","End":"04:24.570","Text":"3x is 2x and then on the left we have minus 5,"},{"Start":"04:24.570 ","End":"04:27.300","Text":"minus 1 is minus 6."},{"Start":"04:27.300 ","End":"04:30.940","Text":"On the right, we have plus 2."},{"Start":"04:31.040 ","End":"04:35.310","Text":"Plus 2 and then the minus 6 brought over,"},{"Start":"04:35.310 ","End":"04:37.650","Text":"so I get 8,"},{"Start":"04:37.650 ","End":"04:42.015","Text":"which means that x is equal to 4."},{"Start":"04:42.015 ","End":"04:45.860","Text":"But, 4 is not in the range x less than minus"},{"Start":"04:45.860 ","End":"04:49.715","Text":"5 so this gets crossed off for being out of the range."},{"Start":"04:49.715 ","End":"04:52.459","Text":"Now the next one in this range,"},{"Start":"04:52.459 ","End":"04:55.190","Text":"x is bigger or equal to minus 5."},{"Start":"04:55.190 ","End":"05:00.170","Text":"The first absolute value is the top one but it\u0027s still"},{"Start":"05:00.170 ","End":"05:05.780","Text":"less than 2/3 so the second absolute value is from the bottom line."},{"Start":"05:05.780 ","End":"05:11.300","Text":"So what I get is that plus 5 plus x this time,"},{"Start":"05:11.300 ","End":"05:16.832","Text":"I mean just regular 5 plus x then minus 1 is equal to,"},{"Start":"05:16.832 ","End":"05:18.950","Text":"this one is still the same,"},{"Start":"05:18.950 ","End":"05:26.060","Text":"so it\u0027s minus 3x minus 2 because we\u0027re still below the 2/3,"},{"Start":"05:26.060 ","End":"05:30.035","Text":"and now we just have to simplify that one."},{"Start":"05:30.035 ","End":"05:33.440","Text":"Let\u0027s see x plus 3x,"},{"Start":"05:33.440 ","End":"05:35.785","Text":"this time it\u0027s 4x,"},{"Start":"05:35.785 ","End":"05:40.905","Text":"and numbers 5 minus 1 is 4,"},{"Start":"05:40.905 ","End":"05:46.910","Text":"and the full goes over to the other side to join as a minus 4 and it joins the 2,"},{"Start":"05:46.910 ","End":"05:51.515","Text":"2 minus 4 is minus 2."},{"Start":"05:51.515 ","End":"05:56.255","Text":"This gives us that x is equal to minus 2 over 4,"},{"Start":"05:56.255 ","End":"05:58.655","Text":"which is minus 1/2."},{"Start":"05:58.655 ","End":"06:00.350","Text":"This is in range."},{"Start":"06:00.350 ","End":"06:04.160","Text":"Minus 1/2 is somewhere in between these 2 numbers so"},{"Start":"06:04.160 ","End":"06:09.230","Text":"this looks promising but I still need a final check on that."},{"Start":"06:09.230 ","End":"06:13.040","Text":"Let\u0027s substitute in the original equation."},{"Start":"06:13.040 ","End":"06:14.660","Text":"Let\u0027s see if x is minus 1/2,"},{"Start":"06:14.660 ","End":"06:20.540","Text":"here I have 4 and 1/2 an absolute value, it\u0027s still 4 and 1/2."},{"Start":"06:20.540 ","End":"06:25.580","Text":"Less 1 is 3 and 1/2 here."},{"Start":"06:25.580 ","End":"06:27.560","Text":"Let\u0027s see what I have on the other side."},{"Start":"06:27.560 ","End":"06:31.450","Text":"Minus 1/2 times 3 is minus 1 and 1/2,"},{"Start":"06:31.450 ","End":"06:34.655","Text":"minus 2, minus 3 and 1/2."},{"Start":"06:34.655 ","End":"06:36.110","Text":"But in absolute value,"},{"Start":"06:36.110 ","End":"06:37.760","Text":"I also get 3 and 1/2."},{"Start":"06:37.760 ","End":"06:40.450","Text":"So 3 and 1/2 on both sides were equal,"},{"Start":"06:40.450 ","End":"06:43.670","Text":"and so like I said, this really does work."},{"Start":"06:43.670 ","End":"06:46.040","Text":"Now the last range where x is bigger,"},{"Start":"06:46.040 ","End":"06:47.630","Text":"or equal to 2/3."},{"Start":"06:47.630 ","End":"06:51.500","Text":"In this case, it\u0027s also bigger or equal to minus 5."},{"Start":"06:51.500 ","End":"06:56.355","Text":"I\u0027m taking the top definition from each of the absolute values,"},{"Start":"06:56.355 ","End":"07:05.405","Text":"and so I\u0027ve got the absolute value here is 5 plus x then minus 1,"},{"Start":"07:05.405 ","End":"07:09.800","Text":"and here just 3x minus 2 as is."},{"Start":"07:09.800 ","End":"07:12.530","Text":"What does that give me on the left,"},{"Start":"07:12.530 ","End":"07:17.415","Text":"minus 2x, and let\u0027s see,"},{"Start":"07:17.415 ","End":"07:19.780","Text":"5 minus 1 is 4,"},{"Start":"07:19.780 ","End":"07:22.820","Text":"thrown over to the other side is minus 4,"},{"Start":"07:22.820 ","End":"07:25.235","Text":"so it\u0027s minus 6,"},{"Start":"07:25.235 ","End":"07:32.640","Text":"which gives me the x is equal to 6 over 2 is 3."},{"Start":"07:32.640 ","End":"07:34.980","Text":"It\u0027s certainly in the range,"},{"Start":"07:34.980 ","End":"07:39.380","Text":"3 is certainly bigger than 2/3 so it looks promising,"},{"Start":"07:39.380 ","End":"07:46.075","Text":"but we\u0027ll do that final check of substituting x equals 3 in our heads,"},{"Start":"07:46.075 ","End":"07:47.760","Text":"5 plus 3 is 8,"},{"Start":"07:47.760 ","End":"07:50.915","Text":"absolute value of 8 is 8 minus 1 is 7."},{"Start":"07:50.915 ","End":"07:52.895","Text":"Left-hand side is 7."},{"Start":"07:52.895 ","End":"07:56.295","Text":"Right-hand side, 3 times 3 is 9,"},{"Start":"07:56.295 ","End":"07:57.840","Text":"9 minus 2 is 7."},{"Start":"07:57.840 ","End":"07:59.550","Text":"Absolute value of 7 is 7."},{"Start":"07:59.550 ","End":"08:02.855","Text":"Yes. We have 2 solutions,"},{"Start":"08:02.855 ","End":"08:04.970","Text":"minus 1/2 and 3,"},{"Start":"08:04.970 ","End":"08:07.520","Text":"and that\u0027s it for part A,"},{"Start":"08:07.520 ","End":"08:11.060","Text":"and now we move on to part B,"},{"Start":"08:11.060 ","End":"08:16.010","Text":"which also looks like it has 2 absolute values."},{"Start":"08:16.010 ","End":"08:18.935","Text":"But if you\u0027re sharp-eyed,"},{"Start":"08:18.935 ","End":"08:21.775","Text":"you\u0027ll notice that x minus 4,"},{"Start":"08:21.775 ","End":"08:23.900","Text":"and 4 minus x are very similar."},{"Start":"08:23.900 ","End":"08:27.755","Text":"In fact, when you have a subtraction in the opposite order,"},{"Start":"08:27.755 ","End":"08:29.990","Text":"one is just the minus of the other."},{"Start":"08:29.990 ","End":"08:32.555","Text":"Actually, this equals this."},{"Start":"08:32.555 ","End":"08:34.490","Text":"If you didn\u0027t notice that you\u0027d hit"},{"Start":"08:34.490 ","End":"08:38.045","Text":"a polytomy way because when you did the definition of this,"},{"Start":"08:38.045 ","End":"08:41.960","Text":"you\u0027d get x minus 4 is bigger than 0, or less than 0."},{"Start":"08:41.960 ","End":"08:44.120","Text":"In any event, you\u0027d get x bigger than 4,"},{"Start":"08:44.120 ","End":"08:45.455","Text":"or x less than 4."},{"Start":"08:45.455 ","End":"08:47.150","Text":"I didn\u0027t get the same thing here."},{"Start":"08:47.150 ","End":"08:49.790","Text":"The only be one special number to put on"},{"Start":"08:49.790 ","End":"08:52.550","Text":"the number line that would be 4 you\u0027d come across it,"},{"Start":"08:52.550 ","End":"08:54.080","Text":"maybe not as elegantly."},{"Start":"08:54.080 ","End":"08:58.189","Text":"I\u0027m saying that the side, I\u0027ll write that."},{"Start":"08:58.189 ","End":"09:04.730","Text":"In general, the absolute value of a is the same as the absolute value of minus a."},{"Start":"09:04.730 ","End":"09:06.770","Text":"Because attaching a sign to a number,"},{"Start":"09:06.770 ","End":"09:08.720","Text":"changing it from plus to minus over minus to"},{"Start":"09:08.720 ","End":"09:11.210","Text":"plus is not going to change the absolute value."},{"Start":"09:11.210 ","End":"09:12.770","Text":"But more than that,"},{"Start":"09:12.770 ","End":"09:16.670","Text":"I\u0027m also saying that if you subtract things in the other order,"},{"Start":"09:16.670 ","End":"09:18.920","Text":"one is the minus of the other."},{"Start":"09:18.920 ","End":"09:24.080","Text":"In general, the same a necessarily but a minus b,"},{"Start":"09:24.080 ","End":"09:28.340","Text":"in general, is minus of b minus a."},{"Start":"09:28.340 ","End":"09:29.750","Text":"If you think of it, I don\u0027t know,"},{"Start":"09:29.750 ","End":"09:32.940","Text":"say 2 minus 5 and 5 minus 2,"},{"Start":"09:32.940 ","End":"09:36.615","Text":"they\u0027re just one is the negative of the other,"},{"Start":"09:36.615 ","End":"09:39.230","Text":"3 and minus 3 or the other way round."},{"Start":"09:39.230 ","End":"09:42.620","Text":"I can conclude from this."},{"Start":"09:42.620 ","End":"09:48.890","Text":"What I have here is x minus 4 and the absolute value of 4 minus x is also"},{"Start":"09:48.890 ","End":"09:55.370","Text":"the same thing as x minus 4 because it\u0027s the minus of it like we said, equals 4."},{"Start":"09:55.370 ","End":"09:57.980","Text":"Now I have one of these and another one of these."},{"Start":"09:57.980 ","End":"09:59.405","Text":"I have 2 of these."},{"Start":"09:59.405 ","End":"10:04.340","Text":"So 2 of these absolute values of x minus 4 makes 4."},{"Start":"10:04.340 ","End":"10:10.715","Text":"Each one is 2 just divided both sides of the equation by 2."},{"Start":"10:10.715 ","End":"10:16.040","Text":"Now, we\u0027re on our usual simple case where we have absolute value"},{"Start":"10:16.040 ","End":"10:21.305","Text":"of something equals a positive number then we know that the answers are,"},{"Start":"10:21.305 ","End":"10:27.070","Text":"x minus 4 is 2 or x minus 4 is minus 2."},{"Start":"10:27.070 ","End":"10:30.320","Text":"I mean, both of these, whatever solutions they gave will be okay."},{"Start":"10:30.320 ","End":"10:33.335","Text":"You don\u0027t even have to check by substitution."},{"Start":"10:33.335 ","End":"10:36.330","Text":"Here we get x equals 6,"},{"Start":"10:36.330 ","End":"10:39.180","Text":"and here we get minus 2 plus 4."},{"Start":"10:39.180 ","End":"10:41.260","Text":"We get x equals 2."},{"Start":"10:41.260 ","End":"10:46.880","Text":"These are bound to be our answers unless we made a mistake and for that reason,"},{"Start":"10:46.880 ","End":"10:48.710","Text":"it is worth substituting."},{"Start":"10:48.710 ","End":"10:50.650","Text":"If we substitute 6,"},{"Start":"10:50.650 ","End":"10:52.220","Text":"6 minus 4 is 2,"},{"Start":"10:52.220 ","End":"10:55.010","Text":"4 minus 6 is minus 2."},{"Start":"10:55.010 ","End":"10:58.322","Text":"But in both cases, the absolute value is 2,"},{"Start":"10:58.322 ","End":"11:00.160","Text":"and 2 is 4."},{"Start":"11:00.160 ","End":"11:03.900","Text":"Same thing for the 2, 2 minus 4 is minus 24,"},{"Start":"11:03.900 ","End":"11:07.430","Text":"minus 2 is plus 2 but the absolute value of each of them is"},{"Start":"11:07.430 ","End":"11:11.450","Text":"2 and 2 and 2 is 4 so that we didn\u0027t have to,"},{"Start":"11:11.450 ","End":"11:14.930","Text":"we did verify these solutions and that\u0027s the answer to part B."},{"Start":"11:14.930 ","End":"11:17.040","Text":"We are done."}],"ID":5044}],"Thumbnail":null,"ID":3874}]

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