WEBVTT 00:00:01.630 --> 00:00:06.274 The purpose of this video is to look at the solution of 00:00:06.274 --> 00:00:07.048 elementary simultaneous 00:00:07.048 --> 00:00:12.884 equations. Before we do that, let's just have a look at a 00:00:12.884 --> 00:00:13.732 relatively straightforward 00:00:13.732 --> 00:00:20.162 single equation. Equation we're going to look at 2X minus 00:00:20.162 --> 00:00:21.899 Y equals 3. 00:00:22.900 --> 00:00:26.651 This is a linear equation. It's a linear equation because there 00:00:26.651 --> 00:00:31.766 are no terms in it that are X squared, Y squared or X times by 00:00:31.766 --> 00:00:36.540 Y or indeed ex cubes. The only terms we've got a terms in X 00:00:36.540 --> 00:00:38.586 terms in Y and some numbers. 00:00:39.700 --> 00:00:42.775 So this represents a linear 00:00:42.775 --> 00:00:48.260 equation. We can rearrange it so that it says why equal 00:00:48.260 --> 00:00:53.330 something, so let's just do that. We can add Y to each side 00:00:53.330 --> 00:00:55.280 so that we get 2X. 00:00:55.960 --> 00:01:01.028 Equals 3 plus Yi. Did say add why two each side and you might 00:01:01.028 --> 00:01:05.010 have wondered what happened here. Well, if I've got minus Y 00:01:05.010 --> 00:01:10.078 and I add why to it, I end up with no wise at all. 00:01:11.400 --> 00:01:18.414 Here we've got two X equals 3 plus Y, so let's take the three 00:01:18.414 --> 00:01:22.923 away from each side. 2X minus three equals Y. 00:01:23.700 --> 00:01:29.556 So there I've got a nice expression for why. If I take 00:01:29.556 --> 00:01:36.876 any value of X. Let's say I take X equals 1, then why will be 00:01:36.876 --> 00:01:44.196 equal to two times by 1 - 3, which gives us minus one. So for 00:01:44.196 --> 00:01:51.028 this value of XI, get that value of Yi can take another value of 00:01:51.028 --> 00:01:53.956 XX equals 2, Y will equal. 00:01:54.020 --> 00:01:58.340 2 * 2 - 3, which is plus one. 00:02:00.720 --> 00:02:06.245 Another value of XX equals 0 Y equals 2 times by 0 - 00:02:06.245 --> 00:02:11.770 3 will 2 * 0 is 0 and that gives me minus three. 00:02:13.280 --> 00:02:17.924 So for every value of XI can generate a value of Y. 00:02:19.260 --> 00:02:24.083 I can plot these as point so I can plot this as the 00:02:24.083 --> 00:02:29.648 .1 - 1 and I can plot this one on a graph as the .21 00:02:29.648 --> 00:02:34.100 and this one on a graph as the point nort minus three. 00:02:34.100 --> 00:02:36.326 So let's just set that up. 00:02:39.700 --> 00:02:40.828 Pair of axes. 00:02:43.560 --> 00:02:47.733 Let's mark the values of X that we've been having a look at. 00:02:49.610 --> 00:02:53.378 So that was X on there and why 00:02:53.378 --> 00:02:57.372 there? And let's put on the values of why that we got. 00:02:59.650 --> 00:03:06.074 When X was zero hour value of why was minus three? 00:03:06.910 --> 00:03:08.770 So that's. There. 00:03:10.500 --> 00:03:13.979 When X was one hour value was 00:03:13.979 --> 00:03:19.977 minus one. And when X was two hour value was one. 00:03:21.430 --> 00:03:25.814 Those three points lie on a straight line. 00:03:27.780 --> 00:03:32.900 Y equals 2X minus three, and that's another reason for 00:03:32.900 --> 00:03:38.532 calling this a linear equation. It gives us a straight line. 00:03:39.650 --> 00:03:40.360 OK. 00:03:42.190 --> 00:03:49.613 You've got that one. Y equals 2X minus three. Supposing we take a 00:03:49.613 --> 00:03:57.036 second one 3X plus two Y equals 8, a second linear equation, and 00:03:57.036 --> 00:03:59.891 supposing we say these two. 00:04:00.920 --> 00:04:03.386 Are true at the same time. 00:04:05.320 --> 00:04:06.480 What does that mean? 00:04:07.060 --> 00:04:11.176 Well, we can plot this as a straight line. Again, it's a 00:04:11.176 --> 00:04:15.292 linear equation, so it's going to give us a straight line. Now 00:04:15.292 --> 00:04:20.094 I don't want to have to workout lots of points for this, So what 00:04:20.094 --> 00:04:24.896 I'm going to do is just sketch it in quickly on the graph. I'm 00:04:24.896 --> 00:04:27.297 going to say when X is 0. 00:04:27.910 --> 00:04:32.875 And cover up the Exterm 2 Y is equal to 8, so why must be 00:04:32.875 --> 00:04:36.516 equal to four which is going to be up there somewhere? 00:04:38.030 --> 00:04:45.204 And when? Why is 03 X is equal to 8 and so X is 8 / 3 00:04:45.204 --> 00:04:50.268 which gives us 2 and 2/3. So somewhere about their two 2/3 00:04:50.268 --> 00:04:55.754 and we know it's a straight line, so we can get that by 00:04:55.754 --> 00:04:57.020 joining up there. 00:04:58.400 --> 00:05:05.480 This is the equation 3X plus two Y equals 8. So what does it mean 00:05:05.480 --> 00:05:12.088 for these two to be true at the same time? Well, it must mean 00:05:12.088 --> 00:05:13.976 it's this point here. 00:05:14.560 --> 00:05:19.756 Where the two lines cross. So when we solve a pair of 00:05:19.756 --> 00:05:23.220 simultaneous equations, what we're actually looking for is 00:05:23.220 --> 00:05:25.818 the intersection of two straight lines. 00:05:26.850 --> 00:05:32.430 Of course, it could happen that we have one line like that. 00:05:33.400 --> 00:05:34.828 And apparel line. 00:05:35.970 --> 00:05:37.218 They would never meet. 00:05:38.170 --> 00:05:42.190 And one of the examples that we're going to be looking at 00:05:42.190 --> 00:05:45.540 later will show what happens in terms of the arithmetic 00:05:45.540 --> 00:05:49.225 when we have this particular case. But for now, let's go 00:05:49.225 --> 00:05:52.910 back and think about these two. How can we handle these 00:05:52.910 --> 00:05:56.595 two algebraically so that we don't have to draw graphs? We 00:05:56.595 --> 00:05:59.610 don't have to rely on sketching, we can calculate 00:05:59.610 --> 00:06:02.960 which is so much easier in most cases that actually 00:06:02.960 --> 00:06:03.965 drawing a graph. 00:06:05.250 --> 00:06:08.868 So let's take these two equations. 00:06:12.850 --> 00:06:18.202 And we're going to look at two methods of solution, so I'm 00:06:18.202 --> 00:06:23.108 going to look at method one. Now, let's begin with the 00:06:23.108 --> 00:06:28.906 original equation that we had two X minus. Y is equal to three 00:06:28.906 --> 00:06:35.596 and then the one that we put with it 3X plus two Y equals 8. 00:06:37.360 --> 00:06:41.728 Our first method of solution, well, one of the things to do is 00:06:41.728 --> 00:06:46.096 to do what we did in the very first case with this and 00:06:46.096 --> 00:06:48.784 Rearrange one of these equations. It doesn't matter 00:06:48.784 --> 00:06:50.800 which one, but we'll take this 00:06:50.800 --> 00:06:54.760 one. So that we get Y 00:06:54.760 --> 00:07:01.424 equals. And we know what the result for that one is. It's Y 00:07:01.424 --> 00:07:03.152 equals 2X minus three. 00:07:05.230 --> 00:07:07.518 So that's equation one. 00:07:08.170 --> 00:07:15.720 That's equation. Two, so this is now, let's call it equation 3 00:07:15.720 --> 00:07:18.870 and we got it by rearranging. 00:07:21.150 --> 00:07:21.940 1. 00:07:23.610 --> 00:07:29.325 What we're going to do with this is if these two have to be true 00:07:29.325 --> 00:07:33.516 at the same time, then this relationship must be true in 00:07:33.516 --> 00:07:37.326 this equation, so we can substitute it in, so let's. 00:07:38.350 --> 00:07:40.990 Substitute 3 00:07:42.460 --> 00:07:48.816 until two so we have 3X plus. 00:07:50.290 --> 00:07:58.222 Two times Y. But why is 2X minus three that's equal to 00:07:58.222 --> 00:08:02.510 8? And you can see that what we've done is we've reduced. 00:08:03.190 --> 00:08:08.480 This. To this equation giving us a single equation in one 00:08:08.480 --> 00:08:12.583 unknown, which is a simple linear equation and we can solve 00:08:12.583 --> 00:08:19.922 it. Multiply out the brackets 3X plus 4X minus 6 equals 8. 00:08:21.150 --> 00:08:26.694 Gather the excess together. 7X minus 6 equals 8. 00:08:27.470 --> 00:08:33.454 At the six to each side, Seven X equals 14, and so X must be 2. 00:08:35.020 --> 00:08:37.090 That's only given as one value. 00:08:38.060 --> 00:08:42.870 We need a value of Y, but up here we've got an expression 00:08:42.870 --> 00:08:48.050 which says Y equals and if we take the value of X that we've 00:08:48.050 --> 00:08:49.530 got and substituted in. 00:08:50.420 --> 00:08:55.033 Therefore, why will be equal to 2 00:08:55.033 --> 00:08:58.987 * 2 - 3 gives US1? 00:09:00.600 --> 00:09:05.472 And so we've got a solution X equals 2, Y equals 1. 00:09:06.910 --> 00:09:08.280 Are we sure it's right? 00:09:09.040 --> 00:09:13.891 Well, we used this equation which came from equation one to 00:09:13.891 --> 00:09:16.096 generate the value of Y. 00:09:16.740 --> 00:09:22.270 So if we take the values of X&Y and put them back into here, 00:09:22.270 --> 00:09:27.010 they should work, should give us the right answer. So let's try 00:09:27.010 --> 00:09:28.590 that. X is 2. 00:09:29.350 --> 00:09:37.036 3 times by two is 6 plus, Y is 1 two times by one is 2 six and 00:09:37.036 --> 00:09:42.160 two gives us eight. Yes, this works. This is a solution of 00:09:42.160 --> 00:09:47.711 that equation and of that one. So this is our answer to the 00:09:47.711 --> 00:09:49.419 pair of simultaneous equations. 00:09:50.320 --> 00:09:54.489 Let's have a look at another one using this particular method. 00:09:57.420 --> 00:10:00.038 The example we're going to use is going to be said. 00:10:00.110 --> 00:10:07.195 Open X. +2 Y equals 47 and 00:10:07.195 --> 00:10:13.922 five X minus four Y equals 1. 00:10:15.930 --> 00:10:21.290 Now. We need to make a choice. We need to choose one of these 00:10:21.290 --> 00:10:27.503 two equations. And Rearrange it so that it says Y equals or if 00:10:27.503 --> 00:10:29.235 we want X equals. 00:10:31.190 --> 00:10:35.740 The choice is entirely ours and we have to make the choice based 00:10:35.740 --> 00:10:40.290 upon what we feel will be the simplest and looking at a pair 00:10:40.290 --> 00:10:44.140 of equations like this often difficult to know which is the 00:10:44.140 --> 00:10:47.640 simplest. Well, let's pick at random. Let's choose this one 00:10:47.640 --> 00:10:51.490 and let's Rearrange Equation too. So we'll start by getting X 00:10:51.490 --> 00:10:56.740 equals this time. So we say 5X is equal to 1 and I'm going to 00:10:56.740 --> 00:10:59.540 add 4 Y to each side plus 4Y. 00:11:00.580 --> 00:11:04.756 Now I'm going to divide throughout by the five so that I 00:11:04.756 --> 00:11:10.494 have. X on its own. Now I've got to divide everything by 5. 00:11:11.130 --> 00:11:15.066 Everything so I had to put that line there to show that 00:11:15.066 --> 00:11:19.330 I'm dividing the one and the four Y. So this is a fraction. 00:11:19.330 --> 00:11:24.250 I'm sure you can tell this is not going to be as easy as the 00:11:24.250 --> 00:11:27.530 previous question was. In fact, it's going to be quite 00:11:27.530 --> 00:11:31.138 difficult because I have to take this now and because it 00:11:31.138 --> 00:11:32.450 came from equation too. 00:11:37.660 --> 00:11:42.626 I'm going to have to take it and substitute it back into equation 00:11:42.626 --> 00:11:47.592 one, and this isn't looking very pretty, so let's give it a try 00:11:47.592 --> 00:11:49.460 sub. 3. 00:11:51.180 --> 00:11:59.002 Until one. So I have 7X but X is this 00:11:59.002 --> 00:12:06.086 lump of algebra here 1 + 4 Y all over 5. 00:12:06.730 --> 00:12:10.501 +2 Y equals 00:12:10.501 --> 00:12:15.604 47. I can see this is becoming quite horrific. 00:12:16.620 --> 00:12:20.547 Multiply throughout by 5 why? Because we're dividing by 5. We 00:12:20.547 --> 00:12:25.545 want to get rid of the fraction. The way to do that is to 00:12:25.545 --> 00:12:30.186 multiply everything by 5 and it has to be everything. So if we 00:12:30.186 --> 00:12:34.827 multiply that by 5 because we're dividing by 5, it's as though we 00:12:34.827 --> 00:12:41.253 actually do nothing to the 1 + 4 Y. That leaves a 7 * 1 + 4 Y. 00:12:41.910 --> 00:12:47.678 We need five times that that's ten Y and we have to have five 00:12:47.678 --> 00:12:51.386 times that remember, an equation is a balance. What 00:12:51.386 --> 00:12:57.154 you do to one side of the balance you have to do to the 00:12:57.154 --> 00:13:00.450 other. If you don't, it's unbalanced. So we're 00:13:00.450 --> 00:13:05.394 multiplying everything by 5. So 5 * 47 five 735, five falls 00:13:05.394 --> 00:13:06.630 of 22135 altogether. 00:13:07.770 --> 00:13:15.386 Now we need to multiply out the brackets 7 + 28 Y plus 10 00:13:15.386 --> 00:13:19.194 Y equals 235. So we take this 00:13:19.194 --> 00:13:23.692 equation. Write it down again so that we can see it clearly. 00:13:30.920 --> 00:13:35.240 Now we can gather these two together gives us 38Y. 00:13:36.310 --> 00:13:41.920 And we can take Seven away from each side, which will 00:13:41.920 --> 00:13:43.450 give us 228. 00:13:44.910 --> 00:13:49.827 Exactly big numbers coming in here 228 / 38 'cause we're 00:13:49.827 --> 00:13:54.744 looking for the number which when we multiplied by 38 will 00:13:54.744 --> 00:13:57.873 give us 228 and that's going to 00:13:57.873 --> 00:14:03.020 be 6. So we've established Y is equal to 6. 00:14:03.950 --> 00:14:09.202 Having done that, we can take it and we can substitute it back 00:14:09.202 --> 00:14:14.454 into the equation that we first had for X. So remember that for 00:14:14.454 --> 00:14:17.282 that we had X was equal to. 00:14:18.100 --> 00:14:25.300 And what we had for that was 1 + 4 Y all over five. We 00:14:25.300 --> 00:14:32.020 substitute in the six, so we have 1 + 24 or over 5 and 00:14:32.020 --> 00:14:39.220 quickly we can see that's 25 / 5. So we have X equals 5. So 00:14:39.220 --> 00:14:44.980 again we've got our pair of values. Our answer to the pair 00:14:44.980 --> 00:14:48.820 of simultaneous equations. We haven't checked it though. 00:14:49.390 --> 00:14:53.845 Now remember that this came from the second equation, so really 00:14:53.845 --> 00:14:59.515 to check it we've got to go back to the very first equation that 00:14:59.515 --> 00:15:05.185 we had written down that one. If you remember was Seven X +2, Y 00:15:05.185 --> 00:15:06.805 is equal to 47. 00:15:07.880 --> 00:15:15.344 So let's just check 7 * 5. That gives us 35 + 00:15:15.344 --> 00:15:17.210 2 * 6. 00:15:17.900 --> 00:15:23.340 That gives us 12, so we 35 + 12 equals 47. And yes, that is what 00:15:23.340 --> 00:15:28.100 we wanted, so we now know that this is correct, but I just stop 00:15:28.100 --> 00:15:29.460 and think about it. 00:15:30.040 --> 00:15:34.876 We got all those fractions to work with. We got this lump of 00:15:34.876 --> 00:15:39.712 algebra to carry around with us. Is there not an easier way of 00:15:39.712 --> 00:15:44.820 doing these? Yes there is. It's useful to have seen the method 00:15:44.820 --> 00:15:49.760 that we have got simply because we will need it again when we 00:15:49.760 --> 00:15:53.180 look at the second video of simultaneous equations, but. 00:15:54.030 --> 00:15:58.566 That is a simple way of handling these, so let's go on now and 00:15:58.566 --> 00:16:00.510 have a look at method 2. 00:16:09.300 --> 00:16:12.850 Now this method is sometimes called elimination and we can 00:16:12.850 --> 00:16:17.465 see why it gets that name and this is the method that you 00:16:17.465 --> 00:16:21.370 really do need to practice and become accustomed to. So let's 00:16:21.370 --> 00:16:24.920 start with the same equations that we had last time. 00:16:26.000 --> 00:16:26.980 And see. 00:16:28.540 --> 00:16:32.630 How it works and how much easier it actually is? 00:16:33.780 --> 00:16:37.316 OK method of elimination. What do we do? 00:16:38.180 --> 00:16:42.428 What we do is we seek to make the 00:16:42.428 --> 00:16:45.260 coefficients in front of the wise. 00:16:46.410 --> 00:16:48.570 Or in front of the axes. 00:16:49.430 --> 00:16:50.210 The same. 00:16:51.650 --> 00:16:56.094 Once we've gotten the same, then we can either add the 00:16:56.094 --> 00:16:59.326 two equations together or subtract them according to 00:16:59.326 --> 00:17:01.346 the signs that are there. 00:17:02.780 --> 00:17:08.300 By doing that, we will get rid of that particular unknown, the 00:17:08.300 --> 00:17:10.140 one that we chose. 00:17:10.750 --> 00:17:14.306 To make the coefficients numerically the same. 00:17:15.720 --> 00:17:18.886 So. This one what would we do? 00:17:19.560 --> 00:17:22.731 Well, if we look at this and 00:17:22.731 --> 00:17:28.910 this. Here we have two Y and here we have minus four Y. 00:17:29.540 --> 00:17:34.398 So if I were to double that, I'd have four. Why there? And it's 00:17:34.398 --> 00:17:38.909 plus four Y Ana minus four Y there, and that seems are pretty 00:17:38.909 --> 00:17:43.420 good thing to do, because then they're both for Y. One of them 00:17:43.420 --> 00:17:47.931 is plus and one of them is minus. And if I add them 00:17:47.931 --> 00:17:51.748 together they will disappear. So let me just number the equations 00:17:51.748 --> 00:17:52.789 one and two. 00:17:53.650 --> 00:17:59.050 And then I can keep a record of what I'm doing. So I'm going to 00:17:59.050 --> 00:18:03.010 multiply the first equation by two and that's going to lead 00:18:03.010 --> 00:18:05.170 Maine to a new equation 3. 00:18:05.780 --> 00:18:07.000 So let's do that. 00:18:07.820 --> 00:18:14.792 2 * 7 X is 14X plus two times, that is 4 00:18:14.792 --> 00:18:21.183 Y equals 2 times that, and 2 * 47 is 94. 00:18:22.190 --> 00:18:27.949 Now equation two. I'm leaving as it is not going to touch it. 00:18:30.510 --> 00:18:32.620 Now I've got two equations. 00:18:34.110 --> 00:18:41.194 This is plus four Y and this is minus four Y. So if I 00:18:41.194 --> 00:18:47.266 add the two equations together, what happens? I get 14X plus 5X. 00:18:47.266 --> 00:18:48.784 That's 19 X. 00:18:50.880 --> 00:18:56.775 No whys. 'cause I've plus four Y add it to minus four Y know wise 00:18:56.775 --> 00:18:57.954 at all equals. 00:18:58.820 --> 00:18:59.850 95 00:19:01.010 --> 00:19:06.386 and so X is 95 over 19, which gives me 5 which if you 00:19:06.386 --> 00:19:10.226 remember, is the answer we have to the last question. 00:19:11.820 --> 00:19:15.285 Now we need to take this and substitute it back. Doesn't 00:19:15.285 --> 00:19:18.435 matter which equation we choose to substitute it back into. 00:19:18.435 --> 00:19:19.695 Let's take this one. 00:19:20.350 --> 00:19:27.562 X is 5, so five times by 5 - 4 Y equals 00:19:27.562 --> 00:19:28.163 1. 00:19:29.200 --> 00:19:35.090 And so we have 25 - 4 Y equals 1. 00:19:37.200 --> 00:19:39.930 Take the four way over to that side by adding 00:19:39.930 --> 00:19:41.295 four Y to each side. 00:19:43.260 --> 00:19:49.116 So that will give us 25 is equal to four Y plus one. Take the one 00:19:49.116 --> 00:19:54.972 away from its side, 24 is 4 Y and so why is equal to 6 and 00:19:54.972 --> 00:19:58.998 we've got exactly the same answer as we had before. And 00:19:58.998 --> 00:20:00.096 let's just look. 00:20:01.400 --> 00:20:05.426 How much simpler that is? How much quicker that answer came 00:20:05.426 --> 00:20:09.818 out. One thing to notice. Well, two things in actual fact. First 00:20:09.818 --> 00:20:14.210 of all, I try to keep the equal signs underneath each other. 00:20:15.060 --> 00:20:19.246 This is not only makes it look neat, it enables you to see what 00:20:19.246 --> 00:20:20.442 it is you're doing. 00:20:21.540 --> 00:20:25.082 Keep the equations together so the setting out of this work 00:20:25.082 --> 00:20:27.336 actually helps you to be able to 00:20:27.336 --> 00:20:32.783 check it. Second thing to notice is down this side. I've kept a 00:20:32.783 --> 00:20:36.886 record of exactly what I've done multiplying the equation by two, 00:20:36.886 --> 00:20:39.870 adding the two equations together. That's very helpful 00:20:39.870 --> 00:20:45.092 when you want to check your work. What did I do? How did I 00:20:45.092 --> 00:20:48.822 actually work this out? By having this record down the 00:20:48.822 --> 00:20:53.671 side, you don't have to work it out again. You can see exactly 00:20:53.671 --> 00:20:58.520 what it is that you did. Now let's take a third example and 00:20:58.520 --> 00:21:03.218 again. Will solve it by means of the method of elimination. Just 00:21:03.218 --> 00:21:08.650 so we've got a second example of that method to look at three X 00:21:08.650 --> 00:21:10.978 +7 Y is 27 and 5X. 00:21:12.620 --> 00:21:14.968 +2 Y is 60. 00:21:16.430 --> 00:21:20.234 OK, we've got a choice to make. We can make either the 00:21:20.234 --> 00:21:23.721 coefficients in front of the axe numerically the same, or the 00:21:23.721 --> 00:21:27.525 coefficients in front of the wise. Well, in order to do that, 00:21:27.525 --> 00:21:31.012 I'd have to multiply the Y. Certainly have to multiply this 00:21:31.012 --> 00:21:35.450 equation by two to give me 14 there and this one by 7:00 to 00:21:35.450 --> 00:21:36.718 give me 14 there. 00:21:38.310 --> 00:21:39.934 How do I make that choice? Well? 00:21:40.990 --> 00:21:45.742 Fairly clearly 2 times by 7 is 14, so 1 by 1, one by the other. 00:21:46.660 --> 00:21:49.918 But I don't really like multiplying by 7 difficult 00:21:49.918 --> 00:21:54.262 number. I prefer to multiply by three and five, so my choices 00:21:54.262 --> 00:21:58.244 actually governed by how well I think I can handle the 00:21:58.244 --> 00:22:02.588 arithmetic. So let's multiply this one by 5 and this one by 00:22:02.588 --> 00:22:06.570 three will give us 15X an 15X number. The equations one. 00:22:08.240 --> 00:22:14.923 2. And I'll take equation one and I will multiply it by 5 and 00:22:14.923 --> 00:22:17.800 that will give me a new equation 00:22:17.800 --> 00:22:24.080 3. So multiplying it by 5:15 00:22:24.080 --> 00:22:30.152 X plus 35 Y is equal 00:22:30.152 --> 00:22:36.378 to 135. And then equation two, I will multiply by 00:22:36.378 --> 00:22:40.788 three and that will give me a new equation for. 00:22:41.610 --> 00:22:46.047 Oh, here we go. Multiplying this by three 15X. 00:22:46.750 --> 00:22:52.056 Plus six Y is equal to 48. 00:22:53.520 --> 00:22:59.870 These are now both 15X and they're both plus 15X. 00:23:00.560 --> 00:23:06.320 So if I take this equation away from that equation, I'll have 00:23:06.320 --> 00:23:13.040 15X minus 15X no X is at all. I live eliminative, the X, I'll 00:23:13.040 --> 00:23:18.800 just have the Wise left, so let's do that equation 3 minus 00:23:18.800 --> 00:23:25.405 equation 4. 15X takeaway 15X no axis 35 Y 00:23:25.405 --> 00:23:30.965 takeaway six Y that gives us 29 Y. 00:23:32.250 --> 00:23:38.663 And then 135 takeaway 48? And that's going to give us 00:23:38.663 --> 00:23:45.076 A7 their 87 altogether. And so why is 87 over 29, 00:23:45.076 --> 00:23:47.408 which gives us 3? 00:23:48.600 --> 00:23:55.410 Having got that, I need to know the value of X so I can take 00:23:55.410 --> 00:24:00.858 Y equals 3 and substituted back let's say into equation one. So 00:24:00.858 --> 00:24:08.122 I have 3X plus Seven times Y 7 threes are 21 is equal to 27 and 00:24:08.122 --> 00:24:14.024 so 3X is 6 taking 21 away from each side and access 2. 00:24:15.530 --> 00:24:21.800 Check this in here 5 twos are ten 2306 ten and six gives me 16 00:24:21.800 --> 00:24:27.652 which is what I want so I know that this is my answer. My 00:24:27.652 --> 00:24:31.414 solution to this pair of simultaneous equations and again 00:24:31.414 --> 00:24:35.594 look how straightforward that is. Much, much easier than the 00:24:35.594 --> 00:24:40.192 first method that we saw. Also think about using letters as 00:24:40.192 --> 00:24:44.790 well. If we've got letters to use instead of coefficients the 00:24:44.790 --> 00:24:50.095 numbers here. So we might have a X plus BY. Again, this is a much 00:24:50.095 --> 00:24:53.445 better method to use. Again, notice the setting down keeping 00:24:53.445 --> 00:24:56.795 it compact, keeping the equal signs under each other and 00:24:56.795 --> 00:25:00.815 keeping a record of what we've done. So do something comes out 00:25:00.815 --> 00:25:04.165 wrong, we can check it, see what we are doing. 00:25:05.130 --> 00:25:09.303 Now all the examples that we've looked at so far of all had 00:25:09.303 --> 00:25:12.513 whole number coefficients. They might have been, plus they might 00:25:12.513 --> 00:25:15.723 be minus, but they've been whole number, and everything that 00:25:15.723 --> 00:25:19.896 we've looked at as being in this sort of form XY number XY 00:25:19.896 --> 00:25:23.427 number. Well, not all equations come like that, so let's just 00:25:23.427 --> 00:25:25.674 have a look at a couple of 00:25:25.674 --> 00:25:29.260 examples that. Don't look like the ones we've just done. 00:25:31.370 --> 00:25:36.320 First of all, let's have this One X equals 3 Y. 00:25:37.000 --> 00:25:42.918 And X over 3 minus Y equals 34 pair of simultaneous 00:25:42.918 --> 00:25:47.222 equations. Linear simultaneous equations again 'cause they both 00:25:47.222 --> 00:25:53.678 got just X&Y in an numbers, nothing else, no X squared's now 00:25:53.678 --> 00:25:55.292 ex wise etc. 00:25:56.030 --> 00:26:01.985 We need to get them into a form that we can use and that would 00:26:01.985 --> 00:26:07.543 be nice to have XY number. So let's do that with this One X 00:26:07.543 --> 00:26:12.307 equals 3 Y, so will have X minus three Y equals 0. 00:26:13.990 --> 00:26:18.682 This one got a fraction in it. Fractions we don't like, can't 00:26:18.682 --> 00:26:22.592 handle fractions. Let's get rid of the three by multiplying 00:26:22.592 --> 00:26:26.893 everything in this equation by three. So will do that three 00:26:26.893 --> 00:26:29.630 times X over 3 just leaves us 00:26:29.630 --> 00:26:37.120 with X. Three times the Y minus three Y equals 3 times. 00:26:37.120 --> 00:26:40.075 This going to be 102. 00:26:42.280 --> 00:26:42.970 Problem. 00:26:44.920 --> 00:26:48.670 These two bits here are exactly 00:26:48.670 --> 00:26:52.408 the same. But these two bits are different. 00:26:54.140 --> 00:26:55.928 What's going to happen? 00:26:56.800 --> 00:27:00.220 Well, clearly if we subtract these two equations one from 00:27:00.220 --> 00:27:03.640 the other, there won't be anything left this side when 00:27:03.640 --> 00:27:06.718 we've done the subtraction X from X, no access. 00:27:07.750 --> 00:27:11.533 Minus 3 Y takeaway minus three Y know why is left, and yet 00:27:11.533 --> 00:27:15.316 we're going to have 0 - 102 equals minus 102 at this side. 00:27:15.316 --> 00:27:17.935 In other words, we're gonna end up with that. 00:27:21.210 --> 00:27:22.938 Which is a wee bit strange. 00:27:23.890 --> 00:27:27.130 What's the problem? What's the difficulty? Remember right back 00:27:27.130 --> 00:27:30.730 at the beginning when we drew a couple of graphs? 00:27:32.100 --> 00:27:37.140 In the first case we had two lines that actually crossed, but 00:27:37.140 --> 00:27:41.760 in the second case I drew 2 lines that were parallel. 00:27:42.510 --> 00:27:47.574 And that's exactly what we've got here. We have got 2 lines 00:27:47.574 --> 00:27:51.794 that are parallel because they've got this same form. They 00:27:51.794 --> 00:27:56.858 are parallel lines so they don't meet. And what this is telling 00:27:56.858 --> 00:28:01.922 us is there in fact is no solution to this pair of 00:28:01.922 --> 00:28:06.564 equations because they come from 2 parallel lines that do not 00:28:06.564 --> 00:28:11.206 meet no solution. There isn't one fixed point, so we would 00:28:11.206 --> 00:28:12.472 write that down. 00:28:12.580 --> 00:28:15.480 Simply say no solutions. 00:28:17.660 --> 00:28:19.660 And it's important to keep an eye out for that. 00:28:20.740 --> 00:28:24.630 Check back, make sure the arithmetic's correct yes, but do 00:28:24.630 --> 00:28:26.186 remember that can happen. 00:28:27.140 --> 00:28:33.720 Let's take just one more final example, X over 5. 00:28:35.340 --> 00:28:38.646 Minus Y over 4 equals 0. 00:28:40.200 --> 00:28:47.890 3X plus 1/2 Y equals 70. Now for this one. 00:28:48.800 --> 00:28:52.170 We've got fractions with dominators five and four, and we 00:28:52.170 --> 00:28:56.214 need to get rid of those. So we need a common denominator. 00:28:57.050 --> 00:29:01.160 With which we can multiply everything in the equation and 00:29:01.160 --> 00:29:06.914 those get rid of the five in the fall. The obvious one to choose 00:29:06.914 --> 00:29:13.079 is 20, because 20 is 5 times by 4. Let us write that down in 00:29:13.079 --> 00:29:18.833 falls 20 times X over 5 - 20 times Y over 4 equals 0 00:29:18.833 --> 00:29:21.710 be'cause. 20 * 0 is still 0. 00:29:23.050 --> 00:29:29.206 Little bit of counseling 5 into 20 goes 4. 00:29:30.050 --> 00:29:33.630 4 into 20 goes 5. 00:29:34.700 --> 00:29:40.208 So we have 4X minus five Y equals 0. 00:29:42.000 --> 00:29:46.708 So that was our first equation that was our second equation. 00:29:46.708 --> 00:29:51.844 This one is now become our third equation. So equation one has 00:29:51.844 --> 00:29:53.556 gone to equation 3. 00:29:54.430 --> 00:29:57.420 Let's look at equation two. Now that we need to deal with it, 00:29:57.420 --> 00:29:59.490 it's got a half way in it. So if 00:29:59.490 --> 00:30:03.626 I multiply every. Anything by two. This will become just why? 00:30:04.350 --> 00:30:05.950 So we have 6X. 00:30:06.830 --> 00:30:13.980 Plus Y equals 34 and so equation two has become. Now 00:30:13.980 --> 00:30:19.240 equation for. We want to eliminate one of the variables 00:30:19.240 --> 00:30:24.050 OK, which one well I'd have to do quite a bit of multiplication 00:30:24.050 --> 00:30:30.340 by 6:00 AM by 4. If it was, the ex is that I wanted to get rid 00:30:30.340 --> 00:30:35.150 of look, there's a minus five here and one there, so to speak. 00:30:35.150 --> 00:30:39.960 So if we multiply this one by five, will get these two the 00:30:39.960 --> 00:30:45.140 same. So let's do that 4X minus five Y equals 0, and then times 00:30:45.140 --> 00:30:46.620 in this by 5. 00:30:46.930 --> 00:30:54.085 30X plus five Y equals and then we do this by 5, five, 420, not 00:30:54.085 --> 00:31:01.240 down and two to carry 5 threes are 15 and the two is 17, so 00:31:01.240 --> 00:31:07.918 that gives us 170 and now we can just add these two together. So 00:31:07.918 --> 00:31:13.642 equation three state as it was equation for we multiplied by 5. 00:31:13.642 --> 00:31:16.981 So that's gone to equation 5 and 00:31:16.981 --> 00:31:21.920 now. Finally, we're going to add together equations three 00:31:21.920 --> 00:31:29.070 and five, and so we have 34 X equals 170 and wise have 00:31:29.070 --> 00:31:30.170 been illuminated. 00:31:33.760 --> 00:31:41.752 34 X is 170 and so X is 170 / 34 and 00:31:41.752 --> 00:31:44.416 that gives us 5. 00:31:45.430 --> 00:31:50.050 We need to go back and substituting to one of our two 00:31:50.050 --> 00:31:53.930 equations. It's just have a look which one? 00:31:55.110 --> 00:31:57.954 Doesn't really matter, I think. Actually choose to go 00:31:57.954 --> 00:32:02.062 for that one. Why? because I can see that five over 5 gives 00:32:02.062 --> 00:32:05.538 me one, and that's a very simple number. Might make the 00:32:05.538 --> 00:32:06.802 arithmetic so much easier. 00:32:08.080 --> 00:32:15.304 So we'll have X over 5 minus Y. Over 4 equals 0. Take the 00:32:15.304 --> 00:32:17.368 Five and substituted in. 00:32:21.520 --> 00:32:27.895 5 over 5. That's just one, and so I have one takeaway Y over 4 00:32:27.895 --> 00:32:33.845 equals 0, so one must be equal to Y over 4. If I multiply 00:32:33.845 --> 00:32:38.945 everything by 4I end up with four equals Y. So there's my 00:32:38.945 --> 00:32:44.470 pair of answers X equals 5, Y equals 4 and I really should 00:32:44.470 --> 00:32:48.720 just check by looking at the second equation now, remember. 00:32:49.050 --> 00:32:54.117 2nd equation was 3X plus 1/2 Y equals 17. 00:32:55.690 --> 00:32:59.102 3X also, half Y 00:32:59.102 --> 00:33:05.975 equals 17. So let's substitute these in. X is 5, three X is. 00:33:05.975 --> 00:33:11.785 Therefore AR15, three fives plus 1/2 of Y. But why is 4 so 1/2 00:33:11.785 --> 00:33:18.010 of it is 2. That gives me 17, which is what I want. Yes, this 00:33:18.010 --> 00:33:23.946 is correct. Let's just recap for a moment. Apparel simultaneous 00:33:23.946 --> 00:33:29.066 equations. They represent two straight lines in effect when we 00:33:29.066 --> 00:33:34.698 solve them together, we are looking for the point where the 00:33:34.698 --> 00:33:36.746 two straight lines intersect. 00:33:38.870 --> 00:33:43.017 The method of elimination is much, much better to use than 00:33:43.017 --> 00:33:45.279 the first method that we saw. 00:33:46.270 --> 00:33:50.287 Remember also in the way that we've set this one out. Keep a 00:33:50.287 --> 00:33:52.450 record of what it is that you 00:33:52.450 --> 00:33:56.636 do. Set you workout so that the equal signs come under each 00:33:56.636 --> 00:34:00.354 other and so that at a glance you can look at what you've 00:34:00.354 --> 00:34:01.498 done. Check your working. 00:34:02.080 --> 00:34:06.460 Finally, remember the answer that you get can always be 00:34:06.460 --> 00:34:10.840 checked by substituting the pair of values into the equations 00:34:10.840 --> 00:34:15.658 that you began with. That means strictly you should never get 00:34:15.658 --> 00:34:19.162 one of these wrong. However, mistakes do happen.