1 00:00:00,773 --> 00:00:05,033 Use substitution to solve for x and y and we have a system of equations here. 2 00:00:05,033 --> 00:00:13,571 The first equation is 2y=x+7 and the second equation here is x=y-4. 3 00:00:13,571 --> 00:00:16,638 So what we want to do when we say substitution is we want to 4 00:00:16,638 --> 00:00:19,448 substitute one of the variables with an expression so that we 5 00:00:19,448 --> 00:00:22,410 have an equation in only one variable then we can solve for it. 6 00:00:22,410 --> 00:00:25,462 So let me show you what I'm talking about. Let me write this first degree equation. 7 00:00:25,462 --> 00:00:29,253 2y=x+7 8 00:00:29,253 --> 00:00:36,173 and we have the second equation over her that x=y-4 9 00:00:36,173 --> 00:00:40,671 So if we are looking for an x and y that satisfy both constraints, what we can see is well look.. 10 00:00:40,671 --> 00:00:44,105 if the x and the y's have to satisfy both restraints, both of these 11 00:00:44,105 --> 00:00:50,035 restraints have to be true. x must be equal to y-4. So anywhere in this 12 00:00:50,035 --> 00:00:52,103 top equation where we see an x 13 00:00:52,103 --> 00:00:57,018 we say well look,that x by the second constraint, 14 00:00:57,018 --> 00:00:59,764 that x has to be equal to y-4. 15 00:00:59,764 --> 00:01:03,087 So everywhere we see an x, we can substitute it by a 16 00:01:03,087 --> 00:01:05,168 y-4. 17 00:01:05,168 --> 00:01:09,843 So let's do that. So if we substitute y-4 for x in this top equation, 18 00:01:09,843 --> 00:01:14,383 the top equation becomes 2y is equal to-- instead of an x-- 19 00:01:14,383 --> 00:01:19,253 we'll write a y-4. 20 00:01:19,253 --> 00:01:25,445 And then we have a plus seven. 21 00:01:25,445 --> 00:01:32,780 All I did here was I substited y-4 for x. 22 00:01:32,780 --> 00:01:35,169 The constraint tells us that we need to do it. 23 00:01:35,169 --> 00:01:39,244 Y-4 needs to be equal to x, or x needs to be equal to y-4. 24 00:01:39,244 --> 00:01:42,304 The value here is now we have an equation. 25 00:01:42,304 --> 00:01:44,457 One equation with one variable; we can just 26 00:01:44,457 --> 00:01:46,176 solve for y. 27 00:01:46,176 --> 00:01:55,343 So we get 2y=y, and then we have minus 4 plus 7 so y+3, 28 00:01:55,343 --> 00:02:00,594 we can subtract y from both sides of this equation. 29 00:02:00,594 --> 00:02:05,504 The left hand side is just 2y-y=y. 30 00:02:05,504 --> 00:02:08,929 y is equal to-- these two cancel out; y is equal to three. 31 00:02:08,929 --> 00:02:10,438 And we can go back and substitute into either of these 32 00:02:10,438 --> 00:02:12,051 equations and solve for x. 33 00:02:12,051 --> 00:02:13,666 And it's easier right over here... 34 00:02:13,666 --> 00:02:15,452 so we can substitute, right over here. 35 00:02:15,452 --> 00:02:17,270 x needs to be equal to y-4. 36 00:02:17,270 --> 00:02:19,205 So we can say that x 37 00:02:19,205 --> 00:02:28,743 is equal to 3-4=-1. 38 00:02:28,743 --> 00:02:37,172 So the solution to this system is x=-1, and y=3. 39 00:02:37,172 --> 00:02:39,701 You can verify that this works in this top equation right 40 00:02:39,701 --> 00:02:41,117 over her. 41 00:02:41,117 --> 00:02:47,169 2*3 is 6, which is indeed equal to -1+7. Now I want to show 42 00:02:47,169 --> 00:02:49,329 you that over here we substituted. 43 00:02:49,329 --> 00:02:51,633 We had an expression or we had an equation 44 00:02:51,633 --> 00:02:55,175 that explicitly solved for x so we were able to substitute the x's. 45 00:02:55,175 --> 00:02:57,248 What I want to show you is that we could have done it the other way around. 46 00:02:57,248 --> 00:03:00,410 We could have solved for y and then substituted for the y's. 47 00:03:00,410 --> 00:03:03,013 So let's do that. We could have substituted 48 00:03:03,013 --> 00:03:05,839 from one constraint into the other constraint or vice-versa. 49 00:03:05,839 --> 00:03:08,071 Either way, we would have gotten the same answer. 50 00:03:08,071 --> 00:03:12,655 So instead of saying that x=y-4, in that second equation 51 00:03:12,655 --> 00:03:21,175 if we add 4 to both sides of this equation, we get x+4=y. 52 00:03:21,175 --> 00:03:24,111 This and this is the exact same constraint. 53 00:03:24,111 --> 00:03:27,333 I just added four to both sides to get this constraint over her. 54 00:03:27,333 --> 00:03:31,170 And now since we've solved the equation explicitly for y, 55 00:03:31,170 --> 00:03:35,581 we can use the first constraint-- the first equation-- 56 00:03:35,581 --> 00:03:38,433 and everywhere we see y, we can substitute it with 57 00:03:38,433 --> 00:03:40,773 x+4. 58 00:03:40,773 --> 00:03:43,444 So it's 2 times--instead of two times y-- we can write 59 00:03:43,444 --> 00:03:49,345 it's 2 times (x+4). 60 00:03:49,345 --> 00:03:53,147 which is equal to x+7. 61 00:03:53,147 --> 00:04:01,919 We can distribute this two, so we get 2x+8=x+7 62 00:04:01,919 --> 00:04:04,668 We can subtract x from both sides of this equation. 63 00:04:04,668 --> 00:04:11,841 and then we can subtract 8 from both sides of this equation 64 00:04:11,841 --> 00:04:14,409 subtract 8. 65 00:04:14,409 --> 00:04:15,844 The left hand side, that cancels out 66 00:04:15,844 --> 00:04:18,173 On the right hand side, that cancels out, and we are left with a -1. 67 00:04:18,173 --> 00:04:22,500 Then, we can substitute back over here. We have y=x+4 68 00:04:22,500 --> 00:04:28,503 so we have y=-1+4 which is equal to 3. 69 00:04:28,503 --> 00:04:32,666 So once again we got the same answer, 70 00:04:32,666 --> 00:04:34,001 even though this time we substituted for y, 71 00:04:34,001 --> 00:04:35,730 instead of substituting for x. 72 00:04:35,730 --> 00:04:39,000 Hopefully you find that interesting.