1 00:00:00,000 --> 00:00:00,360 2 00:00:00,360 --> 00:00:04,530 In the last video, we saw what a system of equations is. 3 00:00:04,530 --> 00:00:06,970 And in this video, I'm going to show you one algebraic 4 00:00:06,969 --> 00:00:09,390 technique for solving systems of equations, where you don't 5 00:00:09,390 --> 00:00:12,630 have to graph the two lines and try to figure out exactly 6 00:00:12,630 --> 00:00:13,560 where they intersect. 7 00:00:13,560 --> 00:00:15,609 This will give you an exact algebraic answer. 8 00:00:15,609 --> 00:00:17,289 And in future videos, we'll see more 9 00:00:17,289 --> 00:00:18,289 methods of doing this. 10 00:00:18,289 --> 00:00:19,759 So let's say you had two equations. 11 00:00:19,760 --> 00:00:27,910 One is x plus 2y is equal to 9, and the other equation is 12 00:00:27,910 --> 00:00:35,140 3x plus 5y is equal to 20. 13 00:00:35,140 --> 00:00:37,810 Now, if we did what we did in the last video, we could graph 14 00:00:37,810 --> 00:00:38,350 each of these. 15 00:00:38,350 --> 00:00:39,009 These are lines. 16 00:00:39,009 --> 00:00:42,629 You could put them in either slope-intercept form or 17 00:00:42,630 --> 00:00:43,560 point-slope form. 18 00:00:43,560 --> 00:00:45,289 They're in standard form right now. 19 00:00:45,289 --> 00:00:47,530 And then you could graph each of these lines, figure out 20 00:00:47,530 --> 00:00:50,039 where they intersect, and that would be a solution to that. 21 00:00:50,039 --> 00:00:52,710 But it's sometimes hard to find, to just by looking, 22 00:00:52,710 --> 00:00:54,280 figure out exactly where they intersect. 23 00:00:54,280 --> 00:00:56,890 So let's figure out a way to algebraically do this. 24 00:00:56,890 --> 00:00:59,149 And what I'm going to do is the substitution method. 25 00:00:59,149 --> 00:01:02,399 I'm going to use one of the equations to solve for one of 26 00:01:02,399 --> 00:01:05,439 the variables, and then I'm going to substitute back in 27 00:01:05,439 --> 00:01:06,879 for that variable over here. 28 00:01:06,879 --> 00:01:08,569 So let me show you what I'm talking about. 29 00:01:08,569 --> 00:01:11,500 So let me solve for x using this top equation. 30 00:01:11,500 --> 00:01:16,439 So the top equation says x plus 2y is equal to 9. 31 00:01:16,439 --> 00:01:20,409 I want to solve for x, so let's subtract 2y from both 32 00:01:20,409 --> 00:01:22,519 sides of this equation. 33 00:01:22,519 --> 00:01:28,500 So I'm left with x is equal to 9 minus 2y. 34 00:01:28,500 --> 00:01:30,780 This is what this first equation is telling me. 35 00:01:30,780 --> 00:01:32,960 I just rearranged it a little bit. 36 00:01:32,959 --> 00:01:34,709 The first equation is saying that. 37 00:01:34,709 --> 00:01:39,349 So in order to satisfy both of these equations, x has to 38 00:01:39,349 --> 00:01:41,890 satisfy this constraint right here. 39 00:01:41,890 --> 00:01:47,079 So I can substitute this back in for x. 40 00:01:47,079 --> 00:01:49,429 We're saying, this top equation says, x has to be 41 00:01:49,430 --> 00:01:50,320 equal to this. 42 00:01:50,319 --> 00:01:52,689 Well, if x has to be equal to that, let's 43 00:01:52,689 --> 00:01:54,819 substitute this in for x. 44 00:01:54,819 --> 00:01:59,989 So this second equation will become 3 times x. 45 00:01:59,989 --> 00:02:07,299 And instead of an x, I'll write this thing, 9 minus 2y. 46 00:02:07,299 --> 00:02:15,090 3 times 9 minus 2y, plus 5y is equal to 20. 47 00:02:15,090 --> 00:02:16,539 That's why it's called the substitution method. 48 00:02:16,539 --> 00:02:18,250 I just substituted for x. 49 00:02:18,250 --> 00:02:20,729 And the reason why that's useful is now I have one 50 00:02:20,729 --> 00:02:23,829 equation with one unknown, and I can solve for y. 51 00:02:23,830 --> 00:02:25,060 So let's do that 52 00:02:25,060 --> 00:02:27,810 3 times 9 is 27. 53 00:02:27,810 --> 00:02:35,150 3 times negative 2 is negative 6y, plus 5y is equal to 20. 54 00:02:35,150 --> 00:02:39,180 Add the negative 6y plus the 5y, add those two terms. You 55 00:02:39,180 --> 00:02:45,750 have 27-- let's see, this will be-- minus y is equal to 20. 56 00:02:45,750 --> 00:02:48,870 Let's subtract 27 from both sides. 57 00:02:48,870 --> 00:02:51,259 And you get-- let me write it out here. 58 00:02:51,259 --> 00:02:56,109 So let's subtract 27 from both sides. 59 00:02:56,110 --> 00:02:59,250 The left-hand side, the 27's cancel each other out. 60 00:02:59,250 --> 00:03:03,419 And you're left with negative y is equal to 20 minus 27, is 61 00:03:03,419 --> 00:03:04,769 negative 7. 62 00:03:04,770 --> 00:03:07,170 And then we can multiply both sides of this equation by 63 00:03:07,169 --> 00:03:12,449 negative 1, and we get y is equal to 7. 64 00:03:12,449 --> 00:03:20,319 So we found the y value of the point of intersection of these 65 00:03:20,319 --> 00:03:22,319 two lines. y is equal to 7. 66 00:03:22,319 --> 00:03:26,549 Let me write over here, so I don't have to keep scrolling 67 00:03:26,550 --> 00:03:29,660 down and back up. y is equal to 7. 68 00:03:29,659 --> 00:03:32,289 Well, if we know y, we can now solve for x. x is 69 00:03:32,289 --> 00:03:34,780 equal to 9 minus 2y. 70 00:03:34,780 --> 00:03:36,120 So let's do that. 71 00:03:36,120 --> 00:03:42,009 x is equal to 9 minus 2 times y, 2 times 7. 72 00:03:42,009 --> 00:03:49,209 Or x is equal to 9 minus 14, or x is equal to negative 5. 73 00:03:49,210 --> 00:03:51,640 So we've just, using substitution, we've been able 74 00:03:51,639 --> 00:03:55,599 to find a pair of x and y points that 75 00:03:55,599 --> 00:03:57,019 satisfy these equations. 76 00:03:57,020 --> 00:04:00,990 The point x is equal to negative 5, y is equal to 7, 77 00:04:00,990 --> 00:04:02,520 satisfy both of these. 78 00:04:02,520 --> 00:04:04,830 And you can try it out. 79 00:04:04,830 --> 00:04:10,270 Negative 5 plus 2 times 7, that's negative 5 plus 14, 80 00:04:10,270 --> 00:04:11,990 that is indeed 9. 81 00:04:11,990 --> 00:04:12,909 You do this equation. 82 00:04:12,909 --> 00:04:18,170 3 times negative 5 is negative 15, plus 5 times y, 83 00:04:18,170 --> 00:04:19,670 plus 5 times 7. 84 00:04:19,670 --> 00:04:24,310 So negative 15 plus 35 is indeed 20. 85 00:04:24,310 --> 00:04:26,350 So this satisfies both equations. 86 00:04:26,350 --> 00:04:28,640 If you were to graph both of these equations, they would 87 00:04:28,639 --> 00:04:33,089 intersect at the point negative 5 comma 7. 88 00:04:33,089 --> 00:04:38,000 Now let's use our newly found skill to do 89 00:04:38,000 --> 00:04:40,829 an actual word problem. 90 00:04:40,829 --> 00:04:46,529 Let's say that they tell us that the sum of 91 00:04:46,529 --> 00:04:55,129 two numbers is 70. 92 00:04:55,129 --> 00:05:00,709 And they differ-- or maybe we could say their difference-- 93 00:05:00,709 --> 00:05:06,819 they differ by 11. 94 00:05:06,819 --> 00:05:08,069 What are the numbers? 95 00:05:08,069 --> 00:05:14,899 96 00:05:14,899 --> 00:05:16,979 So let's do this word problem. 97 00:05:16,980 --> 00:05:18,460 So let's define some variables. 98 00:05:18,459 --> 00:05:29,209 Let's let x be the larger number, and let y be the 99 00:05:29,209 --> 00:05:30,419 smaller number. 100 00:05:30,420 --> 00:05:33,180 I'm just arbitrarily creating these variables. 101 00:05:33,180 --> 00:05:34,639 One of them is larger than the other. 102 00:05:34,639 --> 00:05:39,939 They differ by 11. 103 00:05:39,939 --> 00:05:44,019 Now, this first statement, the sum of the two numbers is 70. 104 00:05:44,019 --> 00:05:51,339 That tells us that x plus y must be equal to 70. 105 00:05:51,339 --> 00:05:56,349 That second statement, that they differ by 11. 106 00:05:56,350 --> 00:05:58,450 That means the larger number minus the smaller 107 00:05:58,449 --> 00:06:00,289 number must be 11. 108 00:06:00,290 --> 00:06:06,319 That tells us that x minus y must be equal to 11. 109 00:06:06,319 --> 00:06:06,959 So there we have it. 110 00:06:06,959 --> 00:06:08,509 We have two equations and two unknowns. 111 00:06:08,509 --> 00:06:10,529 We have a system of two equations. 112 00:06:10,529 --> 00:06:15,039 We can now solve it using the substitution method. 113 00:06:15,040 --> 00:06:18,189 So let's solve for x on this equation right here. 114 00:06:18,189 --> 00:06:21,160 So if you add y to both sides of this 115 00:06:21,160 --> 00:06:26,610 equation, what do you get? 116 00:06:26,610 --> 00:06:28,699 On the left-hand side, you just get an x, because these 117 00:06:28,699 --> 00:06:29,889 cancel out. 118 00:06:29,889 --> 00:06:33,000 And then on the right-hand side, you get x is equal to 11 119 00:06:33,000 --> 00:06:36,629 plus y, or y plus 11. 120 00:06:36,629 --> 00:06:39,240 So we get x is equal to 11 plus y 121 00:06:39,240 --> 00:06:40,829 using the second equation. 122 00:06:40,829 --> 00:06:44,069 And then we can substitute it back into this top equation. 123 00:06:44,069 --> 00:06:46,769 So instead of writing x plus y is equal to 70, we can 124 00:06:46,769 --> 00:06:49,849 substitute this in for x. 125 00:06:49,850 --> 00:06:53,210 We've already used the second equation, the magenta one, now 126 00:06:53,209 --> 00:06:55,359 we have to use the top constraint. 127 00:06:55,360 --> 00:07:00,600 So if we substitute this in, we get y plus 11-- remember, 128 00:07:00,600 --> 00:07:04,990 this is what x was, we're substituting that in for x-- 129 00:07:04,990 --> 00:07:08,699 plus y is equal to 70. 130 00:07:08,699 --> 00:07:11,709 This is x. 131 00:07:11,709 --> 00:07:14,870 And that constraint was given to us by this second equation, 132 00:07:14,870 --> 00:07:16,649 or by this second statement. 133 00:07:16,649 --> 00:07:20,019 I just substituted this x with y plus 11, and I was able to 134 00:07:20,019 --> 00:07:22,479 do that because that's the constraint the second 135 00:07:22,480 --> 00:07:24,420 equation gave us. 136 00:07:24,420 --> 00:07:26,280 So now let's just solve for y. 137 00:07:26,279 --> 00:07:32,229 We get y plus 11, plus y is equal to 70. 138 00:07:32,230 --> 00:07:36,470 That's 2y plus 11 is equal to 70. 139 00:07:36,470 --> 00:07:45,430 And then if we subtract 11 from both sides, we get 2y is 140 00:07:45,430 --> 00:07:46,569 equal to-- what is that? 141 00:07:46,569 --> 00:07:48,219 59? 142 00:07:48,220 --> 00:07:50,160 You subtract 10 from 70, you get 60, so 143 00:07:50,160 --> 00:07:51,530 it's going to be 59. 144 00:07:51,529 --> 00:07:56,119 So y is equal to 59 over 2. 145 00:07:56,120 --> 00:08:00,240 Or another way to write it, you could write that as 59 146 00:08:00,240 --> 00:08:10,110 over 2 is the same thing as-- let's see-- 25-- 29.5. 147 00:08:10,110 --> 00:08:15,379 y is equal to 29.5. 148 00:08:15,379 --> 00:08:17,649 Now, what is x going to be equal to? 149 00:08:17,649 --> 00:08:21,089 Well, we already figured out x is equal to y plus 11. 150 00:08:21,089 --> 00:08:26,119 So x is going to be equal to 29.5-- that's what y is, we 151 00:08:26,120 --> 00:08:30,834 just figured that out-- plus 11, which is equal to-- so you 152 00:08:30,834 --> 00:08:32,548 add 10, you get 39.5. 153 00:08:32,548 --> 00:08:36,808 You add another 1, you get 40.5. 154 00:08:36,808 --> 00:08:37,990 And we're done. 155 00:08:37,990 --> 00:08:40,690 If you wanted to find the intersection of these two 156 00:08:40,690 --> 00:08:48,140 lines, it would intersect at the point 40.5 comma 29.5. 157 00:08:48,139 --> 00:08:51,434 And you could have used this equation to solve for x and 158 00:08:51,434 --> 00:08:52,199 then substituted in this one. 159 00:08:52,200 --> 00:08:53,756 You could have used this equation to solve for y and 160 00:08:53,755 --> 00:08:55,250 then substituted in this one. 161 00:08:55,250 --> 00:08:57,245 You could use this equation to solve for y and then 162 00:08:57,245 --> 00:08:58,759 substitute into that equation. 163 00:08:58,759 --> 00:09:02,769 The important thing is, is you use both constraints. 164 00:09:02,769 --> 00:09:05,850 Now let's just verify that this actually works out. 165 00:09:05,850 --> 00:09:07,330 What's the sum of these two numbers? 166 00:09:07,330 --> 00:09:11,600 40.5 plus 29.5, that indeed is 70. 167 00:09:11,600 --> 00:09:15,250 And the difference between the two is indeed 11. 168 00:09:15,250 --> 00:09:17,549 They're exactly 11 apart. 169 00:09:17,549 --> 00:09:20,319 Anyway, hopefully you found that useful.