1 00:00:00,000 --> 00:00:00,720 2 00:00:00,720 --> 00:00:03,719 We're asked to solve and graph this system of equations here. 3 00:00:03,720 --> 00:00:07,810 And just as a bit of a review, solving a system of equations 4 00:00:07,809 --> 00:00:11,089 really just means figuring out the x and y value that will 5 00:00:11,089 --> 00:00:13,910 satisfy both of these equations. 6 00:00:13,910 --> 00:00:17,230 And one way to do it is to use one of the equations to solve 7 00:00:17,230 --> 00:00:20,990 for either the x or the y, and then substitute for that value 8 00:00:20,989 --> 00:00:21,819 in the other one. 9 00:00:21,820 --> 00:00:23,870 That makes sure that you're making use of both 10 00:00:23,870 --> 00:00:24,780 constraints. 11 00:00:24,780 --> 00:00:26,830 So let's start with this bottom equation right here. 12 00:00:26,829 --> 00:00:30,899 So we have y minus x is equal to 5. 13 00:00:30,899 --> 00:00:33,100 It's pretty straightforward to solve for y here. 14 00:00:33,100 --> 00:00:36,350 We just have to add x to both sides of this equation. 15 00:00:36,350 --> 00:00:37,850 So add x. 16 00:00:37,850 --> 00:00:40,800 And so the left-hand side, these x's cancel out, the 17 00:00:40,799 --> 00:00:43,979 negative x and the positive x, and we're left with y is equal 18 00:00:43,979 --> 00:00:46,519 to 5 plus x. 19 00:00:46,520 --> 00:00:49,780 20 00:00:49,780 --> 00:00:52,969 Now, the whole point of me doing that, is now any time we 21 00:00:52,969 --> 00:00:56,339 see a y in the other equation, we can replace it 22 00:00:56,340 --> 00:00:57,970 with a 5 plus x. 23 00:00:57,969 --> 00:01:01,009 So the other equation was-- let me do it in this orange 24 00:01:01,009 --> 00:01:06,939 color-- 9x plus 3y is equal to 15. 25 00:01:06,939 --> 00:01:10,019 This second equation told us, if we just rearranged it, that 26 00:01:10,019 --> 00:01:14,840 y is equal to 5 plus x, so we can replace y in the second 27 00:01:14,840 --> 00:01:16,740 equation with 5 plus x. 28 00:01:16,739 --> 00:01:19,459 That makes sure we're making use of both constraints. 29 00:01:19,459 --> 00:01:20,259 So let's do that. 30 00:01:20,260 --> 00:01:23,100 We're going to replace y with 5 plus x. 31 00:01:23,099 --> 00:01:30,379 So this 9x plus 3y equals 15 becomes 9x plus 3 times y. 32 00:01:30,379 --> 00:01:32,539 The second equation says y is 5 plus x. 33 00:01:32,540 --> 00:01:35,510 So we're going to put 5 plus x there instead of a y. 34 00:01:35,510 --> 00:01:42,020 3 times 5 plus x is equal to 15. 35 00:01:42,019 --> 00:01:43,519 And now we can just solve for x. 36 00:01:43,519 --> 00:01:49,969 We get 9x plus 3 times 5 is 15 plus 3 times x is 37 00:01:49,969 --> 00:01:53,719 3x is equal to 15. 38 00:01:53,719 --> 00:02:04,870 So we can add the 9x and the 3x, so we get 12x, plus 15, is 39 00:02:04,870 --> 00:02:07,570 equal to 15. 40 00:02:07,569 --> 00:02:13,819 Now we can subtract 15 from both sides, just so you get 41 00:02:13,819 --> 00:02:15,870 only x terms on the left-hand side. 42 00:02:15,870 --> 00:02:19,270 These guys cancel each other out, and you're left with 12x 43 00:02:19,270 --> 00:02:21,210 is equal to 0. 44 00:02:21,210 --> 00:02:25,480 Now you divide both sides by 12, and you get x is equal to 45 00:02:25,479 --> 00:02:28,379 0/12, or x is equal to 0. 46 00:02:28,379 --> 00:02:30,159 So let me scroll down a little bit. 47 00:02:30,159 --> 00:02:31,879 So x is equal to 0. 48 00:02:31,879 --> 00:02:34,430 49 00:02:34,430 --> 00:02:36,520 Now if x equals 0, what is y? 50 00:02:36,520 --> 00:02:39,850 Well, we could substitute into either one of these 51 00:02:39,849 --> 00:02:41,750 equations up here. 52 00:02:41,750 --> 00:02:45,610 If we substitute x equals 0 in this first equation, you get 9 53 00:02:45,610 --> 00:02:51,340 times 0 plus 3y is equal to 15. 54 00:02:51,340 --> 00:02:55,099 Or that's just a 0, so you get 3y is equal to 15. 55 00:02:55,099 --> 00:03:00,729 Divide both sides by 3, you get y is equal to 15/3 or 5. 56 00:03:00,729 --> 00:03:02,169 y is equal to 5. 57 00:03:02,169 --> 00:03:05,039 And we can test that that also satisfies this equation. 58 00:03:05,039 --> 00:03:09,759 y, 5, minus 0 is also equal to 5. 59 00:03:09,759 --> 00:03:14,719 So the value x is equal to-- I'll do this in green. x is 60 00:03:14,719 --> 00:03:18,689 equal to 0, y is equal to 5 satisfies 61 00:03:18,689 --> 00:03:19,590 both of these equations. 62 00:03:19,590 --> 00:03:21,229 So we've done the first part. 63 00:03:21,229 --> 00:03:24,429 Let's do the second part, where we're asked to graph it. 64 00:03:24,430 --> 00:03:26,510 The second equation is pretty straightforward to graph. 65 00:03:26,509 --> 00:03:29,419 We actually ended up putting it in mx plus 66 00:03:29,419 --> 00:03:30,649 b form right there. 67 00:03:30,650 --> 00:03:32,129 And actually, let me rewrite it. 68 00:03:32,129 --> 00:03:34,329 Let me just switch the x and the 5, so it really is in that 69 00:03:34,330 --> 00:03:35,550 mx plus b form. 70 00:03:35,550 --> 00:03:39,710 So y is equal to x plus 5. 71 00:03:39,710 --> 00:03:45,185 So its y-intercept is 5-- 1, 2, 3, 4, 5-- and 72 00:03:45,185 --> 00:03:46,689 its slope is 1, right? 73 00:03:46,689 --> 00:03:50,039 There's a 1 implicitly being multiplied, or the x is being 74 00:03:50,039 --> 00:03:51,650 multiplied by 1. 75 00:03:51,650 --> 00:03:55,830 So it looks like Let me see how well I can draw it. 76 00:03:55,830 --> 00:03:58,450 The line will look like that. 77 00:03:58,449 --> 00:03:59,639 It has a slope of 1. 78 00:03:59,639 --> 00:04:02,559 You move back 1, you go down 1. 79 00:04:02,560 --> 00:04:04,960 You move forward 1, you go up 1. 80 00:04:04,960 --> 00:04:06,750 That's a pretty good job. 81 00:04:06,750 --> 00:04:09,669 So that right here is this equation. 82 00:04:09,669 --> 00:04:11,759 Now let's graph that top equation. 83 00:04:11,759 --> 00:04:14,530 And we just have to put it in mx plus b form, or 84 00:04:14,530 --> 00:04:16,079 slope-intercept form. 85 00:04:16,079 --> 00:04:18,110 And I'll do that in green. 86 00:04:18,110 --> 00:04:22,620 So we have 9x plus 3y is equal to 15. 87 00:04:22,620 --> 00:04:26,629 One simplification we can do right from the get-go is every 88 00:04:26,629 --> 00:04:28,620 number here is divisible by 3, so let's just divide 89 00:04:28,620 --> 00:04:31,459 everything by 3 to make things simpler. 90 00:04:31,459 --> 00:04:37,180 So we get 3x plus y is equal to 5. 91 00:04:37,180 --> 00:04:39,430 Now we can subtract 3x from both sides. 92 00:04:39,430 --> 00:04:43,379 93 00:04:43,379 --> 00:04:48,899 We are left with y is equal to negative 3x plus 5. 94 00:04:48,899 --> 00:04:51,954 So that's what this first equation gets turned into, if 95 00:04:51,954 --> 00:04:54,740 you put it in slope-intercept form. y is equal to 96 00:04:54,740 --> 00:04:56,850 negative 3x plus 5. 97 00:04:56,850 --> 00:04:59,100 So if you were to graph it, the y-intercept is 5. 98 00:04:59,100 --> 00:05:00,900 0, 5. 99 00:05:00,899 --> 00:05:02,349 And then its slope is negative 3. 100 00:05:02,350 --> 00:05:04,570 So you move 1 in the x-direction, you move down 3 101 00:05:04,569 --> 00:05:05,969 in the y-direction. 102 00:05:05,970 --> 00:05:08,210 Move 2, you would move down 6. 103 00:05:08,209 --> 00:05:10,870 2, 4, 6. 104 00:05:10,870 --> 00:05:14,120 Move 2, you go 2, 4, 6. 105 00:05:14,120 --> 00:05:17,189 So this line is going to look something like this. 106 00:05:17,189 --> 00:05:23,410 It's going to look something like that right there. 107 00:05:23,410 --> 00:05:27,160 As you can see, the solution to this system is the point of 108 00:05:27,160 --> 00:05:29,090 intersection of these two lines. 109 00:05:29,089 --> 00:05:32,899 It's the combination of x and y that satisfy both of these. 110 00:05:32,899 --> 00:05:36,089 Remember, this pink line, or this red line, is all of the 111 00:05:36,089 --> 00:05:38,669 x's and y's that satisfy this equation: y minus 112 00:05:38,670 --> 00:05:40,060 x is equal to 5. 113 00:05:40,060 --> 00:05:43,060 This green line is all of the x's and y's, or all the 114 00:05:43,060 --> 00:05:46,810 combinations of them, that satisfy this first equation. 115 00:05:46,810 --> 00:05:49,949 Now, the one x and y combination that satisfies 116 00:05:49,949 --> 00:05:52,420 both is their point of intersection. 117 00:05:52,420 --> 00:05:54,370 And we figured it out algebraically using 118 00:05:54,370 --> 00:05:55,319 substitution. 119 00:05:55,319 --> 00:05:58,779 That happens at x is equal to 0, y is equal to 5. x is equal 120 00:05:58,779 --> 00:06:03,599 to 0-- this is the x-axis-- y is equal to 5 right there.