Solving systems with substitution
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We're asked to solve and graph this system of equations here. And just as a bit of a review, solving a system of equations really just means figuring out the x and y value that will satisfy both of these equations. And one way to do it is to use one of the equations to solve for either the x or the y, and then substitute for that value in the other one. That makes sure that you're making use of both constraints. So let's start with this bottom equation right here. So we have y minus x is equal to 5. It's pretty straightforward to solve for y here. We just have to add x to both sides of this equation. So add x. And so the left-hand side, these x's cancel out, the negative x and the positive x, and we're left with y is equal to 5 plus x. Now, the whole point of me doing that, is now any time we see a y in the other equation, we can replace it with a 5 plus x. So the other equation was-- let me do it in this orange color-- 9x plus 3y is equal to 15. This second equation told us, if we just rearranged it, that y is equal to 5 plus x, so we can replace y in the second equation with 5 plus x. That makes sure we're making use of both constraints. So let's do that. We're going to replace y with 5 plus x. So this 9x plus 3y equals 15 becomes 9x plus 3 times y. The second equation says y is 5 plus x. So we're going to put 5 plus x there instead of a y. 3 times 5 plus x is equal to 15. And now we can just solve for x. We get 9x plus 3 times 5 is 15 plus 3 times x is 3x is equal to 15. So we can add the 9x and the 3x, so we get 12x, plus 15, is equal to 15. Now we can subtract 15 from both sides, just so you get only x terms on the left-hand side. These guys cancel each other out, and you're left with 12x is equal to 0. Now you divide both sides by 12, and you get x is equal to 0/12, or x is equal to 0. So let me scroll down a little bit. So x is equal to 0. Now if x equals 0, what is y? Well, we could substitute into either one of these equations up here. If we substitute x equals 0 in this first equation, you get 9 times 0 plus 3y is equal to 15. Or that's just a 0, so you get 3y is equal to 15. Divide both sides by 3, you get y is equal to 15/3 or 5. y is equal to 5. And we can test that that also satisfies this equation. y, 5, minus 0 is also equal to 5. So the value x is equal to-- I'll do this in green. x is equal to 0, y is equal to 5 satisfies both of these equations. So we've done the first part. Let's do the second part, where we're asked to graph it. The second equation is pretty straightforward to graph. We actually ended up putting it in mx plus b form right there. And actually, let me rewrite it. Let me just switch the x and the 5, so it really is in that mx plus b form. So y is equal to x plus 5. So its y-intercept is 5-- 1, 2, 3, 4, 5-- and its slope is 1, right? There's a 1 implicitly being multiplied, or the x is being multiplied by 1. So it looks like Let me see how well I can draw it. The line will look like that. It has a slope of 1. You move back 1, you go down 1. You move forward 1, you go up 1. That's a pretty good job. So that right here is this equation. Now let's graph that top equation. And we just have to put it in mx plus b form, or slope-intercept form. And I'll do that in green. So we have 9x plus 3y is equal to 15. One simplification we can do right from the get-go is every number here is divisible by 3, so let's just divide everything by 3 to make things simpler. So we get 3x plus y is equal to 5. Now we can subtract 3x from both sides. We are left with y is equal to negative 3x plus 5. So that's what this first equation gets turned into, if you put it in slope-intercept form. y is equal to negative 3x plus 5. So if you were to graph it, the y-intercept is 5. 0, 5. And then its slope is negative 3. So you move 1 in the x-direction, you move down 3 in the y-direction. Move 2, you would move down 6. 2, 4, 6. Move 2, you go 2, 4, 6. So this line is going to look something like this. It's going to look something like that right there. As you can see, the solution to this system is the point of intersection of these two lines. It's the combination of x and y that satisfy both of these. Remember, this pink line, or this red line, is all of the x's and y's that satisfy this equation: y minus x is equal to 5. This green line is all of the x's and y's, or all the combinations of them, that satisfy this first equation. Now, the one x and y combination that satisfies both is their point of intersection. And we figured it out algebraically using substitution. That happens at x is equal to 0, y is equal to 5. x is equal to 0-- this is the x-axis-- y is equal to 5 right there.