Systems of equations with elimination: 6x-6y=-24 & -5x-5y=-60

We never know when we might have to do a little bit more party planning. So it doesn't hurt to have some practice. And that's what this exercise is doing for us, is generating problems so that we can try solving systems of equations with elimination. And so in this first problem, it says solve for x and y using elimination. And then this is what they have-- 6x minus 6y is equal to negative 24. Negative 5x minus 5y is equal to negative 60. So let me get my scratch pad out to solve this. Let me rewrite it. So they gave us 6x minus 6y is equal to negative 24. And negative 5x minus 5y is equal to negative 60. So what we have to think about, and we saw this in several of the other videos, is when we want to eliminate a variable, we want to manipulate these two equations. And if we were to add the corresponding sides, that variable might disappear. So if we just added a 6x to a negative 5x, that's not going to cancel it out. If this was a negative 6x, that would work out. Or if this was a positive 5x, that would work out. But this isn't exactly right. So if I want to eliminate the x, I have to manipulate these equations so that these two characters might cancel out. And one thing that pops into my brain is it looks like all of this stuff up here is divisible by 6, and all of this stuff down here is divisible by 5. And if we were to divide all this stuff up here by 6, we'd be left with an x over here. And if we were to divide all this bottom stuff by 5, we'd be left with a negative x right over here. And then they just might cancel out. So let's try that out. Let's take this first equation. And we're going to multiply both sides by 1/6. Or another way you could think about it is we're dividing both sides by 6. And as long as we do the same thing to both sides, the equation holds. The equality holds. So if you multiply everything by 1/6, 6x times 1/6 is just going to be x. 6y times 1/6 is just y. So it's negative y. Negative 24 times 1/6 is negative 4. Or you could just view it as negative 24 divided by 6 is negative 4. So this equation, the blue one, we've simplified as x minus y is equal to negative 4. Let's do something similar with the second one. Here we could say we're going to multiply everything times 1/5. Or you could say that we're dividing everything by 5. If we do that, negative 5x divided by 5 is just negative x. Negative 5y divided by 5 is negative y. And then negative 60 divided by 5 is negative 12. And now, this looks pretty interesting. If we add the two left-hand sides-- and remember, we can keep the equality, because we're essentially adding the same thing to both sides. You can imagine we're starting with the blue equation. And on the left-hand side, we're adding negative x minus y. And on the right-hand side, we're adding negative 12. But the second equation tells us that those two things are equal. So we're doing the same principle that we saw when we first started looking to algebra, that you can maintain your equality as long as you add the same thing. On the left-hand side, we're going to add this. And on the right-hand side, we're going to add this. But this second equation tells us that those two things are equal. So we can maintain our equality. So let's do that. What do we get on the left-hand side? Well, you have a positive x and a negative x. They cancel out. That was the whole point behind manipulating them in this way. And then you have negative y minus y, which is negative 2y. And then on the right-hand side, you have negative 4 minus 12, which is negative 16. And these are going to be equal to each other. Once again, we're adding the same thing to both sides. To solve for y, we can divide both sides by negative 2. And we are left with y is equal to positive 8. But we are not done yet. We want to go and substitute back into one of the equations. And we can substitute back into this one and to this one, or this one and this one. The solutions need to satisfy all of these essentially. This blue one is another way of expressing this blue equation. This green equation is another way of expressing this green equation. So I'll go for whichever one seems to be the simplest. And this one seems to be pretty simple right over here. So let's take x minus y-- we just solved that y would be positive 8-- is equal to negative 4. And now to solve for x, we just have to add 8 to both sides. And we are left with, on the left-hand side, negative 8 plus 8 cancels out. You're just left with an x. And negative 4 plus 8 is equal to positive 4. So you get x is equal to 4, y is equal to 8. And you can verify that it would work with either one of these equations. 6 times 4 is 24 minus 6 times 8-- so it's 24 minus 48-- is, indeed, negative 24. Negative 5 times 4 is negative 20, minus negative 40, if y is equal to 8, does, indeed, get you negative 60. So it works out for both of these. And we can try it out by inputting our answers. So x is 4, y is 8. So let's do that. So let me type this in. x is going to be equal to 4. y is going to be equal to 8. And let's check our answer. It is correct. Very good.