Topic D: Systems of linear equations and their solutions
Systems of equations with elimination: Sum/difference of numbers
Solve using the elimination method. And they tell us the sum of two numbers is 70. Their difference is 24. What are the two numbers? So let's use this first sentence. Let's construct an equation from this first sentence. Let's construct a constraint. The sum of two numbers. Let's call those numbers x and y. So their sum, x plus y, is equal to 70. That's what this first sentence tells us. The second sentence says their difference is 24. So that means that x minus y is equal to 24. We're just going to assume that x is the larger of the two numbers, and y is the smaller one. So when you take their difference like this, you get positive 24. So we have a system of two equations with two unknowns. And they want us to solve it using the elimination method. So let's do that. So if we were to just add these two equations, on the left-hand side, we would have a positive y, and we would have a negative y over here. And they would cancel out. So if we were to just add these two equations, we would be able to eliminate the y's. So let's do that. So x plus y plus x minus y. Well, the plus y and the minus y cancel out. And you're just left with an x plus an x, which is 2x. And then that is going to be equal to 70 plus 24. 70 plus 24 is 94. And I want to make it very clear-- and I mentioned it in previous videos-- that this process of adding the equations to each other, this is nothing new. We're really just adding the same thing to both sides. We could do it as to both sides of this equation. You could say we're adding 24 to both sides of this equation. Over here were explicitly adding 24 to the 70. And over here you could say we could add 24 to x plus y, but the second constraint tells us that x minus y is the same thing as 24. So we're adding the same thing to both sides. Here we're calling it 24. Here we're calling it x minus y. And we were able to eliminate the y. So we get 2x is equal to 94. Now we can divide both sides of this equation by 2. And we are left with x is equal to-- what is that? 47. And now we can substitute back into either one of these equations to solve for y. So let's try this first one over here. So we have 47 plus y is equal to 70. We can subtract 47 from both sides of this equation. So we subtract 47. And we are left with y is equal to-- what is this? 23. y is equal to 23. And you can verify that it works. If you add the two numbers, 47 plus 23, you definitely get 70. And then if you take 47 minus 23, you definitely get 24. So it definitely meets both constraints.