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## Topic D: Systems of linear equations and their solutions

# Systems of equations with elimination: Sum/difference of numbers

## Video transcript

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.