The sum of two numbers is 45. one number is 4 times as large as the other. what are the numbers?

We can define two variables:
x = large number
y = small number

Now, set up a system of equations:



Solve the system. I'm going to use the second equation to substitute 'x'  into the first one (x = 4y):




Plug that back into 2nd equation to find both numbers:



So, x = 36 and y = 9.

We can check. 36 + 9 actually does equal 45 and 4 times 9 is 36. We match our initial criteria, so our answer is right.