Systems of equations: two numbers have a difference of 14

Question

Two numbers have a difference of 14. If the larger number is increased by 5
and the smaller number is squared, the sum will be 61. Write a system of
equations for this question and then find the numbers

Answer 1

Let's use x and y, where x is larger.

 

x - y = 14 and 

x + 5 + y² = 61

 

The trick is to focus on only one variable. So, x = y + 14. Plug that into the 2nd equation. 

 

So, y + 14 + 5 + y² = 61

Lumping the constants together, y² + y - 42 = 0

Factorizing, (y + 7)(y - 6) = 0, which means y = -7 or 6.

 

If y = -7, then x = 7

If y = 6, then x = 20

 

Both sets of solutions work when plugged back into the equations.