The sum of 3 times the larger of two integers and twice the smaller is 10. The difference between the two integers is 10. What are the two integers?

Question

This is an 8th grade AM math practice question. It's: solving a number problem that can be represented by two linear equations in two variables.

Answer 1

Let x be the larger integer

Let y be the smaller integer

 

3 times the larger:  3x

twice the smaller:  2y

The sum of 3 times the larger and twice the smaller:  3x + 2y

is 10:  = 10

 

3x + 2y = 10     equation 1

 

The difference between the two integers is 10:

x - y = 10         equation 2

 

Let's use elimination.  Multiply equation 2 by 2 and add to eliminate y.

 

3x + 2y = 10

2x - 2xy = 20

-----------------

5x = 30

x = 6

 

Substitute that back into one of the original equations and solve for y

 

x - y = 10

6 - y = 10

-y = 4

y = -4

 

Your numbers are -4 and 6