Here are some examples for solving number problems with two variables.

##### Example 1

The sum of two numbers is 15. The difference of the same two numbers is 7. What are the two numbers?

First, circle what you're looking for— *the two numbers.* Let *x* stand for the larger number and *y* stand for the second number. Now, set up two equations.

The sum of the two numbers is 15.

*x* + *y* = 15

The difference is 7.

*x* – *y* = 7

Now, solve by adding the two equations.

Now, plugging into the first equation gives

The numbers are 11 and 4.

##### Example 2

The sum of twice one number and three times another number is 23 and their product is 20. Find the numbers.

First, circle what you must find— *the numbers*. Let *x* stand for the number that is being multiplied by 2 and *y* stand for the number being multiplied by 3.

Now set up two equations.

The sum of twice a number and three times another number is 23.

2 *x* + 3 *y* = 23

Their product is 20.

*x*( *y*) = 20

Rearranging the first equation gives

3 *y* = 23 – 2 *x*

Dividing each side of the equation by 3 gives

Now, substituting the first equation into the second gives

Multiplying each side of the equation by 3 gives

23 *x* – 2 *x* ^{2} = 60

Rewriting this equation in standard quadratic form gives

2 *x* ^{2} – 23 *x* + 60 = 0

Solving this quadratic equation using factoring gives

(2 *x* – 15)( *x* – 4) = 0

Setting each factor equal to 0 and solving gives

With each *x* value we can find its corresponding *y* value.

If , then or .

If *x* = 4, then or .

Therefore, this problem has two sets of solutions.

The number being multiplied by 2 is , and the number being multiplied by 3 is , or the number being multiplied by 2 is 4 and the number being multiplied by 3 is 5.