Hi, there--

THE PROBLEM:
What are the two numbers whose sum is 23 and its absolute difference is 5?

A SOLUTION:
Let x be the first number.
Let y be the second number.

The sum of two numbers is 23, so
x + y = 23

The absolute difference between the numbers is 5, so
|x - y| = 5

Now solve this system of equations using substitution.
Rewrite the first equation in "x=" form.
x = 23 - y

Substitute 23-y for win the second equation.
|x - y| = 5
|(23 - y) - y| = 5

Simplify.
|23 - 2y| = 5

Then, either 23 - 2y = 5 OR -(23 - 2y) = 5. Solve both sides of the inequality for y.

23 - 2y = 5
-2y = 5 - 23
-2y = 18
y = -9

OR

-(23-2y) = 5
-23 + 2y = 5
2y = 28
y = 14.

If y = -9, then x + (-9) = 23 and x = 34 because the sum of the numbers is 23. However 
the absolute difference between -9 and 34 is 43 because |34 - (-9)| = 43. The pair, -9 and 
34, is not a solution.

If y = 14, then x + 14 = 23 and x = 9 because the sum of the numbers 23. We see that the 
absolute difference between 9 and 14 is 5 because |9 - 14| = 5.

The two numbers are 9 and 14.

Hope this helps!

Mrs. Figgy
math.in.the.vortex@gmail.com