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
