Question

The sum of three numbers is 12. If the second number is subtracted from the sum of the first and third numbers, the result is 2. If the third number is subtracted from the sum of the first and second numbers, the result is 4. Find the three numbers.

Answer 1

the numbers are 3, 5, 4

Explanation:

The given relations can be written as a system of equations. These are conveniently solved using elimination.

Setup

Let the three numbers be represented by x, y, z. The given relations are ...

x +y +z = 12

x -y +z = 2

x +y -z = 4

Solution

Subtracting the second equation from the first gives ...

(x +y +z) -(x -y +z) = (12) -(2)

2y = 10 . . . . . . . . . simplify

y = 5 . . . . . . . . divide by 2

Subtracting the third equation from the first gives ...

(x +y +z) -(x +y -z) = (12) -(4)

2z = 8 . . . . . . . . . simplify

z = 4 . . . . . . . . divide by 2

Using these values of y and z in the first equation lets us find x:

x +5 +4 = 12

x = 3 . . . . . . . . . subtract 9

The numbers, in order, are (3, 5, 4).