The sum of 3 numbers is 15. two times the smallest is 1 less than the largest, while the sum of the largest and smallest is 10. use a linear system in three variables to find the three numbers.

Let x = smallest number

y = middle number

z = largest number

Equations:

sum of 3 numbers: x + y + z = 15

two times smallest is 1 less than largest: 2x = z - 1

sum of largest and smallest is 10: z + x = 10

z + x = 10 can be solved for z ⇒ z = 10 - x

substitute into 2x = z - 1 ⇒ 2x = 10 - x - 1

3x = 9

x = 3 (divided both sides of equation by 3)

Since z + x = 10 ⇒ z + 3 = 10 ⇒ z = 7

Substitute values for x and z into x + y + z = 15 ⇒ 3 + y + 7 = 15

y + 10 = 15

y = 5

answer: 3, 5, 7