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
3, 5, 7