Answer
$(-1,2,5)$
Work Step by Step
Given Equations,
$x-2y+z=0$ Equation $(1)$
$3x-y-2z=-15$ Equation $(2)$
$2x-3y+3z=7$ Equation $(3)$
Multiply Equation $(1)$ by $2$ and add with Equation $(2)$
$2(x-2y+z)+3x-y-2z=2(0)-15$
$2x-4y+2z+3x-y-2z=-15$
$5x-5y=-15$
$x-y=-3$ Equation $(4)$
Multiply Equation $(2)$ by $3$
$3(3x-y-2z)=3(-15)$
$9x-3y-6z=-45$ Equation $(5)$
Multiply Equation $(3)$ by $2$
$2(2x-3y+3z)=2(7)$
$4x-6y+6z=14$ Equation $(6)$
Add Equation $(5)$ and Equation $(6)$
$9x-3y-6z+4x-6y+6z=-45+14$
$13x-9y=-31$ Equation $(7)$
Multiply Equation $(4)$ by $9$ and subtract from Equation $(7)$
$13x-9y-9(x-y)=-31-9(-3)$
$13x-9y-9x+9y=-31+27$
$4x=-4$
$x=-1$
Substituting $x$ value in Equation $(4)$
$x-y=-3$
$-1-y=-3$
$-y=-3+1$
$-y=-2$
$y=2$
Substituting $x$ and $y$ value in Equation $(1)$
$x-2y+z=0$
$-1-2(2)+z=0$
$-1-4+z=0$
$-5+z=0$
$z=5$
Solution: $(-1,2,5)$