Answer
$(-2,3,-1)$
Work Step by Step
$2x+2y+z=1$ Equation $(1)$
$-x+y+2z=3$ Equation $(2)$
$x+2y+4z=0$ Equation $(3)$
Add Equation $(2)$ and Equation $(3)$
$-x+y+2z+x+2y+4z=3+0$
$3y+6z = 3$
$y+2z = 1$ Equation $(4)$
Multiply Equation $(2)$ by $2$ and add with Equation $(1)$
$2(-x+y+2z)+2x+2y+z=2(3)+1$
$-2x+2y+4z+2x+2y+z=6+1$
$4y+5z = 7$ Equation $(5)$
Multiply Equation $(4)$ by $4$ and subtract from Equation $(5)$
$4y+5z-4(y+2z) = 7-4(1)$
$4y+5z-4y-8z=7-4$
$-3z=3$
$z=-1$
Substitute $z$ value in Equation $(4)$
$y+2z = 1$
$y+2(-1) = 1$
$y-2=1$
$y=3$
Substitute $z$ and $y$ values in Equation $(3)$
$x+2y+4z=0$
$x+2(3)+4(-1)=0$
$x+6-4=0$
$x+2=0$
$x=-2$
The solution is $(-2,3,-1)$