EXPLANATION
First, isolate x for x+y-z=4:
[tex]x=4-y+z[/tex][tex]\mathrm{Substitute\:}x=4-y+z[/tex][tex]\begin{bmatrix}2\left(4-y+z\right)+3y-z=8\\ 4-y+z-y=-z\end{bmatrix}[/tex]Simplifying the equations by applying the distributive property and adding like terms:
[tex]\begin{bmatrix}y+z+8=8\\ -2y+z+4=-z\end{bmatrix}[/tex]Isolate y for y+z+8=8
[tex]y=-z[/tex][tex]\mathrm{Substitute\:}y=-z[/tex][tex]\begin{bmatrix}-2\left(-z\right)+z+4=-z\end{bmatrix}[/tex]Simplify:
[tex]\begin{bmatrix}3z+4=-z\end{bmatrix}[/tex][tex]z=-1[/tex][tex]\mathrm{For\:}y=-z[/tex][tex]\mathrm{Substitute\:}z=-1[/tex][tex]y=-\left(-1\right)[/tex][tex]\mathrm{Simplify}[/tex][tex]y=1[/tex][tex]\mathrm{For\:}x=4-y+z[/tex][tex]\mathrm{Substitute\:}z=-1,\:y=1[/tex][tex]x=4-1-1[/tex][tex]\mathrm{Simplify}[/tex][tex]x=2[/tex][tex]\mathrm{The\:solutions\:to\:the\:system\:of\:equations\:are:}[/tex][tex]x=2,\:z=-1,\:y=1[/tex]Expressing as an ordered triple:
[tex](2,1,-1)[/tex]