$\begin{align*} x + 2y - z &= -3\\ 2x - 4y + z &= -7\\ -2x + 2y - 3z &= 4 \end{align*}$
I multiply (1) by $-2$ and get $-2x-4y+2z=6$ and then add it to (2) which gives me $-8y+3z = -1$.
I add (2) and (3) and get $-2y-2z=-3$
I take the first result and get $y=\frac{-1-3z}{8}$ putting that into $-2y-2z=-3$ I get $\frac{-2(-1-3z)}{8-2z}=-3$ which gives me $2+6z-16z = -24$, which gives me $10z=-26$ which I know isn't right already. I messed up the math somewhere, but I have done this problem about a dozen times and always get the wrong answer no matter what.