Solve system of equation using matrices (4 variables) The Question:

Solve using matrices.
  $$2w-2x-2y+2z=10\\w+x+y+z=-5\\3w+x-y+4z=-2\\w+3x-2y+2z=-6$$

My work:

$$
\begin{bmatrix}
2&-2&-2&2&10\\
1&1&1&1&-5\\
3&1&-1&4&-2\\
1&3&-2&2&-6\\
\end{bmatrix}
\rightarrow
\begin{bmatrix}
1&-1&-1&1&5\\
0&2&2&0&-10\\
0&4&-1&1&-11\\
0&-2&3&-1&1\\
\end{bmatrix}
\rightarrow\\
\begin{bmatrix}
1&-1&-1&1&5\\
0&2&2&0&-10\\
0&0&-5&0&9\\
0&0&5&-1&-9\\
\end{bmatrix}
\rightarrow
\begin{bmatrix}
1&-1&-1&1&5\\
0&2&2&0&-10\\
0&0&-5&0&9\\
0&0&0&-1&0\\
\end{bmatrix}\\[6ex] 
z=0, \;y=\frac{-9}5,\; x=\frac{34}5,\; w=10
$$

The correct answer is $z=-1, \;y=-2,\; x=-3,\; w=1$. What did I do wrong?

In the second matrix of your work, entry $a_{3,3}$ should be $2$, not $-1$, and entry $a_{3,5}$ should be $-17$, not $-11$. Also, the whole fourth row seems wrong. It appears that the four row operations performed in that step should have been:
\begin{align}
&1.\text{ Replace R1 with $\frac12\times$ R1.} \\
&2.\text{ Replace R2 with R2 - R1.} \\
&3.\text{ Replace R3 with R3 - 3$\times$R1.} \\
&4.\text{ Replace R4 with R4 - R1.}
\end{align}
It appears that you did the first two of those correctly, but not the third and fourth ones.
