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?

The second matrix should be:

$$\begin{bmatrix}1&-1&-1&1&5\\0&2&2&0&-10\\0&4&2&1&-17\\0&4&-1&1&-11\end{bmatrix}$$

Note the row operations were:

$1)\; -R_1+R_2\to R_2\\
2) \;-3R_1+R_3\to R_3\\
3) \;-R_1+R_4\to R_4$ 

Also, in your "correct" answer, $y$ should be $-2$.
