Solve system of equation using matrices (4 variables)

Question

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?

Answer 1

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.

Answer 2

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 $-2$.