Answer
$$x=1, \quad y=0, \quad z= 4, \quad w=-2.$$
Work Step by Step
The augmented matrix is given by
$$\left[ \begin {array}{ccccc} 2&1&-1&2&-6\\ 3&4&0&1&
1\\ 1&5&2&6&-3\\ 5&2&-1&-1&3
\end {array} \right]
.
$$
Adding $-3$ times the first row to $2$ times the second row, adding the first row to $-2$ times the third row, adding $-5$ times the first row to $2$ times the fourth row, we get
$$ \left[ \begin {array}{ccccc} 2&1&-1&2&-6\\ 0&5 &3&-4&20\\ 0&-9&-5 &-10&0
\\ 0&-1&3&-12&36\end {array} \right]
.
$$
Adding $9$ times the second row to $5$ times the third row, adding the second row to $5$ times the fourth row,
$$ \left[ \begin {array}{ccccc} 2&1&-1&2&-6\\ 0&5 &3&-4&20\\ 0&0&2 &-86&180
\\ 0&0&18&-64&200\end {array} \right]
.
$$
Adding $-9$ times the third row to the fourth row, we have
$$ \left[ \begin {array}{ccccc} 2&1&-1&2&-6\\ 0&5 &3&-4&20\\ 0&0&2 &-86&180
\\ 0&0&0&71&-142\end {array} \right]
.
$$
Now, the crossposting system is given by
$$
\begin{align*}
3x+3y+12z&=6 \\
3y+12z&=6.
\end{align*}
$$
The above system has the solution
$$x=1, \quad y=0, \quad z= 4, \quad w=-2.$$
