The augmented matrix for the system can be written as ...
[tex] \left[\begin{array}{ccccc}1&1&5&0&-2\\0&1&6&1&3\\1&3&11&2&1\\1&1&5&1&-2\end{array}\right] [/tex]
where the order of the variables is x, y, z, w. After subtracting the first row from the 3rd and 4th rows, this becomes ...
[tex] \left[\begin{array}{ccccc}1&1&5&0&-2\\0&1&6&1&3\\0&2&6&2&3\\0&0&0&1&0\end{array}\right] [/tex]
Then subtracting twice the second row from the third, you have ...
[tex] \left[\begin{array}{ccccc}1&1&5&0&-2\\0&1&6&1&3\\0&0&-6&0&-3\\0&0&0&1&0\end{array}\right] [/tex]
The 4th row tells you w = 0.
The third row tells you z = -3/-6 = 1/2.
Substituting these values into the equation for y gives y +6/2 +0 = 3, so y = 0.
Finally, substituting into the equation for x, we get x +0 +5/2 = -2, so x = -9/2.
The solution is (x, y, z, w) = (-9/2, 0, 1/2, 0).
