The matrics of coeficients of the equations are
[tex] \left[\begin{array}{cccc}1&2&2&1\\1&3&-2&-1\\-2&-1&3&3\\1&4&1&-2\end{array}\right]\left[\begin{array}{c}w&x&y&z\end{array}\right] = \left[\begin{array}{c}-2&-6&6&-14\end{array}\right] \ \ \ \ \ \ \ \left\begin{array}{ccc}-R_1+R_2 \rightarrow R_2\\2R_1+R_3\rightarrow R_3\\-R_1+R_4 \rightarrow R_4\end{array}\right[/tex]
[tex] \left[\begin{array}{cccc}1&2&2&1\\0&1&-4&-2\\0&3&7&5\\0&2&-1&-3\end{array}\right]\left[\begin{array}{c}w&x&y&z\end{array}\right] = \left[\begin{array}{c}-2&-4&2&-12\end{array}\right] \ \ \ \ \ \ \ \left\begin{array}{ccc}-2R_2+R_1 \rightarrow R_1\\-3R_2+R_3\rightarrow R_3\\-2R_2+R_4 \rightarrow R_4\end{array}\right[/tex]
[tex] \left[\begin{array}{cccc}1&0&10&5\\0&1&-4&-2\\0&0&19&11\\0&0&7&1\end{array}\right]\left[\begin{array}{c}w&x&y&z\end{array}\right] =\left[\begin{array}{c}6&-4&14&-4\end{array}\right] \ \ \ \ \ \ \ \left\begin{array}{c}\frac{1}{19}R_3\rightarrow R_3\end{array}\right\\ \left[\begin{array}{cccc}1&0&10&5\\0&1&-4&-2\\0&0&1&\frac{11}{19} \\0&0&7&1\end{array}\right]\left[\begin{array}{c}w&x&y&z\end{array}\right]=\left[\begin{array}{c}6&-4&\frac{14}{19} &-4\end{array}\right] \ \ \ \ \ \ \ \left\begin{array}{ccc}-10R_3+R_1 \rightarrow R_1\\4R_3+R_2\rightarrow R_2\\-7R_3+R_4 \rightarrow R_4\end{array}\right[/tex]
[tex]\left[\begin{array}{cccc}1&0&0&- \frac{15}{19} \\0&1&0& \frac{6}{19} \\0&0&1& \frac{11}{19} \\0&0&0&- \frac{58}{19} \end{array}\right]\left[\begin{array}{c}w&x&y&z\end{array}\right] =\left[\begin{array}{c} -\frac{26}{19} &- \frac{20}{19} & \frac{14}{19} &- \frac{174}{19} \end{array}\right] \ \ \ \ \ \ \ \left\begin{array}{c}-\frac{19}{58}R_4\rightarrow R_4\end{array}\right[/tex]
[tex]\left[\begin{array}{cccc}1&0&0&- \frac{15}{19} \\0&1&0& \frac{6}{19} \\0&0&1& \frac{11}{19} \\0&0&0&1\end{array}\right] \left[\begin{array}{c}w&x&y&z\end{array}\right] =\left[\begin{array}{c} -\frac{26}{19} &- \frac{20}{19} & \frac{14}{19} &3\end{array}\right] \ \ \ \ \ \ \ \left\begin{array}{ccc} \frac{15}{19} R_4+R_1 \rightarrow R_1\\ -\frac{6}{19} R_4+R_2\rightarrow R_2\\ -\frac{11}{19} R_4+R_3 \rightarrow R_3\end{array}\right[/tex]
[tex]\left[\begin{array}{cccc}1&0&0&0 \\0&1&0&0 \\0&0&1&0\\0&0&0&1\end{array}\right] \left[\begin{array}{c}w&x&y&z\end{array}\right] =\left[\begin{array}{c}1 &-2 &-1&3\end{array}\right][/tex]
Therefore, w = 1, x = -2, y = -1 and z = 3
