Solve the following system of linear equations using matrix method: $3x+y+z=1,2x+2z=0,5x+5y+2z=2$.

The system of linear equations are

$3x+y+z=1$

$2x+2z=0$

$5x+5y+2x=2$

This corresponds to $\left[\begin{array}{ccc}3& 1& 1\\ 2& 0& 2\\ 5& 5& 2\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 2\end{array}\right]$

The augmented matrix is $\left[\begin{array}{cccc}3& 1& 1& 1\\ 2& 0& 2& 0\\ 5& 5& 2& 2\end{array}\right]$

Applying elementary row operations,

$\left[\begin{array}{cccc}3& 1& 1& 1\\ 2& 0& 2& 0\\ 5& 5& 2& 2\end{array}\right]\stackrel{R_1\to R_1-R_2}{\to }\left[\begin{array}{cccc}1& 1& -1& 1\\ 2& 0& 2& 0\\ 5& 5& 2& 2\end{array}\right]$

$\downarrow R_2\to R_2-2R_1$

$R_3\to R_3-5R_1$

$\left[\begin{array}{cccc}1& 1& -1& 1\\ 0& 1& -2& 1\\ 0& 0& 7& -3\end{array}\right]\stackrel{R_2\to \frac{R_2}{-2}}{\leftarrow }\left[\begin{array}{cccc}1& 1& 1& -1\\ 0& -2& 4& -2\\ 0& 0& 7& -3\end{array}\right]$

$\downarrow R_1\to R_1-R_2$

$\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& -2& 1\\ 0& 0& 7& -3\end{array}\right]\stackrel{R_3\to \frac{R_3}{7}}{\to }\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& -2& 1\\ 0& 0& 1& \frac{-3}{7}\end{array}\right]$

$\downarrow R_2\to R_2+2R_3$

$R_1\to R_1-R_3$

$\left[\begin{array}{cccc}1& 0& 0& \frac{3}{7}\\ 0& 1& 0& \frac{1}{7}\\ 0& 0& 1& \frac{-3}{7}\end{array}\right]$

So, $x=\frac{3}{7},y=\frac{1}{7},z=\frac{-3}{7}$