Answer
$$x_1=5 , \quad x_2=2 , \quad x_3=-6$$.
Work Step by Step
The system can written on the form $Ax=b$ as follows
$$x_1\left[ \begin {array}{cc} {2} \\{2}\\{4} \end {array} \right]+x_2\left[ \begin {array}{cc} {3}\\ {-3} \\{-2}\end {array} \right]+x_3\left[ \begin {array}{cc} {1}\\ {-3} \\{3}\end {array} \right]=\left[ \begin {array}{cc} {10}\\ {22}\\{-2} \end {array} \right].$$
To solve system using Gaussian elimination, we form the augmented matrix as follows
$$\left[ \begin {array}{cccc} 2&3&1&10\\ 2&-3&-3&22
\\ 4&-2&3&-2\end {array} \right]
.
$$
The reduced row echelon form is given by
$$ \left[ \begin {array}{cccc} 1&0&0&5\\ 0&1&0&2
\\ 0&0&1&-6\end {array} \right]
.
$$
From which, we have
$$x_1=5 , \quad x_2=2 , \quad x_3=-6$$.
