Answer
$$x_1=\frac{2}{3} , \quad x_2=-{\frac {17}{3}}, \quad x_3=-\frac{11}{3}$$.
Work Step by Step
The system can written on the form $Ax=b$ as follows
$$x_1\left[ \begin {array}{cc} {-3} \\{2}\\{1} \end {array} \right]+x_2\left[ \begin {array}{cc} {-1}\\ {4} \\{-2}\end {array} \right]+x_3\left[ \begin {array}{cc} {1}\\ {-5} \\{3}\end {array} \right]=\left[ \begin {array}{cc} {0}\\ {-3}\\{1} \end {array} \right].$$
To solve system using Gaussian elimination, we form the augmented matrix as follows
$$\left[ \begin {array}{cccc} -3&-1&1&0\\ 2&4&-5&-3
\\ 1&-2&3&1\end {array} \right]
.
$$
The reduced row echelon form is given by
$$\left[ \begin {array}{cccc} 1&0&0&\frac{2}{3}\\ 0&1&0&-{
\frac {17}{3}}\\ 0&0&1&-\frac{11}{3}\end {array} \right]
.
$$
From which, we have
$$x_1=\frac{2}{3} , \quad x_2=-{
\frac {17}{3}}, \quad x_3=-\frac{11}{3}$$.
