Answer
$$x_1=2, \quad x_2=\frac {3}{2}, \quad x_3=\frac {3}{2}.$$
Work Step by Step
Given $$ \begin{aligned} x_{1}+ x_{2}-3 x_{3} &=-1 \\ -x_{1}+2 x_{2} &=1 \\ x_{1}- x_{2}+ x_{3} &=2 \end{aligned}. $$ The augmented matrix is given by $$ \left[ \begin {array}{cccc} 1&1&-3&1\\ -1&2&0&1 \\ 1&-1&1&2\end {array} \right] . $$ Using Gauss-Jordan elimination, we get the row-reduced echelon form as follows $$\left[ \begin {array}{cccc} 1&0&0&2\\ 0&1&0&{ \frac {3}{2}}\\ 0&0&1&\frac{3}{2}\end{array} \right].$$ From which the solution is $$x_1=2, \quad x_2=\frac {3}{2}, \quad x_3=\frac {3}{2}.$$
