Answer
The system has a unique solution, that is,
$$x_1=1, \quad x_2=-1, \quad x_3=2.$$
Work Step by Step
Given
$$
\begin{aligned}
x_{1}-2 x_{2}+3 x_{3} &=9 \\
-x_{1}+3 x_{2}-x_{3} &=-6 \\
2 x_{1}-5 x_{2}+5 x_{3} &=17
\end{aligned}.
$$
The augmented matrix is given by
$$
\left[\begin{array}{rrrr}
{1} & {-2} & {3} & {9} \\
{-1} & {3} & {-1} & {-6} \\
{2} & {-5} & {5} & {17}
\end{array}\right].
$$
Using Gauss-Jordan elimination, we get the row-reduced echelon form as follows
$$\left[ \begin {array}{cccc} 1&0&0&1\\ 0&1&0&-1
\\ 0&0&1&2\end {array} \right].
$$
The system has a unique solution, that is,
$$x_1=1, \quad x_2=-1, \quad x_3=2.$$
