Answer
$$x_1=-1, \quad x_2=3, \quad x_3=-2.$$
Work Step by Step
Given
$$
\begin{aligned}
x_{1}-5 x_{2}+2 x_{3} &=-20 \\
-3x_{1}+ x_{2}-x_3 &=8 \\
- 2x_{2}+5 x_{3} &=-16
\end{aligned}.
$$
The augmented matrix is given by
$$
\left[ \begin {array}{cccc} 1&-5&2&-20\\ -3&1&-1&8
\\ 0&-2&5&-16\end {array} \right]
.
$$
Using Gauss-Jordan elimination, we get the row-reduced echelon form as follows
$$\left[ \begin {array}{cccc} 1&0&0&-1\\ 0&1&0&3
\\ 0&0&1&-2\end {array} \right]
.$$
From which the solution is
$$x_1=-1, \quad x_2=3, \quad x_3=-2.$$