Answer
$$x=2-3t, \quad y=6+5t , \quad z=t.$$
Work Step by Step
The augmented matrix is given by
$$
\left[ \begin {array}{cccc} 2&5&-19&34\\ 3&8&-31&54
\end {array} \right]
.
$$
Using Gauss-Jordan elimination, we get the row-reduced echelon form as follows
$$\left[ \begin {array}{cccc} 1&0&3&2\\ 0&1&-5&6
\end {array} \right]
.$$
Putting, $z=t$ and by the back substitution we get the solution
$$x=2-3t, \quad y=6+5t , \quad z=t.$$
