$\left\{\begin{array} { l } x=-13-y+3z \\ 5x+4y+z=17 \\ -3x+y-4z=-34\end{array} \right.$
Substitute the given value of $x$ into the equation $5x+4y+z=17$$\left\{\begin{array} { l } 5\left( -13-y+3z \right)+4y+z=17 \\ -3x+y-4z=-34\end{array} \right.$
Substitute the given value of $x$ into the equation $-3x+y-4z=-34$$\left\{\begin{array} { l } 5\left( -13-y+3z \right)+4y+z=17 \\ -3\left( -13-y+3z \right)+y-4z=-34\end{array} \right.$
Simplify the expression$\left\{\begin{array} { l } -y+16z=82 \\ -3\left( -13-y+3z \right)+y-4z=-34\end{array} \right.$
Simplify the expression$\left\{\begin{array} { l } -y+16z=82 \\ 4y-13z=-73\end{array} \right.$
Multiply both sides of the equation by $4$$\left\{\begin{array} { l } -4y+64z=328 \\ 4y-13z=-73\end{array} \right.$
Sum the equations vertically to eliminate at least one variable$51z=255$
Divide both sides of the equation by $51$$z=5$
Substitute the given value of $z$ into the equation $4y-13z=-73$$4y-13 \times 5=-73$
Solve the equation for $y$$y=-2$
Substitute the given values of $\begin{array} { l }y,& z\end{array}$ into the equation $x=-13-y+3z$$x=-13-\left( -2 \right)+3 \times 5$
Simplify the expression$x=4$
The possible solution of the system is the ordered triple $\left( x, y, z\right)$$\left( x, y, z\right)=\left( 4, -2, 5\right)$
Check if the given ordered triple is a solution of the system of equations$\left\{\begin{array} { l } 4+\left( -2 \right)-3 \times 5=-13 \\ 5 \times 4+4 \times \left( -2 \right)+5=17 \\ -3 \times 4+\left( -2 \right)-4 \times 5=-34\end{array} \right.$
Simplify the equalities$\left\{\begin{array} { l } -13=-13 \\ 17=17 \\ -34=-34\end{array} \right.$
Since all of the equalities are true, the ordered triple is the solution of the system$\left( x, y, z\right)=\left( 4, -2, 5\right)$