Calculate: {\text{begin}array l x+y-3z=-13 } 5x+4y+z=17-3x+y-4z=-34\text{end}array .

Expression: $\left\{\begin{array} { l } x+y-3z=-13 \\ 5x+4y+z=17 \\ -3x+y-4z=-34\end{array} \right.$

Solve the equation for $x$

$\left\{\begin{array} { l } x=-13-y+3z \\ 5x+4y+z=17 \\ -3x+y-4z=-34\end{array} \right.$

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

$\left( x, y, z\right)=\left( 4, -2, 5\right)$