Solve for: {\text{begin}array l x-y+z=-8 } 4x+2y-z=12-2x+4y-z=36\text{end}array .

Expression: $\left\{\begin{array} { l } x-y+z=-8 \\ 4x+2y-z=12 \\ -2x+4y-z=36\end{array} \right.$

Rewrite the system as two systems, each consisting of two equations

$\begin{array} { l }\left\{\begin{array} { l } x-y+z=-8 \\ 4x+2y-z=12\end{array} \right.,\\\left\{\begin{array} { l } x-y+z=-8 \\ -2x+4y-z=36\end{array} \right.\end{array}$

Solve the system of equations

$\begin{array} { l }5x+y=4,\\\left\{\begin{array} { l } x-y+z=-8 \\ -2x+4y-z=36\end{array} \right.\end{array}$

Solve the system of equations

$\begin{array} { l }5x+y=4,\\-x+3y=28\end{array}$

Write as a system of equations

$\left\{\begin{array} { l } 5x+y=4 \\ -x+3y=28\end{array} \right.$

Solve the system of equations

$\begin{array} { l }x=-1,\\y=9\end{array}$

Substitute the given values of $\begin{array} { l }x,& y\end{array}$ into the equation $x-y+z=-8$

$-1-9+z=-8$

Solve the equation for $z$

$z=2$

The possible solution of the system is the ordered triple $\left( x, y, z\right)$

$\left( x, y, z\right)=\left( -1, 9, 2\right)$

Check if the given ordered triple is a solution of the system of equations

$\left\{\begin{array} { l } -1-9+2=-8 \\ 4 \times \left( -1 \right)+2 \times 9-2=12 \\ -2 \times \left( -1 \right)+4 \times 9-2=36\end{array} \right.$

Simplify the equalities

$\left\{\begin{array} { l } -8=-8 \\ 12=12 \\ 36=36\end{array} \right.$

Since all of the equalities are true, the ordered triple is the solution of the system

$\left( x, y, z\right)=\left( -1, 9, 2\right)$