Evaluate: {\text{begin}array l x-2y+3z=11 } 4x+y-z=4 2x-y+3z=10\text{end}array .

Expression: $\left\{\begin{array} { l } x-2y+3z=11 \\ 4x+y-z=4 \\ 2x-y+3z=10\end{array} \right.$

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

$\begin{array} { l }\left\{\begin{array} { l } x-2y+3z=11 \\ 4x+y-z=4\end{array} \right.,\\\left\{\begin{array} { l } 4x+y-z=4 \\ 2x-y+3z=10\end{array} \right.\end{array}$

Solve the system of equations

$\begin{array} { l }9x+z=19,\\\left\{\begin{array} { l } 4x+y-z=4 \\ 2x-y+3z=10\end{array} \right.\end{array}$

Solve the system of equations

$\begin{array} { l }9x+z=19,\\6x+2z=14\end{array}$

Write as a system of equations

$\left\{\begin{array} { l } 9x+z=19 \\ 6x+2z=14\end{array} \right.$

Solve the system of equations

$\begin{array} { l }z=1,\\x=2\end{array}$

Substitute the given values of $\begin{array} { l }z,& x\end{array}$ into the equation $4x+y-z=4$

$4 \times 2+y-1=4$

Solve the equation for $y$

$y=-3$

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

$\left( x, y, z\right)=\left( 2, -3, 1\right)$

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

$\left\{\begin{array} { l } 2-2 \times \left( -3 \right)+3 \times 1=11 \\ 4 \times 2+\left( -3 \right)-1=4 \\ 2 \times 2-\left( -3 \right)+3 \times 1=10\end{array} \right.$

Simplify the equalities

$\left\{\begin{array} { l } 11=11 \\ 4=4 \\ 10=10\end{array} \right.$

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

$\left( x, y, z\right)=\left( 2, -3, 1\right)$