Evaluate: {\text{begin}array l 3x+5y-z=-3 } 2x+y-3z=-9 x+3y+2z=7\text{end}array .

Expression: $\left\{\begin{array} { l } 3x+5y-z=-3 \\ 2x+y-3z=-9 \\ x+3y+2z=7\end{array} \right.$

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

$\begin{array} { l }\left\{\begin{array} { l } 3x+5y-z=-3 \\ 2x+y-3z=-9\end{array} \right.,\\\left\{\begin{array} { l } 2x+y-3z=-9 \\ x+3y+2z=7\end{array} \right.\end{array}$

Solve the system of equations

$\begin{array} { l }-7x+14z=42,\\\left\{\begin{array} { l } 2x+y-3z=-9 \\ x+3y+2z=7\end{array} \right.\end{array}$

Solve the system of equations

$\begin{array} { l }-7x+14z=42,\\-5x+11z=34\end{array}$

Write as a system of equations

$\left\{\begin{array} { l } -7x+14z=42 \\ -5x+11z=34\end{array} \right.$

Solve the system of equations

$\begin{array} { l }x=2,\\z=4\end{array}$

Substitute the given values of $\begin{array} { l }x,& z\end{array}$ into the equation $2x+y-3z=-9$

$2 \times 2+y-3 \times 4=-9$

Solve the equation for $y$

$y=-1$

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

$\left( x, y, z\right)=\left( 2, -1, 4\right)$

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

$\left\{\begin{array} { l } 3 \times 2+5 \times \left( -1 \right)-4=-3 \\ 2 \times 2+\left( -1 \right)-3 \times 4=-9 \\ 2+3 \times \left( -1 \right)+2 \times 4=7\end{array} \right.$

Simplify the equalities

$\left\{\begin{array} { l } -3=-3 \\ -9=-9 \\ 7=7\end{array} \right.$

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

$\left( x, y, z\right)=\left( 2, -1, 4\right)$