Evaluate: {\text{begin}array l 4x+9y+12z=-1 } 3x+y+16z=-46 2x+7y+3z=19\text{end}array .

Expression: $\left\{\begin{array} { l } 4x+9y+12z=-1 \\ 3x+y+16z=-46 \\ 2x+7y+3z=19\end{array} \right.$

Solve the equation for $y$

$\left\{\begin{array} { l } 4x+9y+12z=-1 \\ y=-46-3x-16z \\ 2x+7y+3z=19\end{array} \right.$

Substitute the given value of $y$ into the equation $4x+9y+12z=-1$

$\left\{\begin{array} { l } 4x+9\left( -46-3x-16z \right)+12z=-1 \\ 2x+7y+3z=19\end{array} \right.$

Substitute the given value of $y$ into the equation $2x+7y+3z=19$

$\left\{\begin{array} { l } 4x+9\left( -46-3x-16z \right)+12z=-1 \\ 2x+7\left( -46-3x-16z \right)+3z=19\end{array} \right.$

Simplify the expression

$\left\{\begin{array} { l } -23x-132z=413 \\ 2x+7\left( -46-3x-16z \right)+3z=19\end{array} \right.$

Simplify the expression

$\left\{\begin{array} { l } -23x-132z=413 \\ -19x-109z=341\end{array} \right.$

Multiply both sides of the equation by $-109$

$\left\{\begin{array} { l } 2507x+14388z=-45017 \\ -19x-109z=341\end{array} \right.$

Multiply both sides of the equation by $132$

$\left\{\begin{array} { l } 2507x+14388z=-45017 \\ -2508x-14388z=45012\end{array} \right.$

Sum the equations vertically to eliminate at least one variable

$-x=-5$

Change the signs on both sides of the equation

$x=5$

Substitute the given value of $x$ into the equation $-19x-109z=341$

$-19 \times 5-109z=341$

Solve the equation for $z$

$z=-4$

Substitute the given values of $\begin{array} { l }z,& x\end{array}$ into the equation $y=-46-3x-16z$

$y=-46-3 \times 5-16 \times \left( -4 \right)$

Simplify the expression

$y=3$

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

$\left( x, y, z\right)=\left( 5, 3, -4\right)$

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

$\left\{\begin{array} { l } 4 \times 5+9 \times 3+12 \times \left( -4 \right)=-1 \\ 3 \times 5+3+16 \times \left( -4 \right)=-46 \\ 2 \times 5+7 \times 3+3 \times \left( -4 \right)=19\end{array} \right.$

Simplify the equalities

$\left\{\begin{array} { l } -1=-1 \\ -46=-46 \\ 19=19\end{array} \right.$

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

$\left( x, y, z\right)=\left( 5, 3, -4\right)$