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