$\left\{\begin{array} { l } -4r-4s+3t=24 \\ t=10+2r+3s \\ -3r+5s-4t=-7\end{array} \right.$
Substitute the given value of $t$ into the equation $-4r-4s+3t=24$$\left\{\begin{array} { l } -4r-4s+3\left( 10+2r+3s \right)=24 \\ -3r+5s-4t=-7\end{array} \right.$
Substitute the given value of $t$ into the equation $-3r+5s-4t=-7$$\left\{\begin{array} { l } -4r-4s+3\left( 10+2r+3s \right)=24 \\ -3r+5s-4\left( 10+2r+3s \right)=-7\end{array} \right.$
Simplify the expression$\left\{\begin{array} { l } 2r+5s=-6 \\ -3r+5s-4\left( 10+2r+3s \right)=-7\end{array} \right.$
Simplify the expression$\left\{\begin{array} { l } 2r+5s=-6 \\ -11r-7s=33\end{array} \right.$
Multiply both sides of the equation by $7$$\left\{\begin{array} { l } 14r+35s=-42 \\ -11r-7s=33\end{array} \right.$
Multiply both sides of the equation by $5$$\left\{\begin{array} { l } 14r+35s=-42 \\ -55r-35s=165\end{array} \right.$
Sum the equations vertically to eliminate at least one variable$-41r=123$
Divide both sides of the equation by $-41$$r=-3$
Substitute the given value of $r$ into the equation $2r+5s=-6$$2 \times \left( -3 \right)+5s=-6$
Solve the equation for $s$$s=0$
Substitute the given values of $\begin{array} { l }s,& r\end{array}$ into the equation $t=10+2r+3s$$t=10+2 \times \left( -3 \right)+3 \times 0$
Simplify the expression$t=4$
The possible solution of the system is the ordered triple $\left( r, s, t\right)$$\left( r, s, t\right)=\left( -3, 0, 4\right)$
Check if the given ordered triple is a solution of the system of equations$\left\{\begin{array} { l } -4 \times \left( -3 \right)-4 \times 0+3 \times 4=24 \\ -2 \times \left( -3 \right)-3 \times 0+4=10 \\ -3 \times \left( -3 \right)+5 \times 0-4 \times 4=-7\end{array} \right.$
Simplify the equalities$\left\{\begin{array} { l } 24=24 \\ 10=10 \\ -7=-7\end{array} \right.$
Since all of the equalities are true, the ordered triple is the solution of the system$\left( r, s, t\right)=\left( -3, 0, 4\right)$