# Evaluate: {\text{begin}array l-4r-4s+3t=24 }-2r-3s+t=10-3r+5s-4t=-7\text{end}array .

## Expression: $\left\{\begin{array} { l } -4r-4s+3t=24 \\ -2r-3s+t=10 \\ -3r+5s-4t=-7\end{array} \right.$

Solve the equation for $t$

$\left\{\begin{array} { l } -4r-4s+3t=24 \\ t=10+2r+3s \\ -3r+5s-4t=-7\end{array} \right.$

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

$\left( r, s, t\right)=\left( -3, 0, 4\right)$

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