Solve each system of equations

$\begin{array}{c}r+s+t=5\\ 2r-7s-3t=13\\ \frac{1}{2}r-\frac{1}{3}s+\frac{2}{3}t=-1\end{array}$

Multiply the equation $r+s+t=5$by $3$and add the new resultant equation to the equation $2r-7s-3t=13$.

$\begin{array}{l}\underset{\_}{\begin{array}{l}r+s+t=5\\ 2r-7s-3t=13\end{array}}\begin{array}{c}\text{multiplyby3}\to \\ \end{array}\underset{\_}{\begin{array}{l}3r+3s+3t=15\\ 2r-7s-3t=13\end{array}}\\ 5r-4s+0=28\end{array}$

So, the resultant equation is $5r-4s=28$.

Next, multiply the equation $r+s+t=5$by $4$and multiply the equation $\frac{1}{2}r-\frac{1}{3}s+\frac{2}{3}t=-1$by 6.

Then subtract the two new resultant equations

$\begin{array}{l}\underset{\_}{\begin{array}{l}r+s+t=5\\ \frac{1}{2}r-\frac{1}{3}s+\frac{2}{3}t=-1\end{array}}\begin{array}{c}\text{multiplyby}4\to \\ \text{multiplyby}6\to \end{array}\underset{\_}{\begin{array}{l}4r+4s+4t=20\\ (-)3r-2s+4t=-6\end{array}}\\ r+6s+0=26\end{array}$

So, the resultant equation is $r+6s=26$

Multiply $5r-4s=28$by 3 and multiply the equation $r+6s=26$by 2.

Then add the two new resultant equations.

$\begin{array}{l}\underset{\_}{\begin{array}{l}5r-4s=28\\ r+6s=26\end{array}}\begin{array}{c}\text{multiplyby3}\\ \text{multiplyby2}\end{array}\underset{\_}{\begin{array}{l}15r-12s=84\\ 2r+12s=52\end{array}}\\ 17r+0=136\end{array}$

Solve $17r=136$for $r$:

$\begin{array}{c}17r=136\\ \frac{17r}{17}=\frac{136}{17}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Dividebothsidesby}17\\ r=8\end{array}$

Substitute$\$r=8\$in\backslash [r+6s=26\backslash ]$and find the value of $s$.

$\begin{array}{c}r+6s=26\\ 8+6s=26\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Substitute}8\text{for}r\\ 6s=18\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Subtract}8\text{frombothsides}\\ s=3\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Dividebothsidesby6}\end{array}$

Substitute $r=8,s=3inr+s+t=5$and find the value of $t$.

$\begin{array}{c}r+s+t=5\\ 8+3+t=5\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}substitute}8\text{for}r,3\text{for}s\\ 11+t=5\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}simplify}\\ t=-6\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Subtract}11\text{\hspace{0.17em}frombothsides}\end{array}$

Hence, the solution of the given system of equations is$\left(r,s,t\right)=\left(8,3,-6\right)$.