Evaluate: {\text{begin}array l-3x+2y=-7 } 5x+y=2\text{end}array .

Expression: $\left\{\begin{array} { l } -3x+2y=-7 \\ 5x+y=2\end{array} \right.$

Move the variable to the right-hand side and change its sign

$\left\{\begin{array} { l } 2y=-7+3x \\ 5x+y=2\end{array} \right.$

Move the variable to the right-hand side and change its sign

$\left\{\begin{array} { l } 2y=-7+3x \\ y=2-5x\end{array} \right.$

Divide both sides of the equation by $2$

$\left\{\begin{array} { l } y=-\frac{ 7 }{ 2 }+\frac{ 3 }{ 2 }x \\ y=2-5x\end{array} \right.$

Since both expressions $-\frac{ 7 }{ 2 }+\frac{ 3 }{ 2 }x$ and $2-5x$ are equal to $y$, set them equal to each other forming an equation in $x$

$-\frac{ 7 }{ 2 }+\frac{ 3 }{ 2 }x=2-5x$

Solve the equation for $x$

$x=\frac{ 11 }{ 13 }$

Substitute the given value of $x$ into the equation $y=2-5x$

$y=2-5 \times \frac{ 11 }{ 13 }$

Solve the equation for $y$

$y=-\frac{ 29 }{ 13 }$

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

$\left( x, y\right)=\left( \frac{ 11 }{ 13 }, -\frac{ 29 }{ 13 }\right)$

Check if the given ordered pair is the solution of the system of equations

$\left\{\begin{array} { l } -3 \times \frac{ 11 }{ 13 }+2 \times \left( -\frac{ 29 }{ 13 } \right)=-7 \\ 5 \times \frac{ 11 }{ 13 }+\left( -\frac{ 29 }{ 13 } \right)=2\end{array} \right.$

Simplify the equalities

$\left\{\begin{array} { l } -7=-7 \\ 2=2\end{array} \right.$

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

$\left( x, y\right)=\left( \frac{ 11 }{ 13 }, -\frac{ 29 }{ 13 }\right)$