Evaluate: {\text{begin}array l 3x+2y=6 } 5x+4y=8\text{end}array .

Expression: $\left\{\begin{array} { l } 3x+2y=6 \\ 5x+4y=8\end{array} \right.$

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

$\left\{\begin{array} { l } 2y=6-3x \\ 5x+4y=8\end{array} \right.$

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

$\left\{\begin{array} { l } 2y=6-3x \\ 4y=8-5x\end{array} \right.$

Divide both sides of the equation by $2$

$\left\{\begin{array} { l } 2y=6-3x \\ 2y=4-\frac{ 5 }{ 2 }x\end{array} \right.$

Since both expressions $6-3x$ and $4-\frac{ 5 }{ 2 }x$ are equal to $2y$, set them equal to each other forming an equation in $x$

$6-3x=4-\frac{ 5 }{ 2 }x$

Solve the equation for $x$

$x=4$

Substitute the given value of $x$ into the equation $2y=4-\frac{ 5 }{ 2 }x$

$2y=4-\frac{ 5 }{ 2 } \times 4$

Solve the equation for $y$

$y=-3$

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

$\left( x, y\right)=\left( 4, -3\right)$

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

$\left\{\begin{array} { l } 3 \times 4+2 \times \left( -3 \right)=6 \\ 5 \times 4+4 \times \left( -3 \right)=8\end{array} \right.$

Simplify the equalities

$\left\{\begin{array} { l } 6=6 \\ 8=8\end{array} \right.$

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

$\left( x, y\right)=\left( 4, -3\right)$