Evaluate: {\text{begin}array l 4x+5y=-0.5 } 3x+7y=0.6\text{end}array .

Expression: $\left\{\begin{array} { l } 4x+5y=-0.5 \\ 3x+7y=0.6\end{array} \right.$

Multiply both sides of the equation by $3$

$\left\{\begin{array} { l } 12x+15y=-1.5 \\ 3x+7y=0.6\end{array} \right.$

Multiply both sides of the equation by $-4$

$\left\{\begin{array} { l } 12x+15y=-1.5 \\ -12x-28y=-2.4\end{array} \right.$

Sum the equations vertically to eliminate at least one variable

$-13y=-3.9$

Divide both sides of the equation by $-13$

$y=0.3$

Substitute the given value of $y$ into the equation $4x+5y=-0.5$

$4x+5 \times 0.3=-0.5$

Solve the equation for $x$

$x=-\frac{ 1 }{ 2 }$

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

$\left( x, y\right)=\left( -\frac{ 1 }{ 2 }, 0.3\right)$

Convert the decimal into a fraction

$\left( x, y\right)=\left( -\frac{ 1 }{ 2 }, \frac{ 3 }{ 10 }\right)$

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

$\left\{\begin{array} { l } 4 \times \left( -\frac{ 1 }{ 2 } \right)+5 \times \frac{ 3 }{ 10 }=-0.5 \\ 3 \times \left( -\frac{ 1 }{ 2 } \right)+7 \times \frac{ 3 }{ 10 }=0.6\end{array} \right.$

Simplify the equalities

$\left\{\begin{array} { l } -\frac{ 1 }{ 2 }=-\frac{ 1 }{ 2 } \\ \frac{ 3 }{ 5 }=\frac{ 3 }{ 5 }\end{array} \right.$

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

$\left( x, y\right)=\left( -\frac{ 1 }{ 2 }, \frac{ 3 }{ 10 }\right)$