Evaluate: {\text{begin}array l 0.5x+4.6y=1 } 1.8x+1.2y=2\text{end}array .

Expression: $\left\{\begin{array} { l } 0.5x+4.6y=1 \\ 1.8x+1.2y=2\end{array} \right.$

Multiply both sides of the equation by $10$

$\left\{\begin{array} { l } 5x+46y=10 \\ 1.8x+1.2y=2\end{array} \right.$

Multiply both sides of the equation by $10$

$\left\{\begin{array} { l } 5x+46y=10 \\ 18x+12y=20\end{array} \right.$

Multiply both sides of the equation by $18$

$\left\{\begin{array} { l } 90x+828y=180 \\ 18x+12y=20\end{array} \right.$

Multiply both sides of the equation by $-5$

$\left\{\begin{array} { l } 90x+828y=180 \\ -90x-60y=-100\end{array} \right.$

Sum the equations vertically to eliminate at least one variable

$768y=80$

Divide both sides of the equation by $768$

$y=\frac{ 5 }{ 48 }$

Substitute the given value of $y$ into the equation $18x+12y=20$

$18x+12 \times \frac{ 5 }{ 48 }=20$

Solve the equation for $x$

$x=\frac{ 25 }{ 24 }$

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

$\left( x, y\right)=\left( \frac{ 25 }{ 24 }, \frac{ 5 }{ 48 }\right)$

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

$\left\{\begin{array} { l } 0.5 \times \frac{ 25 }{ 24 }+4.6 \times \frac{ 5 }{ 48 }=1 \\ 1.8 \times \frac{ 25 }{ 24 }+1.2 \times \frac{ 5 }{ 48 }=2\end{array} \right.$

Simplify the equalities

$\left\{\begin{array} { l } 1=1 \\ 2=2\end{array} \right.$

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

$\left( x, y\right)=\left( \frac{ 25 }{ 24 }, \frac{ 5 }{ 48 }\right)$