$\left\{\begin{array} { l } 5x+46y=10 \\ 1.8x+1.2y=2\end{array} \right.$
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$\left( x, y\right)=\left( \frac{ 25 }{ 24 }, \frac{ 5 }{ 48 }\right)$