Eliminating the variable x by multiplying the first equation with -3.

$ \displaystyle x+2y=2\Leftrightarrow -3x-6y=-6$

We obtain equivalent systems.

$ \displaystyle \left\{ {\begin{array}{*{20}{l}} {x+2y=2} \\ {3x+y=1} \end{array}} \right.\Leftrightarrow \left\{ \begin{array}{l}-3x-6y=-6\\3x+y=1\end{array} \right.$

Then we add the equations to eliminate the variable x and obtain one equation with one variable.

$\displaystyle \frac{\begin{array}{l}-3x-6y=-6\\~~~3x+~~y~~=~~1\end{array}}{{~~~~~~-5y=-5}}$