jmc

If $x<0,$ then $|x|=-x,$ so from the first equation, $y=10.$ But then the second equation gives us $x=12,$ contradiction, so $x\ge 0,$ which means $|x|=x.$

If $y>0,$ then $|y|=y,$ so from the second equation, $x=12.$ But the the first equation gives us $y=-14,$ contradiction, so $y\le 0,$ which means $|y|=-y.$

Thus, the given equations become $2x+y=10$ and $x-2y=12.$ Solving, we find $x=\frac{32}{5}$ and $y=-\frac{14}{5},$ so $x+y=\boxed{\frac{18}{5}}.$