jmc

The upper diagonal line is the graph of $x+y=3.$ The lower diagonal line is the graph of $2x+y=2.$ The $y$-axis is the graph of $x=0$ and the $x$-axis is the graph of $y=0.$ The shaded region includes the solutions to the system. The longest side is the upper diagonal side. The length of this side is $\boxed{3\sqrt{2}}.$