We can solve this system of equations using the elimination method. The given system is:

1. $x - y = 2$
2. $-2x + 2y = 2$

Step 1: Add the equations to eliminate one variable.

First, observe that adding the two equations will eliminate the $y$-terms since their coefficients are opposites.

$(x - y) + (-2x + 2y) = 2 + 2$

Simplifying both sides:

$x - y - 2x + 2y = 4$ $- x + y = 4$

Step 2: Substitute or simplify further.

The resulting equation is $-x + y = 4$. However, this doesn't directly eliminate a variable, so we could revisit the first equation to try a simpler strategy:

Multiplying