1. **State the problem:** Solve the system of linear equations for $x$ and $y$: $$2x + y = 8$$ $$2x - y = 12$$ 2. **Add the two equations to eliminate $y$:** $$ (2x + y) + (2x - y) = 8 + 12 $$ $$ 2x + y + 2x - y = 20 $$ $$ 4x = 20 $$ 3. **Solve for $x$:** $$ x = \frac{20}{4} $$ $$ x = 5 $$ 4. **Substitute $x=5$ into the first equation to find $y$:** $$ 2(5) + y = 8 $$ $$ 10 + y = 8 $$ $$ y = 8 - 10 $$ $$ y = -2 $$ 5. **Final answer:** $$ x = 5, \quad y = -2 $$