1. **State the problem:** Solve the system of linear equations: $$2x + y = 8$$ $$2x - y = 12$$ 2. **Formula and rules:** To solve for $x$ and $y$, we can use the method of addition (elimination) or substitution. Here, elimination is straightforward. 3. **Add the two equations:** $$ (2x + y) + (2x - y) = 8 + 12 $$ $$ 2x + y + 2x - y = 20 $$ $$ 4x = 20 $$ 4. **Solve for $x$:** $$ x = \frac{20}{4} = 5 $$ 5. **Substitute $x=5$ into the first equation:** $$ 2(5) + y = 8 $$ $$ 10 + y = 8 $$ 6. **Solve for $y$:** $$ y = 8 - 10 = -2 $$ **Final answer:** $$ x = 5, \quad y = -2 $$