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