1. **Problem statement:** Solve the system of linear equations $2x+y=8$ and $2x-y=12$. 2. **Formula/rule used:** To solve a system, we can add the equations to eliminate one variable. 3. **Add the equations:** $$\begin{aligned}2x+y&=8\\2x-y&=12\\\hline 4x+\cancel{y}+\cancel{-y}&=20\end{aligned}$$ 4. **Simplify:** $$4x=20$$ 5. **Divide both sides by $4$:** $$\frac{4x}{4}=\frac{20}{4}$$ $$\cancel{4}x=\cancel{4}5$$ $$x=5$$ 6. **Substitute $x=5$ into $2x+y=8$:** $$2(5)+y=8$$ $$10+y=8$$ $$y=8-10$$ $$y=-2$$ 7. **Final answer:** $x=5$ and $y=-2$.