1. **Problem:** Solve the system of equations $2x+y=8$ and $2x-y=12$, then find $x$ and $y$. 2. **Formula used:** For a system, we can use elimination by adding the equations to remove one variable. 3. Add the equations: $$ (2x+y)+(2x-y)=8+12 $$ 4. Simplify the left side and right side: $$ 4x=20 $$ 5. Divide both sides by $4$: $$ \frac{4x}{4}=\frac{20}{4} $$ $$ \cancel{4}x/\cancel{4}=\frac{20}{4} $$ 6. So, $$ x=5 $$ 7. Substitute $x=5$ into $2x+y=8$: $$ 2(5)+y=8 $$ 8. Simplify: $$ 10+y=8 $$ 9. Subtract $10$ from both sides: $$ 10-y?\text{ No, }10+y=8 $$ $$ 10+\cancel{y}=8 $$ $$ y=8-10 $$ 10. So, $$ y=-2 $$ 11. **Final answer:** $x=5$ and $y=-2$.