1. **Problem:** Solve the system of equations. $$2x+y=8$$ $$2x-y=12$$ We need to find $x$ and $y$. 2. **Use the elimination idea.** When we add the two equations, the $y$ terms cancel. $$\begin{aligned} 2x+y &= 8 \\ 2x-y &= 12 \\ \hline 4x &= 20 \end{aligned}$$ 3. **Solve for $x$.** Divide both sides by $4$. $$\frac{4x}{4}=\frac{20}{4}$$ $$\cancel{4}x=\frac{\cancel{20}}{\cancel{4}}$$ $$x=5$$ 4. **Substitute $x=5$ into one equation to find $y$.** Use $2x+y=8$. $$2(5)+y=8$$ $$10+y=8$$ Subtract $10$ from both sides. $$y=8-10$$ $$y=-2$$ 5. **Check the answer.** Use $2x-y=12$. $$2(5)-(-2)=10+2=12$$ That works. 6. **Final answer:** $$x=5,\quad y=-2$$