1. **State the problem:** Solve the system of linear equations: $$2x + y = 8$$ $$2x - y = 12$$ 2. **Given values:** $$x = 7$$ $$y = 7$$ 3. **Check if the given values satisfy the equations:** For the first equation: $$2(7) + 7 = 14 + 7 = 21 \neq 8$$ For the second equation: $$2(7) - 7 = 14 - 7 = 7 \neq 12$$ 4. Since the given values do not satisfy the system, solve the system for $x$ and $y$. 5. Add the two equations to eliminate $y$: $$\begin{aligned} 2x + y &= 8 \\ 2x - y &= 12 \\ \hline 4x + \cancel{y} - \cancel{y} &= 8 + 12 \\ 4x &= 20 \\ x &= \frac{20}{4} = 5 \end{aligned}$$ 6. Substitute $x=5$ into the first equation to find $y$: $$2(5) + y = 8$$ $$10 + y = 8$$ $$y = 8 - 10 = -2$$ 7. **Final solution:** $$x = 5, \quad y = -2$$