1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x + 2y = 2 \\ x - y = 8 \end{cases}$$ 2. **Use substitution or elimination method.** Here, we will use substitution. 3. From the second equation, solve for $x$: $$x - y = 8 \implies x = y + 8$$ 4. Substitute $x = y + 8$ into the first equation: $$ (y + 8) + 2y = 2 $$ 5. Simplify the equation: $$ y + 8 + 2y = 2 $$ $$ 3y + 8 = 2 $$ 6. Isolate $y$: $$ 3y = 2 - 8 $$ $$ 3y = -6 $$ 7. Divide both sides by 3: $$ y = \frac{\cancel{3}y}{\cancel{3}} = \frac{-6}{3} $$ $$ y = -2 $$ 8. Substitute $y = -2$ back into $x = y + 8$: $$ x = -2 + 8 $$ $$ x = 6 $$ **Final answer:** $$ (x, y) = (6, -2) $$