1. **State the problem:** Solve the system of linear equations: $$2x + 2y = 8$$ $$2x - y = 2$$ 2. **Choose a method:** We will use the substitution or elimination method. Here, elimination is straightforward. 3. **Eliminate one variable:** Multiply the second equation by 2 to align the coefficients of $y$: $$2(2x - y) = 2(2) \Rightarrow 4x - 2y = 4$$ 4. **Add the first equation and the new equation:** $$2x + 2y = 8$$ $$4x - 2y = 4$$ Adding: $$2x + 2y + 4x - 2y = 8 + 4$$ Simplify: $$6x + \cancel{2y} - \cancel{2y} = 12$$ $$6x = 12$$ 5. **Solve for $x$:** $$x = \frac{12}{6}$$ $$x = 2$$ 6. **Substitute $x=2$ into one original equation to find $y$:** Using the second equation: $$2(2) - y = 2$$ $$4 - y = 2$$ 7. **Solve for $y$:** $$-y = 2 - 4$$ $$-y = -2$$ $$y = 2$$ **Final answer:** $$x = 2, \quad y = 2$$