1. **State the problem:** Solve the system of linear equations: $$4x + 8y = 20$$ $$-4x + 2y = -30$$ 2. **Formula and rules:** We can solve this system using the elimination method by adding or subtracting equations to eliminate one variable. 3. **Add the two equations to eliminate $x$:** $$\begin{aligned} &(4x + 8y) + (-4x + 2y) = 20 + (-30) \\ &4x - 4x + 8y + 2y = -10 \\ &0 + 10y = -10 \\ &10y = -10 \end{aligned}$$ 4. **Solve for $y$:** $$y = \frac{-10}{10} = -1$$ 5. **Substitute $y = -1$ into the first equation to find $x$:** $$4x + 8(-1) = 20$$ $$4x - 8 = 20$$ $$4x = 20 + 8$$ $$4x = 28$$ 6. **Divide both sides by 4:** $$x = \frac{28}{4}$$ $$x = 7$$ 7. **Final answer:** $$\boxed{(x, y) = (7, -1)}$$