1. **State the problem:** Solve the system of linear equations: $$\begin{cases}-8x + 2y = 14 \\ y + 2x = -5 \end{cases}$$ 2. **Rewrite the second equation to isolate $y$:** $$y = -5 - 2x$$ 3. **Substitute $y$ from the second equation into the first equation:** $$-8x + 2(-5 - 2x) = 14$$ 4. **Simplify the equation:** $$-8x - 10 - 4x = 14$$ $$-12x - 10 = 14$$ 5. **Add 10 to both sides:** $$-12x - 10 + 10 = 14 + 10$$ $$-12x = 24$$ 6. **Divide both sides by $-12$:** $$\cancel{-12}x = \frac{24}{\cancel{-12}}$$ $$x = -2$$ 7. **Substitute $x = -2$ back into the expression for $y$:** $$y = -5 - 2(-2)$$ $$y = -5 + 4$$ $$y = -1$$ **Final answer:** $$\boxed{(x, y) = (-2, -1)}$$