1. **State the problem:** Solve the system of linear equations using the elimination method: $$-2x - 5y = -12$$ $$5x + 4y = -4$$ 2. **Goal:** Find values of $x$ and $y$ that satisfy both equations simultaneously. 3. **Eliminate one variable:** To eliminate $y$, multiply the first equation by 4 and the second equation by 5 to make the coefficients of $y$ opposites: $$4(-2x - 5y) = 4(-12) \Rightarrow -8x - 20y = -48$$ $$5(5x + 4y) = 5(-4) \Rightarrow 25x + 20y = -20$$ 4. **Add the two new equations:** $$(-8x - 20y) + (25x + 20y) = -48 + (-20)$$ $$(-8x + 25x) + (-20y + 20y) = -68$$ $$17x + 0 = -68$$ $$17x = -68$$ 5. **Solve for $x$:** $$x = \frac{-68}{17} = -4$$ 6. **Substitute $x = -4$ into one original equation to find $y$:** Use the first equation: $$-2(-4) - 5y = -12$$ $$8 - 5y = -12$$ 7. **Solve for $y$:** $$-5y = -12 - 8$$ $$-5y = -20$$ $$y = \frac{-20}{-5} = 4$$ **Final answer:** $$x = -4$$ $$y = 4$$