1. **State the problem:** Solve the system of equations: $$-2x + 7y = 4$$ $$-5x - 9y = 10$$ 2. **Choose a method:** We will use the elimination method to solve for $x$ and $y$. 3. **Eliminate one variable:** Multiply the first equation by 5 and the second equation by 2 to align coefficients of $x$: $$5(-2x + 7y) = 5(4) \Rightarrow -10x + 35y = 20$$ $$2(-5x - 9y) = 2(10) \Rightarrow -10x - 18y = 20$$ 4. **Subtract the second from the first:** $$(-10x + 35y) - (-10x - 18y) = 20 - 20$$ $$-10x + 35y + 10x + 18y = 0$$ $$53y = 0$$ 5. **Solve for $y$:** $$y = \frac{0}{53} = 0$$ 6. **Substitute $y=0$ into the first equation:** $$-2x + 7(0) = 4$$ $$-2x = 4$$ 7. **Solve for $x$:** $$x = \frac{4}{-2} = -2$$ **Final answer:** $$x = -2, \quad y = 0$$