1. **State the problem:** Solve the system of equations using elimination: $$-5x + 6y = 5$$ $$-9x + 10y = 13$$ 2. **Goal:** Eliminate one variable by making the coefficients of either $x$ or $y$ the same (or opposites). 3. **Eliminate $x$:** Multiply the first equation by 9 and the second by 5 to align coefficients of $x$: $$9(-5x + 6y) = 9(5) \Rightarrow -45x + 54y = 45$$ $$5(-9x + 10y) = 5(13) \Rightarrow -45x + 50y = 65$$ 4. **Subtract the second from the first to eliminate $x$:** $$(-45x + 54y) - (-45x + 50y) = 45 - 65$$ $$-45x + 54y + 45x - 50y = -20$$ $$4y = -20$$ 5. **Solve for $y$:** $$y = \frac{-20}{4} = -5$$ 6. **Substitute $y = -5$ into the first original equation:** $$-5x + 6(-5) = 5$$ $$-5x - 30 = 5$$ 7. **Solve for $x$:** $$-5x = 5 + 30$$ $$-5x = 35$$ $$x = \frac{35}{-5} = -7$$ **Final answer:** $$(x, y) = (-7, -5)$$