1. **State the problem:** Solve the system of equations using the elimination method: $$6x - 5y = 50$$ $$2x + 6y = -14$$ 2. **Goal:** Eliminate one variable by making the coefficients of either $x$ or $y$ the same (or opposites) in both equations. 3. **Eliminate $x$:** Multiply the first equation by 2 and the second equation by -6 to align coefficients of $x$: $$2(6x - 5y) = 2(50) \Rightarrow 12x - 10y = 100$$ $$-6(2x + 6y) = -6(-14) \Rightarrow -12x - 36y = 84$$ 4. **Add the two new equations:** $$\cancel{12x} - 10y + \cancel{-12x} - 36y = 100 + 84$$ $$-10y - 36y = 184$$ $$-46y = 184$$ 5. **Solve for $y$:** $$y = \frac{184}{-46} = -4$$ 6. **Substitute $y = -4$ into one original equation to find $x$:** Use the second equation: $$2x + 6(-4) = -14$$ $$2x - 24 = -14$$ $$2x = -14 + 24$$ $$2x = 10$$ 7. **Solve for $x$:** $$x = \frac{10}{2} = 5$$ 8. **Final answer:** The solution to the system is the ordered pair: $$(5, -4)$$