1. **State the problem:** Solve the system of equations using elimination: $$\begin{cases} x - 2y = 10 \\ 3x + y = -12 \end{cases}$$ 2. **Write down the equations:** Equation 1: $x - 2y = 10$ Equation 2: $3x + y = -12$ 3. **Goal:** Eliminate one variable by making the coefficients of $x$ or $y$ opposites. 4. **Multiply Equation 2 by 2** to align the $y$ terms for elimination: $$2 \times (3x + y) = 2 \times (-12)$$ which gives $$6x + 2y = -24$$ 5. **Add Equation 1 and the new equation:** $$\begin{aligned} &(x - 2y) + (6x + 2y) = 10 + (-24) \\ &x - 2y + 6x + 2y = -14 \\ &(x + 6x) + (-2y + 2y) = -14 \\ &7x + \cancel{-2y + 2y} = -14 \\ &7x = -14 \end{aligned}$$ 6. **Solve for $x$:** $$x = \frac{-14}{7} = -2$$ 7. **Substitute $x = -2$ into Equation 1 to find $y$:** $$-2 - 2y = 10$$ 8. **Solve for $y$:** $$-2y = 10 + 2 = 12$$ $$y = \frac{12}{-2} = -6$$ 9. **Final solution:** $$\boxed{x = -2, y = -6}$$ This means the solution to the system is $x = -2$ and $y = -6$.