1. **State the problem:** Solve the system of linear equations: $$3x - 6y = -2$$ $$-4x + 9y = -2$$ 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 3 and the second equation by 2 to align coefficients of $y$: $$3(3x - 6y) = 3(-2) \Rightarrow 9x - 18y = -6$$ $$2(-4x + 9y) = 2(-2) \Rightarrow -8x + 18y = -4$$ 4. **Add the two equations to eliminate $y$:** $$9x - 18y + (-8x + 18y) = -6 + (-4)$$ $$9x - 8x + (-18y + 18y) = -10$$ $$x + \cancel{-18y + 18y} = -10$$ $$x = -10$$ 5. **Substitute $x = -10$ into the first original equation:** $$3(-10) - 6y = -2$$ $$-30 - 6y = -2$$ 6. **Solve for $y$:** $$-6y = -2 + 30$$ $$-6y = 28$$ $$\cancel{-6}y = \cancel{-6} \times -\frac{28}{6}$$ $$y = -\frac{28}{6} = -\frac{14}{3}$$ 7. **Final solution:** $$x = -10, \quad y = -\frac{14}{3}$$