1. **State the problem:** Solve the system of linear equations: $$-3x + 5y = 7$$ $$4x + 2y = -18$$ 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 2 and the second equation by 5 to align coefficients of $y$: $$2(-3x + 5y) = 2(7) \Rightarrow -6x + 10y = 14$$ $$5(4x + 2y) = 5(-18) \Rightarrow 20x + 10y = -90$$ 4. **Subtract the first new equation from the second:** $$\cancel{20x} + 10y - (\cancel{-6x} + 10y) = -90 - 14$$ $$20x + 10y + 6x - 10y = -104$$ $$26x = -104$$ 5. **Solve for $x$:** $$x = \frac{-104}{26} = -4$$ 6. **Substitute $x = -4$ into one of the original equations, for example the first:** $$-3(-4) + 5y = 7$$ $$12 + 5y = 7$$ 7. **Solve for $y$:** $$5y = 7 - 12$$ $$5y = -5$$ $$y = \frac{-5}{5} = -1$$ **Final answer:** $$x = -4, \quad y = -1$$