1. **State the problem:** Solve the system of linear equations: $$4x - 3y = 8$$ $$2x + 5y = -22$$ 2. **Formula and method:** We will use the method of elimination to solve for $x$ and $y$. The goal is to eliminate one variable by making the coefficients of that variable equal in both equations. 3. **Eliminate $x$:** Multiply the second equation by 2 to match the coefficient of $x$ in the first equation: $$2(2x + 5y) = 2(-22)$$ $$4x + 10y = -44$$ 4. **Subtract the first equation from this new equation:** $$(4x + 10y) - (4x - 3y) = -44 - 8$$ $$4x + 10y - 4x + 3y = -52$$ $$13y = -52$$ 5. **Solve for $y$:** $$y = \frac{-52}{13} = -4$$ 6. **Substitute $y = -4$ into the first equation:** $$4x - 3(-4) = 8$$ $$4x + 12 = 8$$ $$4x = 8 - 12$$ $$4x = -4$$ 7. **Solve for $x$:** $$x = \frac{-4}{4} = -1$$ **Final answer:** $$x = -1, \quad y = -4$$