1. **State the problem:** Solve the system of linear equations: $$4x + y = -114$$ $$-3x + 5y = 5$$ 2. **Choose a method:** We will use the substitution or elimination method. Here, elimination is convenient. 3. **Eliminate one variable:** Multiply the first equation by 5 to align coefficients of $y$: $$5(4x + y) = 5(-114)$$ $$20x + 5y = -570$$ 4. **Subtract the second equation from this new equation:** $$ (20x + 5y) - (-3x + 5y) = -570 - 5$$ $$20x + 5y + 3x - 5y = -575$$ $$23x = -575$$ 5. **Solve for $x$:** $$x = \frac{-575}{23}$$ 6. **Simplify the fraction:** $$x = -25$$ 7. **Substitute $x = -25$ into the first equation to find $y$:** $$4(-25) + y = -114$$ $$-100 + y = -114$$ 8. **Solve for $y$:** $$y = -114 + 100$$ $$y = -14$$ **Final answer:** $$x = -25, \quad y = -14$$