1. **State the problem:** Solve the equation $$\frac{15}{x-1} = \frac{19}{x+4}$$ for $x$. 2. **Use the cross-multiplication method:** When two fractions are equal, their cross products are equal. So, $$15(x+4) = 19(x-1)$$ 3. **Expand both sides:** $$15x + 60 = 19x - 19$$ 4. **Bring all terms involving $x$ to one side and constants to the other:** $$15x - 19x = -19 - 60$$ 5. **Simplify:** $$-4x = -79$$ 6. **Divide both sides by $-4$ to solve for $x$:** $$x = \frac{-79}{-4} = \frac{79}{4}$$ 7. **Final answer:** $$x = \frac{79}{4}$$