1. **State the problem:** Solve the equation $$\binom{n}{15} \frac{19}{x-1} = \frac{19}{x+4}$$ for $x$. 2. **Understand the equation:** The binomial coefficient $\binom{n}{15}$ is a constant with respect to $x$, so treat it as a constant $C = \binom{n}{15}$. 3. **Rewrite the equation:** $$C \frac{19}{x-1} = \frac{19}{x+4}$$ 4. **Divide both sides by 19:** $$C \frac{1}{x-1} = \frac{1}{x+4}$$ 5. **Cross-multiply to clear denominators:** $$C (x+4) = x-1$$ 6. **Distribute $C$:** $$C x + 4C = x - 1$$ 7. **Group $x$ terms on one side:** $$C x - x = -1 - 4C$$ 8. **Factor out $x$:** $$x (C - 1) = -1 - 4C$$ 9. **Solve for $x$:** $$x = \frac{-1 - 4C}{C - 1} = \frac{-(1 + 4C)}{C - 1}$$ 10. **Substitute back $C = \binom{n}{15}$:** $$x = \frac{-(1 + 4 \binom{n}{15})}{\binom{n}{15} - 1}$$ **Final answer:** $$x = \frac{-(1 + 4 \binom{n}{15})}{\binom{n}{15} - 1}$$