1. **State the problem:** Solve the equation $\frac{x-5}{3} - 5 = \frac{x-9}{5}$ for $x$. 2. **Write down the equation:** $$\frac{x-5}{3} - 5 = \frac{x-9}{5}$$ 3. **Isolate terms and clear denominators:** To eliminate fractions, multiply every term by the least common denominator (LCD), which is 15. Multiply both sides by 15: $$15 \times \left(\frac{x-5}{3} - 5\right) = 15 \times \frac{x-9}{5}$$ 4. **Distribute multiplication:** $$15 \times \frac{x-5}{3} - 15 \times 5 = 15 \times \frac{x-9}{5}$$ Simplify each term: $$5(x-5) - 75 = 3(x-9)$$ 5. **Expand both sides:** $$5x - 25 - 75 = 3x - 27$$ Simplify left side: $$5x - 100 = 3x - 27$$ 6. **Collect like terms:** Subtract $3x$ from both sides: $$5x - 3x - 100 = -27$$ Simplify: $$2x - 100 = -27$$ Add 100 to both sides: $$2x = 73$$ 7. **Solve for $x$:** Divide both sides by 2: $$x = \frac{73}{2}$$ 8. **Final answer:** $$x = 36.5$$