1. **State the problem:** Solve the equation $$\frac{X}{3} - \frac{13 - 2X}{2} = \frac{1}{6}$$ for $X$. 2. **Identify the common denominator:** The denominators are 3, 2, and 6. The least common denominator (LCD) is 6. 3. **Multiply both sides by the LCD to clear fractions:** $$6 \times \left( \frac{X}{3} - \frac{13 - 2X}{2} \right) = 6 \times \frac{1}{6}$$ 4. **Distribute multiplication:** $$6 \times \frac{X}{3} - 6 \times \frac{13 - 2X}{2} = 1$$ 5. **Simplify each term:** $$2X - 3(13 - 2X) = 1$$ 6. **Distribute the -3:** $$2X - 39 + 6X = 1$$ 7. **Combine like terms:** $$8X - 39 = 1$$ 8. **Add 39 to both sides:** $$8X - 39 + 39 = 1 + 39$$ $$8X = 40$$ 9. **Divide both sides by 8:** $$\cancel{8}X = \frac{40}{\cancel{8}}$$ $$X = 5$$ **Final answer:** $$X = 5$$