1. **State the problem:** Solve the equation $$\frac{15X-35}{10} + \frac{4-X}{3} = \frac{20}{4} + \frac{3X-3}{18}$$ for $X$. 2. **Identify the least common denominator (LCD):** The denominators are 10, 3, 4, and 18. The LCD of these numbers is 180. 3. **Multiply both sides of the equation by the LCD to clear denominators:** $$180 \times \left(\frac{15X-35}{10} + \frac{4-X}{3}\right) = 180 \times \left(\frac{20}{4} + \frac{3X-3}{18}\right)$$ 4. **Distribute multiplication:** $$180 \times \frac{15X-35}{10} + 180 \times \frac{4-X}{3} = 180 \times \frac{20}{4} + 180 \times \frac{3X-3}{18}$$ 5. **Simplify each term:** $$18 \times (15X - 35) + 60 \times (4 - X) = 45 \times 20 + 10 \times (3X - 3)$$ 6. **Expand each term:** $$18 \times 15X - 18 \times 35 + 60 \times 4 - 60 \times X = 900 + 10 \times 3X - 10 \times 3$$ $$270X - 630 + 240 - 60X = 900 + 30X - 30$$ 7. **Combine like terms on the left:** $$ (270X - 60X) + (-630 + 240) = 900 + 30X - 30$$ $$210X - 390 = 900 + 30X - 30$$ 8. **Simplify the right side:** $$210X - 390 = 870 + 30X$$ 9. **Bring all $X$ terms to one side and constants to the other:** $$210X - 30X = 870 + 390$$ $$180X = 1260$$ 10. **Divide both sides by 180 to solve for $X$:** $$\cancel{180}X = \frac{1260}{\cancel{180}}$$ $$X = 7$$ **Final answer:** $$X = 7$$