1. **State the problem:** Solve the equation $$6 - \frac{3y}{7} = \frac{y + 20}{9}$$ for $y$. 2. **Write down the equation:** $$6 - \frac{3y}{7} = \frac{y + 20}{9}$$ 3. **Clear the denominators:** To eliminate fractions, multiply both sides by the least common multiple (LCM) of 7 and 9, which is 63. Multiply both sides by 63: $$63 \times \left(6 - \frac{3y}{7}\right) = 63 \times \frac{y + 20}{9}$$ 4. **Distribute multiplication:** $$63 \times 6 - 63 \times \frac{3y}{7} = 63 \times \frac{y + 20}{9}$$ Calculate each term: $$378 - 63 \times \frac{3y}{7} = 63 \times \frac{y + 20}{9}$$ Since $63 \div 7 = 9$, $$378 - 9 \times 3y = 7 \times (y + 20)$$ Simplify: $$378 - 27y = 7y + 140$$ 5. **Collect like terms:** Add $27y$ to both sides: $$378 = 7y + 140 + 27y$$ Combine like terms: $$378 = 34y + 140$$ 6. **Isolate $y$:** Subtract 140 from both sides: $$378 - 140 = 34y$$ Simplify: $$238 = 34y$$ 7. **Solve for $y$:** Divide both sides by 34: $$\frac{238}{34} = y$$ Show cancellation: $$y = \frac{\cancel{34}7}{\cancel{34}1} = 7$$ **Final answer:** $$y = 7$$