1. **State the problem:** Solve the equation $$\frac{r + 5}{6} = \frac{r + 3}{4} + \frac{r - 1}{9}$$ for $r$. 2. **Identify the formula and rules:** To solve equations with fractions, find a common denominator to eliminate the fractions by multiplying both sides. 3. **Find the least common denominator (LCD):** The denominators are 6, 4, and 9. The LCD is $$\text{lcm}(6,4,9) = 36$$. 4. **Multiply both sides by 36 to clear denominators:** $$36 \times \frac{r + 5}{6} = 36 \times \frac{r + 3}{4} + 36 \times \frac{r - 1}{9}$$ 5. **Simplify each term:** $$6(r + 5) = 9(r + 3) + 4(r - 1)$$ 6. **Expand each term:** $$6r + 30 = 9r + 27 + 4r - 4$$ 7. **Combine like terms on the right:** $$6r + 30 = (9r + 4r) + (27 - 4)$$ $$6r + 30 = 13r + 23$$ 8. **Isolate variable terms on one side:** $$6r + 30 - 13r = 23$$ $$\cancel{6r} + 30 - \cancel{13r} = 23 - 7r$$ $$-7r + 30 = 23$$ 9. **Subtract 30 from both sides:** $$-7r + 30 - 30 = 23 - 30$$ $$-7r = -7$$ 10. **Divide both sides by -7:** $$\frac{-7r}{-7} = \frac{-7}{-7}$$ $$r = 1$$ **Final answer:** $$r = 1$$