1. **State the problem:** Solve the equation $$\frac{3}{4}(x - 7) = 9$$ for $x$. 2. **Formula and rules:** To solve for $x$, we need to isolate $x$ on one side of the equation. This involves undoing multiplication by multiplying both sides by the reciprocal of $\frac{3}{4}$, which is $\frac{4}{3}$. 3. **Multiply both sides by $\frac{4}{3}$:** $$\frac{4}{3} \times \frac{3}{4}(x - 7) = \frac{4}{3} \times 9$$ 4. **Cancel common factors:** $$\cancel{\frac{4}{3}} \times \cancel{\frac{3}{4}} (x - 7) = \frac{4}{3} \times 9$$ This simplifies to: $$x - 7 = \frac{4}{3} \times 9$$ 5. **Calculate the right side:** $$x - 7 = \frac{4}{3} \times 9 = \frac{4}{3} \times \frac{9}{1} = \frac{36}{3} = 12$$ 6. **Add 7 to both sides to isolate $x$:** $$x - 7 + 7 = 12 + 7$$ $$x = 19$$ **Final answer:** $$x = 19$$