1. **State the problem:** Solve the equation $27x = \frac{1}{9}$ for $x$. 2. **Recall the formula and rules:** To isolate $x$, divide both sides of the equation by 27. 3. **Perform the division:** $$x = \frac{1}{9} \div 27$$ 4. **Rewrite division as multiplication by reciprocal:** $$x = \frac{1}{9} \times \frac{1}{27}$$ 5. **Multiply the fractions:** $$x = \frac{1 \times 1}{9 \times 27} = \frac{1}{243}$$ 6. **Final answer:** $$x = \frac{1}{243}$$ This means $x$ is the reciprocal of 243.