1. **State the problem:** Solve the equation $$\frac{a}{x} - \frac{1}{c} = \frac{b}{d}$$ for $x$. 2. **Formula and rules:** To solve for $x$, isolate the term containing $x$ and then solve the resulting equation. Remember to find a common denominator when combining fractions. 3. **Isolate the term with $x$:** $$\frac{a}{x} = \frac{b}{d} + \frac{1}{c}$$ 4. **Find a common denominator on the right side:** $$\frac{b}{d} + \frac{1}{c} = \frac{bc}{dc} + \frac{d}{dc} = \frac{bc + d}{dc}$$ 5. **Rewrite the equation:** $$\frac{a}{x} = \frac{bc + d}{dc}$$ 6. **Cross-multiply to solve for $x$:** $$a \cdot dc = x (bc + d)$$ 7. **Isolate $x$:** $$x = \frac{a \cdot dc}{bc + d}$$ 8. **Final answer:** $$\boxed{x = \frac{a d c}{b c + d}}$$