1. **State the problem:** Solve for $b$ in the equation $$\frac{a}{x} - \frac{1}{c} = \frac{b}{d}.$$\n\n2. **Formula and rules:** To isolate $b$, multiply both sides of the equation by $d$ to eliminate the denominator on the right side.\n\n3. **Multiply both sides by $d$: $$d\left(\frac{a}{x} - \frac{1}{c}\right) = d \cdot \frac{b}{d}$$**\n\n4. **Simplify the right side using cancellation: $$d \cdot \frac{b}{d} = \cancel{d} \cdot \frac{b}{\cancel{d}} = b$$**\n\n5. **Distribute $d$ on the left side: $$d \cdot \frac{a}{x} - d \cdot \frac{1}{c} = b$$**\n\n6. **Rewrite the left side as: $$\frac{ad}{x} - \frac{d}{c} = b$$**\n\n7. **Final answer:** $$b = \frac{ad}{x} - \frac{d}{c}$$\n\nThis expresses $b$ in terms of $a$, $x$, $c$, and $d$.