1. **State the problem:** Solve the equation $$\frac{x}{a} + \frac{x}{b} = 1$$ for $x$. 2. **Formula and rules:** To solve for $x$, we need to combine the fractions on the left side by finding a common denominator and then isolate $x$. 3. **Find common denominator:** The common denominator of $a$ and $b$ is $ab$. Rewrite each fraction: $$\frac{x}{a} = \frac{xb}{ab}, \quad \frac{x}{b} = \frac{xa}{ab}$$ 4. **Combine fractions:** $$\frac{xb}{ab} + \frac{xa}{ab} = \frac{xb + xa}{ab} = \frac{x(b + a)}{ab}$$ 5. **Set equal to 1:** $$\frac{x(b + a)}{ab} = 1$$ 6. **Isolate $x$ by multiplying both sides by $ab$:** $$\cancel{\frac{x(b + a)}{ab}} \times ab = 1 \times ab \implies x(b + a) = ab$$ 7. **Divide both sides by $(b + a)$ to solve for $x$:** $$x = \frac{ab}{b + a}$$ **Final answer:** $$x = \frac{ab}{a + b}$$