1. **State the problem:** Solve the equation $$\frac{1}{x+y} + \frac{1}{x-y} = 0$$ for $x$ and $y$. 2. **Use a common denominator:** The denominators are $x+y$ and $x-y$. The common denominator is $(x+y)(x-y)$. 3. **Rewrite the equation:** $$\frac{x-y}{(x+y)(x-y)} + \frac{x+y}{(x-y)(x+y)} = 0$$ 4. **Combine the fractions:** $$\frac{(x-y) + (x+y)}{(x+y)(x-y)} = 0$$ 5. **Simplify the numerator:** $$(x-y) + (x+y) = x - y + x + y = 2x$$ 6. **Rewrite the equation:** $$\frac{2x}{(x+y)(x-y)} = 0$$ 7. **Set numerator equal to zero (since denominator cannot be zero):** $$2x = 0$$ 8. **Solve for $x$:** $$x = 0$$ 9. **Check denominator for restrictions:** Denominator $(x+y)(x-y)$ cannot be zero, so $x \neq \pm y$. 10. **Final solution:** $$x = 0 \quad \text{and} \quad y \neq 0$$ This means $x$ must be zero and $y$ can be any value except zero to avoid division by zero.