1. **State the problem:** Solve the equation $$\frac{3}{7x - 1} + \frac{1}{x} = 0$$ for $x$. 2. **Identify the formula and rules:** To solve equations with fractions, find a common denominator and combine terms. Then solve the resulting equation. 3. **Find the common denominator:** The denominators are $7x - 1$ and $x$. The common denominator is $$x(7x - 1)$$. 4. **Rewrite each term with the common denominator:** $$\frac{3}{7x - 1} = \frac{3x}{x(7x - 1)}$$ $$\frac{1}{x} = \frac{7x - 1}{x(7x - 1)}$$ 5. **Combine the fractions:** $$\frac{3x + (7x - 1)}{x(7x - 1)} = 0$$ 6. **Simplify the numerator:** $$3x + 7x - 1 = 10x - 1$$ 7. **Set the numerator equal to zero (since denominator cannot be zero):** $$10x - 1 = 0$$ 8. **Solve for $x$:** $$10x = 1$$ $$x = \frac{1}{10}$$ 9. **Check for restrictions:** Denominator terms cannot be zero: $$7x - 1 \neq 0 \Rightarrow x \neq \frac{1}{7}$$ $$x \neq 0$$ Since $x = \frac{1}{10}$ does not violate these, it is valid. **Final answer:** $$x = \frac{1}{10}$$