1. We are asked to solve the equation $$\frac{x - 3}{5} + \frac{3x + 1}{10} = 0$$. 2. To solve this, we first find a common denominator for the fractions, which is 10. 3. Rewrite each term with denominator 10: $$\frac{2(x - 3)}{10} + \frac{3x + 1}{10} = 0$$ 4. Combine the numerators over the common denominator: $$\frac{2(x - 3) + (3x + 1)}{10} = 0$$ 5. Multiply both sides by 10 to clear the denominator: $$2(x - 3) + (3x + 1) = 0$$ 6. Distribute and simplify: $$2x - 6 + 3x + 1 = 0$$ $$5x - 5 = 0$$ 7. Solve for $x$: $$5x = 5$$ $$x = 1$$ Final answer: $x = 1$