1. **State the problem:** Solve the equation $$\frac{2x - 1}{3} - \frac{3x + 1}{5} = 1$$. 2. **Identify the formula and rules:** To solve equations with fractions, find a common denominator to eliminate the fractions by multiplying both sides. 3. **Find the least common denominator (LCD):** The denominators are 3 and 5, so $$\text{LCD} = 15$$. 4. **Multiply both sides of the equation by 15:** $$15 \times \left(\frac{2x - 1}{3} - \frac{3x + 1}{5}\right) = 15 \times 1$$ 5. **Distribute multiplication:** $$15 \times \frac{2x - 1}{3} - 15 \times \frac{3x + 1}{5} = 15$$ 6. **Simplify each term:** $$\cancel{15} \times \frac{2x - 1}{\cancel{3}} = 5(2x - 1)$$ $$\cancel{15} \times \frac{3x + 1}{\cancel{5}} = 3(3x + 1)$$ 7. **Rewrite the equation:** $$5(2x - 1) - 3(3x + 1) = 15$$ 8. **Expand the parentheses:** $$10x - 5 - 9x - 3 = 15$$ 9. **Combine like terms:** $$10x - 9x - 5 - 3 = 15$$ $$x - 8 = 15$$ 10. **Isolate $x$ by adding 8 to both sides:** $$x - 8 + 8 = 15 + 8$$ $$x = 23$$ **Final answer:** $$x = 23$$