1. **State the problem:** Solve the equation $$\frac{3x + 1}{5} + \frac{x - 2}{2} = \frac{47}{10}$$. 2. **Find a common denominator:** The denominators are 5, 2, and 10. The least common denominator (LCD) is 10. 3. **Multiply both sides by 10 to clear denominators:** $$10 \times \left(\frac{3x + 1}{5} + \frac{x - 2}{2}\right) = 10 \times \frac{47}{10}$$ 4. **Distribute multiplication:** $$10 \times \frac{3x + 1}{5} + 10 \times \frac{x - 2}{2} = 47$$ 5. **Simplify each term:** $$2(3x + 1) + 5(x - 2) = 47$$ 6. **Expand the terms:** $$6x + 2 + 5x - 10 = 47$$ 7. **Combine like terms:** $$11x - 8 = 47$$ 8. **Isolate $x$:** $$11x = 47 + 8$$ $$11x = 55$$ 9. **Solve for $x$:** $$x = \frac{55}{11} = 5$$ **Final answer:** $$x = 5$$