1. **Stating the problem:** Solve the equation $$\frac{x-5}{3} = \frac{x-3}{3} + \frac{6x+1}{21}$$ for $x$. 2. **Formula and rules:** To solve equations with fractions, find a common denominator to clear the fractions and simplify the equation. 3. **Find the common denominator:** The denominators are 3, 3, and 21. The least common denominator (LCD) is 21. 4. **Multiply both sides by 21 to clear denominators:** $$21 \times \frac{x-5}{3} = 21 \times \frac{x-3}{3} + 21 \times \frac{6x+1}{21}$$ 5. **Simplify each term:** $$7(x-5) = 7(x-3) + (6x+1)$$ 6. **Expand the terms:** $$7x - 35 = 7x - 21 + 6x + 1$$ 7. **Combine like terms on the right:** $$7x - 35 = 13x - 20$$ 8. **Bring all $x$ terms to one side and constants to the other:** $$7x - 13x = -20 + 35$$ 9. **Simplify:** $$-6x = 15$$ 10. **Divide both sides by -6:** $$x = \frac{15}{-6} = -\frac{5}{2}$$ **Final answer:** $$x = -\frac{5}{2}$$