1. **Problem statement:** Solve the equation $$\frac{x-5}{3} - \frac{x-3}{3} = \frac{6x+1}{21}$$. 2. **Formula and rules:** To solve equations with fractions, first find a common denominator or multiply through by the least common multiple (LCM) to clear denominators. 3. **Step-by-step solution:** - The denominators are 3, 3, and 21. The LCM of 3 and 21 is 21. - Multiply both sides of the equation by 21 to clear denominators: $$21 \times \left(\frac{x-5}{3} - \frac{x-3}{3}\right) = 21 \times \frac{6x+1}{21}$$ - Simplify each term: $$7(x-5) - 7(x-3) = 6x + 1$$ - Distribute 7: $$7x - 35 - 7x + 21 = 6x + 1$$ - Combine like terms on the left: $$-14 = 6x + 1$$ - Subtract 1 from both sides: $$-14 - 1 = 6x$$ $$-15 = 6x$$ - Divide both sides by 6: $$x = \frac{-15}{6} = -\frac{5}{2}$$ 4. **Answer:** $$x = -\frac{5}{2}$$ This means the value of $x$ that satisfies the equation is $-2.5$.