1. **State the problem:** We need to find which expressions are equivalent to the expression $$ (x - y) \frac{5}{8} - \frac{1}{4} x + y $$. 2. **Rewrite the original expression:** $$ (x - y) \frac{5}{8} - \frac{1}{4} x + y = \frac{5}{8} x - \frac{5}{8} y - \frac{1}{4} x + y $$ 3. **Combine like terms:** For the $x$ terms: $$ \frac{5}{8} x - \frac{1}{4} x = \frac{5}{8} x - \frac{2}{8} x = \frac{3}{8} x $$ For the $y$ terms: $$ - \frac{5}{8} y + y = - \frac{5}{8} y + \frac{8}{8} y = \frac{3}{8} y $$ 4. **Simplified expression:** $$ \frac{3}{8} x + \frac{3}{8} y $$ 5. **Check each option:** - A: $$ \frac{3}{8} x + \frac{3}{8} y $$ (matches simplified expression, correct) - B: $$ \frac{3}{8} x + 1 \frac{5}{8} y $$ (does not match, incorrect) - C: $$ \frac{5}{8} x - y - \frac{1}{4} x + y $$ simplifies to $$ \frac{3}{8} x $$ (missing $y$ term, incorrect) - D: $$ \frac{5}{8} x - \frac{5}{8} y - \frac{1}{4} x + y $$ simplifies to original expression, which simplifies to $$ \frac{3}{8} x + \frac{3}{8} y $$ (correct) - E: $$ \frac{5}{8} x - \frac{1}{4} x + y - \frac{5}{8} y $$ also simplifies to $$ \frac{3}{8} x + \frac{3}{8} y $$ (correct) **Final answers:** A, D, and E are equivalent to the original expression.