1. **State the problem:** Simplify the expression $$\frac{1}{3}(18a + 6b) - 2(2a + 4b)$$ and find which given expressions are equivalent to it. 2. **Apply the distributive property:** $$\frac{1}{3}(18a + 6b) = \frac{1}{3} \times 18a + \frac{1}{3} \times 6b = 6a + 2b$$ $$-2(2a + 4b) = -2 \times 2a - 2 \times 4b = -4a - 8b$$ 3. **Combine like terms:** $$6a + 2b - 4a - 8b = (6a - 4a) + (2b - 8b) = 2a - 6b$$ 4. **Check each option:** - $10a + 10b$ is not equal to $2a - 6b$. - $2a - 6b$ matches exactly. - $-2(a - 3b) = -2a + 6b$ which is not equal to $2a - 6b$. - $-2(-a + 3b) = 2a - 6b$ which matches. - $6(\frac{1}{3}a - b) = 6(\frac{1}{3}a) - 6b = 2a - 6b$ which matches. - $6(\frac{1}{3}a - \frac{1}{6}b) = 2a - b$ which is not equal to $2a - 6b$. **Final answer:** The expressions equivalent to $$\frac{1}{3}(18a + 6b) - 2(2a + 4b)$$ are: $$2a - 6b$$ $$-2(-a + 3b)$$ $$6(\frac{1}{3}a - b)$$