For each expression, use a property to write an equivalent expression. Tell which property you used. 10. \( c d= \) \( \qquad \) 11. \( x+13= \) \( \qquad \) \( \qquad \) \( \qquad \) 12. \( 4(2 x-3)= \) \( \qquad \) 13. \( 2+(a+b)= \) \( \qquad \) \( \qquad \) \( \qquad \) 14. Draw algebra tile models to prove that \( 4+8 x \) and \( 4(2 x+1) \) are equivalent. \( \square \)

Let's go through each expression step by step, identifying the properties used to write equivalent expressions. ### 10. \( c d \) **Equivalent Expression:** \( d c \) **Property Used:** Commutative Property of Multiplication This property states that changing the order of the factors does not change the product. ### 11. \( x + 13 \) **Equivalent Expression:** \( 13 + x \) **Property Used:** Commutative Property of Addition This property states that changing the order of the addends does not change the sum. ### 12. \( 4(2x - 3) \) **Equivalent Expression:** \( 8x - 12 \) **Property Used:** Distributive Property This property states that \( a(b + c) = ab + ac \). Here, we distribute \( 4 \) to both \( 2x \) and \( -3 \). ### 13. \( 2 + (a + b) \) **Equivalent Expression:** \( (2 + a) + b \) **Property Used:** Associative Property of Addition This property states that the way in which numbers are grouped in addition does not change the sum. ### 14. Algebra Tile Models for \( 4 + 8x \) and \( 4(2x + 1) \) To prove that \( 4 + 8x \) and \( 4(2x + 1) \) are equivalent using algebra tiles, we can represent each expression with tiles: 1. **For \( 4 + 8x \)**: - Use 4 unit tiles to represent the constant \( 4 \). - Use 8 \( x \) tiles to represent \( 8x \). 2. **For \( 4(2x + 1) \)**: - First, distribute \( 4 \) to both \( 2x \) and \( 1 \): - \( 4 \times 2x = 8x \) (8 \( x \) tiles) - \( 4 \times 1 = 4 \) (4 unit tiles) - Combine these to show that you have 4 unit tiles and 8 \( x \) tiles. Both models will show the same number of unit tiles and \( x \) tiles, thus proving that \( 4 + 8x \) and \( 4(2x + 1) \) are equivalent. ### Summary of Properties Used: - Commutative Property of Multiplication - Commutative Property of Addition - Distributive Property - Associative Property of Addition