For each line of the expression, here are the steps identified along with the properties that justify each transformation:
Line 1: \( 5(4 + 7x) \)
- This line is the original expression, so there is no property needed to justify it.
Line 2: \( 5(7x + 4) \)
- Property: Commutative Property of Addition
- Justification: The expression inside the parentheses changes from \( (4 + 7x) \) to \( (7x + 4) \) by rearranging the order of the terms. The commutative property allows us to change the order of addition.
Line 3: \( 35x + 20 \)
- Property: Distributive Property
- Justification: The expression \( 5(7x + 4) \) is simplified by distributing the \( 5 \) across both terms inside the parentheses, giving us \( 5 \cdot 7x + 5 \cdot 4 \), which simplifies to \( 35x + 20 \).
Thus, the properties that justify each step are:
- Line 2: Commutative Property of Addition
- Line 3: Distributive Property