To identify the properties that justify each step, let us analyze the lines given:
Line 1: \( b(a + 8) \)
Line 2: \( ba + b \cdot 7 \)
Line 3: \( ab + 7b \)
Transition from Line 1 to Line 2:
The transition from Line 1 to Line 2 can be justified by the Distributive Property. The Distributive Property states that for any numbers \( a, b, \) and \( c \):
\[ b(a + c) = ba + bc \]
In this case, with \( c = 8 \):
\[ b(a + 8) = ba + b \cdot 8 \]
Thus, Line 1 to Line 2 is justified by the Distributive Property.
Transition from Line 2 to Line 3:
The transition from Line 2 to Line 3 can be justified by the Commutative Property of Multiplication. The Commutative Property of Multiplication states that for any numbers \( x \) and \( y \):
\[ xy = yx \]
Thus, \( b \cdot 7 = 7b \). So, we can rewrite \( ba + b \cdot 7 \) as \( ab + 7b \).
Therefore, Line 2 to Line 3 is justified by the Commutative Property of Multiplication.
Summary:
- Line 1 to Line 2: Distributive Property
- Line 2 to Line 3: Commutative Property of Multiplication
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