Final answer:
The step from Line 1 to Line 2 is justified by the Commutative Property of Addition, and the step from Line 2 to Line 3 is justified by the Associative Property of Addition.
Step-by-step explanation:
To identify the properties that justify each step from Line 1 to Line 4, let's examine the changes made at each step:
- Line 1 to Line 2: The expression (2x + 9) becomes (9 + 2x). Here, the order of the terms in the parentheses has been changed. This change is justified by the Commutative Property of Addition, which states that you can change the order of the terms when adding and it will not affect the sum (A + B = B + A).
- Line 2 to Line 3: The expression (9 + 2x) + 5x becomes 9 + (2x + 5x). In this step, the association, or grouping, of the terms has been changed while maintaining the order of the terms. This is justified by the Associative Property of Addition, which states that the way in which terms are grouped does not affect the sum (A + (B + C) = (A + B) + C).
- Line 3 to Line 4: The expression 9 + (2x + 5x) simplifies to 9 + 7x by combining like terms. Combining like terms itself is a simplification process but it is based on the understanding of the Distributive Property. However, in this case, we are simply performing an addition of the coefficients of like terms (2x + 5x = 7x), which falls under the Commutative and Associative Properties.
Based on the above, the answers are:
- Line 1 to Line 2: Commutative Property of Addition
- Line 2 to Line 3: Associative Property of Addition