Let's analyze each of the steps and identify the appropriate properties that justify the transitions between them:
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Line 1: (10x)(5y)
Line 2: 10(x*5)y
Justification: This step is justified by the Associative Property of Multiplication. It rearranges the grouping of the factors but does not change their product. -
Line 2: 10(x*5)y
Line 3: 10(5*x)y
Justification: This step is justified by the Commutative Property of Multiplication. It changes the order of the factors inside the parentheses (5 and x). -
Line 3: 10(5*x)y
Line 4: (105)(xy)
Justification: This step is again an application of the Associative Property of Multiplication. It groups the factors 10 and 5 together, while also combining x and y. -
Line 4: (105)(xy)
Line 5: 50xy
Justification: This final step simplifies the product by calculating the multiplication of 10 and 5, which results in 50, and combines that with the product of x and y.
To summarize the properties used for each transition:
- Line 1 to Line 2: Associative Property of Multiplication
- Line 2 to Line 3: Commutative Property of Multiplication
- Line 3 to Line 4: Associative Property of Multiplication
- Line 4 to Line 5: Basic multiplication/combination of terms.
The mention of "distributive property" seems to be a misunderstanding, as it is not applicable in the steps provided.