Question
Use the Commutative Property to determine the missing step in proving the equivalence of 12a+10+a=10+a+12a.
Step 1: [missing]
Step 2: 10+13a=10+13a
Step 3: The expressions are equivalent because they both equal the same value.
A. 10+12a+a=10+13a
B. 12a+10=10+12a
C. 12a+10+a-a=10+a+12a-a
D. 12a+10+a-10=10+a+12a-10
Step 1: [missing]
Step 2: 10+13a=10+13a
Step 3: The expressions are equivalent because they both equal the same value.
A. 10+12a+a=10+13a
B. 12a+10=10+12a
C. 12a+10+a-a=10+a+12a-a
D. 12a+10+a-10=10+a+12a-10
Answers
Answered by
GPT-4o mini
To determine the correct missing step using the Commutative Property, we focus on the rearrangement of the terms in the expression \(12a + 10 + a\) to match the order of terms in \(10 + a + 12a\).
The original expression is \(12a + 10 + a\). We can rearrange this to group the constant and variable terms as needed:
1. Start with \(12a + 10 + a\).
2. Use the Commutative Property, which allows us to change the order of addition.
Given the options:
A. \(10 + 12a + a = 10 + 13a\) - This shows the correct final form but isn’t a rearrangement of just the left-hand side.
B. \(12a + 10 = 10 + 12a\) - This only rearranges part of the original expression and doesn't introduce the \(a\).
C. \(12a + 10 + a - a = 10 + a + 12a - a\) - This adds and subtracts \(a\), which complicates things rather than simplifying.
D. \(12a + 10 + a - 10 = 10 + a + 12a - 10\) - This also adds and subtracts \(10\), complicating the expressions again.
From a reasoning perspective, the most straightforward missing step using the Commutative Property is:
**Step 1: 12a + 10 + a = 10 + a + 12a**
This is not one of the options given, so let's check which of the provided answers matches:
After analyzing the options again, it seems like **B** and **A** are close but don’t directly mirror the step we need.
Looking for something that rearranges correctly, the simplest and most appropriate choice leading us to \(10 + 13a\) through steps involved but within the context of how to employ it effectively would closely relate to:
**Answer: B. 12a + 10 = 10 + 12a**
Even though it doesn't account for the additional \(a\) immediately, it starts rearranging terms correctly which would eventually lead us to step 2 after dealing with all constants and variable behaviors later.
However, the missing link before the jump into summation \(10 + 13a\) still connects us through more systematic handling leading back to option A when reasoning on the aggregate.
Overall, the least convoluted and most reasonable way to address would be to settle on B in reference, moving onward to sum efficiently linking \(10 + a + 12a\).
Thus, the most fitting completion to the proving of equivalence under the mentioned properties from what could correlate towards the assessment correctly links through B’s rearrangement, aligning closure toward \(=10 + 13a\) efficiently structurally, even though the simplified might loop back; final step being, while \(10+12a + a\) encapsulates \(10 + a + 12a\) contextually as it moves to estimation phase.
The original expression is \(12a + 10 + a\). We can rearrange this to group the constant and variable terms as needed:
1. Start with \(12a + 10 + a\).
2. Use the Commutative Property, which allows us to change the order of addition.
Given the options:
A. \(10 + 12a + a = 10 + 13a\) - This shows the correct final form but isn’t a rearrangement of just the left-hand side.
B. \(12a + 10 = 10 + 12a\) - This only rearranges part of the original expression and doesn't introduce the \(a\).
C. \(12a + 10 + a - a = 10 + a + 12a - a\) - This adds and subtracts \(a\), which complicates things rather than simplifying.
D. \(12a + 10 + a - 10 = 10 + a + 12a - 10\) - This also adds and subtracts \(10\), complicating the expressions again.
From a reasoning perspective, the most straightforward missing step using the Commutative Property is:
**Step 1: 12a + 10 + a = 10 + a + 12a**
This is not one of the options given, so let's check which of the provided answers matches:
After analyzing the options again, it seems like **B** and **A** are close but don’t directly mirror the step we need.
Looking for something that rearranges correctly, the simplest and most appropriate choice leading us to \(10 + 13a\) through steps involved but within the context of how to employ it effectively would closely relate to:
**Answer: B. 12a + 10 = 10 + 12a**
Even though it doesn't account for the additional \(a\) immediately, it starts rearranging terms correctly which would eventually lead us to step 2 after dealing with all constants and variable behaviors later.
However, the missing link before the jump into summation \(10 + 13a\) still connects us through more systematic handling leading back to option A when reasoning on the aggregate.
Overall, the least convoluted and most reasonable way to address would be to settle on B in reference, moving onward to sum efficiently linking \(10 + a + 12a\).
Thus, the most fitting completion to the proving of equivalence under the mentioned properties from what could correlate towards the assessment correctly links through B’s rearrangement, aligning closure toward \(=10 + 13a\) efficiently structurally, even though the simplified might loop back; final step being, while \(10+12a + a\) encapsulates \(10 + a + 12a\) contextually as it moves to estimation phase.
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