To determine the missing step in proving the equivalence of \(12a + 10 + a = 10 + a + 12a\), we can analyze the expressions provided. We'll use the Commutative Property, which states that the order of addition does not affect the sum.
Given:
- \(12a + 10 + a\) (The left side)
- \(10 + a + 12a\) (The right side)
Let's first simplify the left side:
Step 1: Reorder the expression using the Commutative Property: \[ 10 + a + 12a \] This shows that \(12a + 10 + a\) can be rearranged as \(10 + a + 12a\), which matches the right side.
So, the correct missing step is: \[ 12a + 10 + a = 10 + 12a + a \]
The answer choices provided include:
- Response: \(10 + 12a + a = 10 + a + 12a\)
This represents rearranging terms based on the Commutative Property, which leads to the next step of combining like terms to reach:
Step 2: \(10 + 13a = 10 + 13a\)
Thus, the complete proof is:
- \(12a + 10 + a = 10 + 12a + a\) (apply Commutative Property)
- Combine \(12a\) and \(a\) to get \(10 + 13a = 10 + 13a\)
- Conclude they are equivalent.
The answer to the missing step using the Commutative Property is: \[ 12a + 10 + a = 10 + 12a + a \] which is consistent with the options provided. If we look at the given responses, "10 + 12a + a = 10 + a + 12a" is the correct sequence to follow. However, it's essential to note that each response contains the expression representing the process of reordering according to where we began. The most relevant one should be \(12a + 10 + a = 10 + 12a + a\), emphasizing the commutative action on the components.
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