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. (1 point) Responses
A. 10+12a+a=10+13a
B. 12a+10+a−10=10+a+12a−10
C. 12a+10+a−a=10+a+12a−a
D. 12a+10=10+12a

Answer 1

To determine the missing step in the proof, we need to consider the Commutative Property, which states that the order of addition does not affect the sum.

Given the expression \(12a + 10 + a\) and wanting to show that it is equivalent to \(10 + a + 12a\), let's analyze what the expressions look like in terms of reordering the terms.

Starting with Step 2:
- We see that \(10 + 13a\) is the simplified version of \(12a + 10 + a\) since \(12a + a = 13a\).

The missing step should rearrange \(12a + 10 + a\) to match the target expression \(10 + a + 12a\) before simplifying.

Looking at the options provided:

A. \(10 + 12a + a = 10 + 13a\) - This is a valid rearrangement, but it doesn't match the format we are looking for since we want to move \(12a\) to the back.

B. \(12a + 10 + a - 10 = 10 + a + 12a - 10\) - This involves subtraction which complicates the proof without need.

C. \(12a + 10 + a - a = 10 + a + 12a - a\) - This also complicates things unnecessarily with subtraction.

D. \(12a + 10 = 10 + 12a\) - This uses the Commutative Property directly on part of the expression and correctly rearranges the terms, but it’s not the whole expression yet.

While options A and D are close, option A provides the necessary transition directly to the setup for simplification followed by comparison.

Therefore, the missing step is:

**A. \(10 + 12a + a = 10 + 13a\)**