To determine the missing step in the proof of equivalence for the expressions \(12a + 10 + a = 10 + a + 12a\), we can apply the Commutative Property, which states that the order of addition does not change the sum.
Given that:
- Step 1 is the missing step
- Step 2 is \(10 + 13a = 10 + 13a\)
- Step 3 concludes that the expressions are equivalent because they equal the same value.
Let's analyze the options provided for Step 1:
- 10 + 12a + a = 10 + 13a: This combines the terms, resulting in the different formulations on both sides.
- d plus 10 plus a equals 10 plus a plus d: This doesn't relate directly to the given context of the problem.
- 12a + 10 = 10 + 12a: This shows a direct application of the Commutative Property but doesn't take into account the second \(a\).
- 12a + 10 + a - a = 10 + a + 12a - a: While this correctly retains the values and transformations, it is an unnecessary complication and does not utilize the properties effectively.
- 12a + 10 + a - 10 = 10 + a + 12a - 10: Similar to the prior option, this makes attempts to simplify which isn't necessary for proving equivalency.
The most straightforward application of the Commutative Property would be to rearrange \(12a + 10 + a\) to \(10 + 12a + a\) first.
Thus, a correct Step 1 using the Commutative Property would be: 10 + 12a + a = 10 + 13a.
That brings us to the equivalence shown in Step 2, which states both sides equal \(10 + 13a\).
The missing step is: 10 + 12a + a = 10 + 13a.
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