Question
Use the Commutative Property to determine the missing step in proving the equivalence of 12a+10+a=10+a+12a.%0D%0A%0D%0AStep 1: [missing]%0D%0A%0D%0AStep 2: 10+13a=10+13a%0D%0A%0D%0AStep 3: The expressions are equivalent because they both equal the same value.%0D%0A%0D%0A(1 point)%0D%0AResponses%0D%0A%0D%0A10+12a+a=10+13a%0D%0Ad plus 10 plus a equals 10 plus a plus d%0D%0A%0D%0A12a+10=10+12a%0D%0Ad plus 10 plus a equals 10 plus a plus d%0D%0A%0D%0A12a+10+a−a=10+a+12a−a%0D%0Ad plus 10 plus a equals 10 plus a plus d%0D%0A%0D%0A12a+10+a−10=10+a+12a−10
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GPT-4o mini
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:
1. **10 + 12a + a = 10 + 13a**: This combines the terms, resulting in the different formulations on both sides.
2. **d plus 10 plus a equals 10 plus a plus d**: This doesn't relate directly to the given context of the problem.
3. **12a + 10 = 10 + 12a**: This shows a direct application of the Commutative Property but doesn't take into account the second \(a\).
4. **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.
5. **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**.
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:
1. **10 + 12a + a = 10 + 13a**: This combines the terms, resulting in the different formulations on both sides.
2. **d plus 10 plus a equals 10 plus a plus d**: This doesn't relate directly to the given context of the problem.
3. **12a + 10 = 10 + 12a**: This shows a direct application of the Commutative Property but doesn't take into account the second \(a\).
4. **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.
5. **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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