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

12a+10=10+12a
d plus 10 plus a equals 10 plus a plus d

12a+10+a−10=10+a+12a−10
d plus 10 plus a equals 10 plus a plus d

12a+10+a−a=10+a+12a−a
d plus 10 plus a equals 10 plus a plus d

10+12a+a=10+13a

Answer 1

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

12a+10=10+12a
d plus 10 plus a equals 10 plus a plus d

12a+10+a−10=10+a+12a−10
d plus 10 plus a equals 10 plus a plus d

12a+10+a−a=10+a+12a−a
d plus 10 plus a equals 10 plus a plus d

10+12a+a=10+13a

Answer 2

Substituting 1 for x in the equation 5x + 3=x⋅5 + 3 is a test case for which property?(1 point)
Responses

The Associative Property of Multiplication
The Associative Property of Multiplication

The Commutative Property of Addition
The Commutative Property of Addition

The Associative Property of Addition
The Associative Property of Addition

The Commutative Property of Multiplication
The Commutative Property of Multiplication

Answer 3

To fill in the missing step in the proof of the equivalence \( 12a + 10 + a = 10 + a + 12a \) using the Commutative Property, we need a step that rearranges the terms in the expression \( 12a + 10 + a \).

The correct missing step is:

**Step 1: 10 + 12a + a = 10 + a + 12a**

This uses the Commutative Property, which allows us to rearrange the terms in an expression.

The final sequence would look like this:

Step 1: 10 + 12a + a = 10 + a + 12a
Step 2: 10 + 13a = 10 + 13a
Step 3: The expressions are equivalent because they both equal the same value.

The first response option, **12a+10=10+12a**, is also a valid use of the Commutative Property, but it doesn't lead directly to rearranging to form the step needed in this proof sequence. The best choice that leads to the sequence in the proof is:

**10 + 12a + a = 10 + a + 12a**.

Answer 4

To determine the missing step in proving the equivalence of \( 12a + 10 + a = 10 + a + 12a \) using the Commutative Property, we want to find a way to rearrange the terms so it leads correctly to Step 2.

Step 1 should rearrange the terms of \( 12a + 10 + a \). The most appropriate response, which clearly shows the application of the Commutative Property to achieve the desired form in Step 2, is:

**10 + 12a + a = 10 + 13a**

This shows that we can rearrange the terms of \( 12a + 10 + a \) to bring similar terms together and then combine them.

So, the correct missing step for proving the equivalence is:

**10 + 12a + a = 10 + 13a**.

Answer 5

The equation \( 5x + 3 = x \cdot 5 + 3 \) is tested by substituting \( x = 1 \) to demonstrate the property of multiplication.

In this case, the substitution shows:

1. On the left side: \( 5(1) + 3 = 5 + 3 = 8 \)
2. On the right side: \( (1) \cdot 5 + 3 = 5 + 3 = 8 \)

Since both sides equal 8, this illustrates that the order of multiplication does not affect the outcome, validating the property involved.

Specifically, this demonstrates the **Commutative Property of Multiplication**, which states that changing the order of the factors does not change the product.

So the correct response is:

**The Commutative Property of Multiplication**.