Use the Commutative Property to determine the missing step in proving the equivalence of 12a+10+a=10+a+12a .

Question

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

10+12a+a=10+13a
d plus 10 plus a equals 10 plus a plus d

12a+10=10+12a
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

12a+10+a−10=10+a+12a−10

Answer 1

To determine the missing step using the Commutative Property, we want to rearrange the terms in the expression \(12a + 10 + a\) to match the form presented in the second step, \(10 + 13a\).

Given this, the missing step can be:

Step 1: \(10 + 12a + a = 10 + 13a\)

This makes use of the Commutative Property, which states that the order of addition does not change the sum.

So, the complete sequence will be:

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

Therefore, the correct response is:

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 Commutative Property of Addition
The Commutative Property of Addition

The Associative Property of Multiplication
The Associative Property of Multiplication

The Associative Property of Addition
The Associative Property of Addition

The Commutative Property of Multiplication

Answer 3

Substituting 1 for \(x\) in the equation \(5x + 3 = x \cdot 5 + 3\) is a test case for the Commutative Property of Multiplication.

This is because the equation shows that the terms can be rearranged while still yielding the same result, specifically focusing on how multiplication (here, \(x \cdot 5\) and \(5x\)) can be commuted.

Thus, the correct response is:

The Commutative Property of Multiplication.

Answer 4

Which step contains an error using the Commutative Property of Addition in verifying 8+x2+17x−x2+4=10+4x2+10x−4x2+2+7x ?

Step 1: 8+4+x2−x2+17x=10+2+10x−4x2+7x +4x2

Step 2: 8+4+17x+x2−x2=10+2+4x2−4x2+10x+7x

Step 3: 12+17x+x2=12−x2+17x

(1 point)
Responses

Step 3
Step 3

No mistake was made.
No mistake was made.

Step 2
Step 2

Step 1

Answer 5

To identify which step contains an error while applying the Commutative Property of Addition, let's analyze each step given:

1. Step 1: \(8 + 4 + x^2 - x^2 + 17x = 10 + 2 + 10x - 4x^2 + 7x + 4x^2\)

   • This step is correct as it rearranges and combines like terms correctly.

2. Step 2: \(8 + 4 + 17x + x^2 - x^2 = 10 + 2 + 4x^2 - 4x^2 + 10x + 7x\)

   • Here, the left-hand side simplifies to \(12 + 17x\) since \(x^2 - x^2 = 0\).
   • The right-hand side simplifies correctly to \(12 + 10x + 7x\) which further combines to \(12 + 17x\).
   • Thus, this step also appears to be correct.

3. Step 3: \(12 + 17x + x^2 = 12 - x^2 + 17x\)

   • This step contains an error. The left-hand side is \(12 + 17x\) and has mistakenly added \(x^2\), while the right-hand side shows \(12 - x^2 + 17x\).
   • The presence of \(x^2\) on one side without it being on the other side indicates a mistake.

Given this analysis, the error occurs in:

Step 3.