Question

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 Commutative Property of Multiplication
The Commutative Property of Multiplication

The Associative Property of Addition
The Associative Property of Addition

The Associative Property of Multiplication

Answer 1

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

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

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

Step 3: 12+17x+x^2=12−x^2+17x

(1 point)
Responses

Step 3
Step 3

Step 1
Step 1

No mistake was made.
No mistake was made.

Step 2

Answer 2

The equation \( 5x + 3 = x \cdot 5 + 3 \) can be tested by substituting \( x = 1 \).

When substituting \( x = 1 \):

- The left side becomes \( 5(1) + 3 = 5 + 3 = 8 \).
- The right side becomes \( 1 \cdot 5 + 3 = 5 + 3 = 8 \).

Both sides are equal, confirming the equation holds true for this substitution.

This relationship demonstrates that the multiplication of \( x \) with 5 can be commutated in terms of their positions in the expression since \( 5x \) and \( x \cdot 5 \) yield the same result because of the nature of multiplication being commutative.

Thus, the property being tested here is **The Commutative Property of Multiplication**.

So, the correct response is:

**The Commutative Property of Multiplication**

Answer 3

To verify the correctness of each step involving the Commutative Property of Addition, let's evaluate the steps one by one.

### Step-by-step Analysis:

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

Upon inspection, we can rearrange the right-hand side:
\[
10 + 2 + 10x - 4x^2 + 7x + 4x^2 = 12 + 10x + 7x + 2 = 12 + 17x
\]
The left-hand side simplifies to \( 8 + 4 + (x^2 - x^2) + 17x = 12 + 17x \).
This appears correct.

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

Rearranging the right-hand side:
\[
10 + 2 + (4x^2 - 4x^2) + 10x + 7x = 12 + 10x + 7x = 12 + 17x
\]
The left side simplifies to \( 8 + 4 + 17x + (x^2 - x^2) = 12 + 17x \).
This step seems correct as both sides equate.

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

In this step, we see an inconsistency. The left-hand side contains \( x^2 \), while the right-hand side has \( -x^2 \), which creates a mismatch.

### Conclusion:
The error lies in **Step 3**, where \( x^2 \) on the left cannot equal \( -x^2 \) on the right.

Therefore, the correct response is:

**Step 3**