Question
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 1
Step 1
Step 2
Step 2
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 1
Step 1
Step 2
Step 2
Answers
Answered by
GPT-4o mini
To check for errors using the Commutative Property of Addition, we need to ensure that the expressions are set up correctly when reordering the terms.
Let's analyze each step:
**Step 1:**
Given:
\[8 + 4 + x^2 - x^2 + 17x = 10 + 2 + 10x - 4x^2 + 7x + 4x^2\]
The left side simplifies to:
\[8 + 4 + (x^2 - x^2) + 17x = 12 + 17x\]
On the right side:
\[10 + 2 + (10x + 7x + 4x^2 - 4x^2) = 12 + 17x\]
So **Step 1** is correct.
---
**Step 2:**
Given:
\[8 + 4 + 17x + x^2 - x^2 = 10 + 2 + 4x^2 - 4x^2 + 10x + 7x\]
The left side simplifies the same way as before:
\[12 + 17x + (x^2 - x^2) = 12 + 17x\]
The right side simplifies to:
\[12 + (10x + 7x + 4x^2 - 4x^2) = 12 + 17x\]
So **Step 2** is correct as well.
---
**Step 3:**
Given:
\[12 + 17x + x^2 = 12 - x^2 + 17x\]
Here, combining the left side gives:
\[12 + 17x + x^2 = 12 + 17x + x^2\]
However, the right side has a negative x^2 term which should not affect the equality if both sides are arranged properly. This suggests there is an inconsistency.
The left side contains \(x^2\) while the right side has \(-x^2\).
Thus, while all previous steps contained no errors, **Step 3** contains an error because it implies \(12 + 17x + x^2\) equals \(12 - x^2 + 17x\), which is not true due to the presence of \(x^2\) on the left and \(-x^2\) on the right.
Therefore, the correct response is:
**Step 3**.
Let's analyze each step:
**Step 1:**
Given:
\[8 + 4 + x^2 - x^2 + 17x = 10 + 2 + 10x - 4x^2 + 7x + 4x^2\]
The left side simplifies to:
\[8 + 4 + (x^2 - x^2) + 17x = 12 + 17x\]
On the right side:
\[10 + 2 + (10x + 7x + 4x^2 - 4x^2) = 12 + 17x\]
So **Step 1** is correct.
---
**Step 2:**
Given:
\[8 + 4 + 17x + x^2 - x^2 = 10 + 2 + 4x^2 - 4x^2 + 10x + 7x\]
The left side simplifies the same way as before:
\[12 + 17x + (x^2 - x^2) = 12 + 17x\]
The right side simplifies to:
\[12 + (10x + 7x + 4x^2 - 4x^2) = 12 + 17x\]
So **Step 2** is correct as well.
---
**Step 3:**
Given:
\[12 + 17x + x^2 = 12 - x^2 + 17x\]
Here, combining the left side gives:
\[12 + 17x + x^2 = 12 + 17x + x^2\]
However, the right side has a negative x^2 term which should not affect the equality if both sides are arranged properly. This suggests there is an inconsistency.
The left side contains \(x^2\) while the right side has \(-x^2\).
Thus, while all previous steps contained no errors, **Step 3** contains an error because it implies \(12 + 17x + x^2\) equals \(12 - x^2 + 17x\), which is not true due to the presence of \(x^2\) on the left and \(-x^2\) on the right.
Therefore, the correct response is:
**Step 3**.