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
Step 1
Step 1
Step 3
Step 3
No mistake was made.
No mistake was made.
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
Step 1
Step 1
Step 3
Step 3
No mistake was made.
No mistake was made.
Step 2
Answers
Answered by
GPT-4o mini
To identify any errors made while applying the Commutative Property of Addition in verifying the equation \( 8+x^2+17x−x^2+4=10+4x^2+10x−4x^2+2+7x \), we will analyze each step carefully.
**Step 1:**
The original expression is rearranged to:
\[ 8+4+x^2−x^2+17x = 10+2+10x−4x^2+7x +4x^2 \]
Here, the terms appear to be repositioned correctly, and the total for each side is valid. The Commutative Property of Addition (which states that \(a + b = b + a\)) has been used correctly since the order of addition or rearrangement of terms is not an issue.
**Step 2:**
Now, we analyze this step:
\[ 8+4+17x+x^2−x^2 = 10+2+4x^2−4x^2+10x+7x \]
In this step:
- On the left side, it rearranges to \(8+4+17x+x^2−x^2\), which seems consistent with previous terms.
- On the right side, we must ensure that all terms are arranged correctly: the \(4x^2\) and \(-4x^2\) cancel but should be added together first before cancellation.
However, the left side translates to \(12 + 17x + x^2 - x^2\), simplifying to \(12 + 17x\), which is correct as it simplifies to \(12 + 17x\). The right side simplifies:
- \(10 + 2 + 4x^2 - 4x^2 + 10x + 7x\) simplifies to \(12 + 17x\) as well after canceling \(4x^2\) with \(-4x^2\).
This structural approach is consistent across steps.
**Step 3:**
We see:
\[ 12 + 17x + x^2 = 12 - x^2 + 17x \]
Here lies the discrepancy; \(x^2\) cannot equal \(-x^2\) unless you are asserting \(x\) to neutralize it inappropriately across both expressions.
So, the error made is between **Step 2** and **Step 3**. Specifically, in **Step 2**, the addition of \(4x^2\) and cancelling \(4x^2\) needs to be placed as a separate valid cancellation before arriving at or transitioning into other steps.
Thus, the step containing an error is:
**Step 3** - the rearranging and assumptions yielding \(x^2\) inappropriately placed versus \(−x^2\).
**Step 1:**
The original expression is rearranged to:
\[ 8+4+x^2−x^2+17x = 10+2+10x−4x^2+7x +4x^2 \]
Here, the terms appear to be repositioned correctly, and the total for each side is valid. The Commutative Property of Addition (which states that \(a + b = b + a\)) has been used correctly since the order of addition or rearrangement of terms is not an issue.
**Step 2:**
Now, we analyze this step:
\[ 8+4+17x+x^2−x^2 = 10+2+4x^2−4x^2+10x+7x \]
In this step:
- On the left side, it rearranges to \(8+4+17x+x^2−x^2\), which seems consistent with previous terms.
- On the right side, we must ensure that all terms are arranged correctly: the \(4x^2\) and \(-4x^2\) cancel but should be added together first before cancellation.
However, the left side translates to \(12 + 17x + x^2 - x^2\), simplifying to \(12 + 17x\), which is correct as it simplifies to \(12 + 17x\). The right side simplifies:
- \(10 + 2 + 4x^2 - 4x^2 + 10x + 7x\) simplifies to \(12 + 17x\) as well after canceling \(4x^2\) with \(-4x^2\).
This structural approach is consistent across steps.
**Step 3:**
We see:
\[ 12 + 17x + x^2 = 12 - x^2 + 17x \]
Here lies the discrepancy; \(x^2\) cannot equal \(-x^2\) unless you are asserting \(x\) to neutralize it inappropriately across both expressions.
So, the error made is between **Step 2** and **Step 3**. Specifically, in **Step 2**, the addition of \(4x^2\) and cancelling \(4x^2\) needs to be placed as a separate valid cancellation before arriving at or transitioning into other steps.
Thus, the step containing an error is:
**Step 3** - the rearranging and assumptions yielding \(x^2\) inappropriately placed versus \(−x^2\).