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: 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
Answers
Answered by
GPT-4o mini
To determine which step contains an error in the application of the Commutative Property of Addition, we will analyze each step carefully.
**Step 1:**
The equation is:
\[ 8 + 4 + x^2 - x^2 + 17x = 10 + 2 + 10x - 4x^2 + 7x + 4x^2 \]
In this step, we should ensure that the terms on both sides of the equation match correctly. The left-hand side simplifies to:
\[ 12 + 17x \]
(using \(x^2 - x^2 = 0\)).
On the right-hand side, the expression simplifies as follows:
\[
10 + 2 + 10x - 4x^2 + 7x + 4x^2 = 12 + 17x
\]
(using \( -4x^2 + 4x^2 = 0\)).
Since both sides are equal in this step, Step 1 is correct.
---
**Step 2:**
The equation is:
\[ 8 + 4 + 17x + x^2 - x^2 = 10 + 2 + 4x^2 - 4x^2 + 10x + 7x \]
Here again, the left-hand side simplifies to:
\[ 12 + 17x \]
On the right-hand side, simplifying gives us:
\[ 10 + 2 + 4x^2 - 4x^2 + 10x + 7x = 12 + 17x \]
Thus, both sides are equal again in this step, so Step 2 is also correct.
---
**Step 3:**
The equation is represented as:
\[ 12 + 17x + x^2 = 12 - x^2 + 17x \]
In this step, the left-hand side is:
\[ 12 + 17x + x^2 \]
The right-hand side is:
\[ 12 - x^2 + 17x \]
This shows that the right-hand side has a term \(-x^2\) whereas the left-hand side has a term \(+x^2\). This is incorrect. The left side contains a positive \(x^2\) while the right side contains a negative \(x^2\).
Therefore, **the error occurs in Step 3** due to the incorrect treatment of the \(x^2\) terms. The correct application should have either matched both sides concerning the \(x^2\) terms or simply stated that both sides must be equivalent.
**Step 1:**
The equation is:
\[ 8 + 4 + x^2 - x^2 + 17x = 10 + 2 + 10x - 4x^2 + 7x + 4x^2 \]
In this step, we should ensure that the terms on both sides of the equation match correctly. The left-hand side simplifies to:
\[ 12 + 17x \]
(using \(x^2 - x^2 = 0\)).
On the right-hand side, the expression simplifies as follows:
\[
10 + 2 + 10x - 4x^2 + 7x + 4x^2 = 12 + 17x
\]
(using \( -4x^2 + 4x^2 = 0\)).
Since both sides are equal in this step, Step 1 is correct.
---
**Step 2:**
The equation is:
\[ 8 + 4 + 17x + x^2 - x^2 = 10 + 2 + 4x^2 - 4x^2 + 10x + 7x \]
Here again, the left-hand side simplifies to:
\[ 12 + 17x \]
On the right-hand side, simplifying gives us:
\[ 10 + 2 + 4x^2 - 4x^2 + 10x + 7x = 12 + 17x \]
Thus, both sides are equal again in this step, so Step 2 is also correct.
---
**Step 3:**
The equation is represented as:
\[ 12 + 17x + x^2 = 12 - x^2 + 17x \]
In this step, the left-hand side is:
\[ 12 + 17x + x^2 \]
The right-hand side is:
\[ 12 - x^2 + 17x \]
This shows that the right-hand side has a term \(-x^2\) whereas the left-hand side has a term \(+x^2\). This is incorrect. The left side contains a positive \(x^2\) while the right side contains a negative \(x^2\).
Therefore, **the error occurs in Step 3** due to the incorrect treatment of the \(x^2\) terms. The correct application should have either matched both sides concerning the \(x^2\) terms or simply stated that both sides must be equivalent.