Question
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 + x ^ 2 - x ^ 2 = 10 + 2 + 4x ^ 2 - 4x ^ 2 + 10x + 7x Step 3 12 + 17x + x ^ 2 = 12 - x ^ 2 + 17x
Answers
Answered by
GPT-4o mini
To find the error in the steps using the Commutative Property of Addition in verifying the equality, let's analyze each step one by one.
Starting with the original expression:
**Left Side:** \( 8 + x^2 + 17x - x^2 + 4 \)
**Right Side:** \( 10 + 4x^2 + 10x - 4x^2 + 2 + 7x \)
### Step 1:
**Expression:**
Left: \( 8 + 4 + x^2 - x^2 + 17x \)
Right: \( 10 + 2 + 10x - 4x^2 + 7x + 4x^2 \)
Here, the commutative property is applied to regroup the terms. This step correctly rearranges the expressions.
### Step 2:
**Expression:**
Left: \( 8 + 4 + 17x + x^2 - x^2 \)
Right: \( 10 + 2 + 4x^2 - 4x^2 + 10x + 7x \)
This also seems to apply the commutative property appropriately as it rearranges the terms.
### Step 3:
**Expression:**
Left: \( 12 + 17x + x^2 \)
Right: \( 12 - x^2 + 17x \)
In this step, the left side of the equation is simplified correctly as it combines \( 8 + 4 = 12\) and recognizes that \( x^2 - x^2 = 0\).
The right side attempts to simplify \(10 + 2 + 4x^2 - 4x^2 + 10x + 7x\) which should result in \( 12 + 0 + (10x + 7x)\) and therefore \( 12 + 17x\).
### Conclusion:
Looking closely at Step 2, it's seen that the right side contains a simplification mistake. The term management involving \(-4x^2 + 4x^2\) is incorrect to manage on the right side because it should yield zero.
Therefore, the error arises in **Step 3** since it incorrectly states the right side without processing the terms correctly. The left side and the right side in this step should actually be identical as \( 12 + 17x\), but instead, it states \( 12 - x^2 + 17x\).
Thus, **Step 3** contains the error.
Starting with the original expression:
**Left Side:** \( 8 + x^2 + 17x - x^2 + 4 \)
**Right Side:** \( 10 + 4x^2 + 10x - 4x^2 + 2 + 7x \)
### Step 1:
**Expression:**
Left: \( 8 + 4 + x^2 - x^2 + 17x \)
Right: \( 10 + 2 + 10x - 4x^2 + 7x + 4x^2 \)
Here, the commutative property is applied to regroup the terms. This step correctly rearranges the expressions.
### Step 2:
**Expression:**
Left: \( 8 + 4 + 17x + x^2 - x^2 \)
Right: \( 10 + 2 + 4x^2 - 4x^2 + 10x + 7x \)
This also seems to apply the commutative property appropriately as it rearranges the terms.
### Step 3:
**Expression:**
Left: \( 12 + 17x + x^2 \)
Right: \( 12 - x^2 + 17x \)
In this step, the left side of the equation is simplified correctly as it combines \( 8 + 4 = 12\) and recognizes that \( x^2 - x^2 = 0\).
The right side attempts to simplify \(10 + 2 + 4x^2 - 4x^2 + 10x + 7x\) which should result in \( 12 + 0 + (10x + 7x)\) and therefore \( 12 + 17x\).
### Conclusion:
Looking closely at Step 2, it's seen that the right side contains a simplification mistake. The term management involving \(-4x^2 + 4x^2\) is incorrect to manage on the right side because it should yield zero.
Therefore, the error arises in **Step 3** since it incorrectly states the right side without processing the terms correctly. The left side and the right side in this step should actually be identical as \( 12 + 17x\), but instead, it states \( 12 - x^2 + 17x\).
Thus, **Step 3** contains the error.