Question
Olivia is trying to simplify the expression 2+(x+5)
using properties of algebra. She has explained her reasoning in steps, but where did she go wrong?
I can use the Distributive Property to rewrite this expression as (2+x)+(2+5)
Then, I can use the Commutative Property to rewrite it again as (2+5)+(2+x)
I can also use the Associative Property to write the expression as (2+5+2)+x
Finally, I can simplify to show that the expression is equivalent to 9+x
(1 point)
Responses
Step 1
Step 1
Step 2
Step 2
Step 3
Step 3
Step 4
using properties of algebra. She has explained her reasoning in steps, but where did she go wrong?
I can use the Distributive Property to rewrite this expression as (2+x)+(2+5)
Then, I can use the Commutative Property to rewrite it again as (2+5)+(2+x)
I can also use the Associative Property to write the expression as (2+5+2)+x
Finally, I can simplify to show that the expression is equivalent to 9+x
(1 point)
Responses
Step 1
Step 1
Step 2
Step 2
Step 3
Step 3
Step 4
Answers
GPT-4o mini
Olivia's reasoning contains an error in her application of the Distributive Property in Step 1. Let's analyze the steps she took:
1. **Step 1**: Olivia claims to use the Distributive Property to rewrite \(2 + (x + 5)\) as \((2 + x) + (2 + 5)\). This is incorrect. The Distributive Property is used when multiplying a term outside of parentheses by the terms inside. This expression does not require distribution; it is simply adding constants and variables.
2. **Step 2**: She then uses the Commutative Property to rewrite it again as \((2 + 5) + (2 + x)\). This step builds on the incorrect first step. The Commutative Property allows for rearranging terms, but since the expression has already been incorrectly modified, the foundation for this step is flawed.
3. **Step 3**: Olivia uses the Associative Property to write the expression as \((2 + 5 + 2) + x\). This step also relies on the previous mistakes, so it is not valid.
4. **Step 4**: Finally, she simplifies the expression to \(9 + x\), which is incorrect. The correct simplification of the original expression \(2 + (x + 5)\) is \(x + 7\) (combining the constants 2 and 5).
In summary, Olivia's error happens in Step 1 with her incorrect use of the Distributive Property. The correct approach to the original expression should simply involve adding the constants together:
**Correct initial step**: \(2 + (x + 5) = 2 + 5 + x = 7 + x\).
1. **Step 1**: Olivia claims to use the Distributive Property to rewrite \(2 + (x + 5)\) as \((2 + x) + (2 + 5)\). This is incorrect. The Distributive Property is used when multiplying a term outside of parentheses by the terms inside. This expression does not require distribution; it is simply adding constants and variables.
2. **Step 2**: She then uses the Commutative Property to rewrite it again as \((2 + 5) + (2 + x)\). This step builds on the incorrect first step. The Commutative Property allows for rearranging terms, but since the expression has already been incorrectly modified, the foundation for this step is flawed.
3. **Step 3**: Olivia uses the Associative Property to write the expression as \((2 + 5 + 2) + x\). This step also relies on the previous mistakes, so it is not valid.
4. **Step 4**: Finally, she simplifies the expression to \(9 + x\), which is incorrect. The correct simplification of the original expression \(2 + (x + 5)\) is \(x + 7\) (combining the constants 2 and 5).
In summary, Olivia's error happens in Step 1 with her incorrect use of the Distributive Property. The correct approach to the original expression should simply involve adding the constants together:
**Correct initial step**: \(2 + (x + 5) = 2 + 5 + x = 7 + x\).