Question

5(2x+6)=

Answer 1

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

Answer 2

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

Answer 3

To simplify the expression \(5(2x + 6)\), you can distribute the 5 to both terms inside the parentheses:

\[
5(2x) + 5(6)
\]

Calculating each term gives:

\[
10x + 30
\]

So, the simplified expression is:

\[
10x + 30
\]

Answer 4

Olivia's reasoning contains an error in her use of the Distributive Property in Step 1. Let's analyze each step of her reasoning:

1. **Step 1**: She states that she can use the Distributive Property to rewrite \(2 + (x + 5)\) as \((2 + x) + (2 + 5)\). This is incorrect because the Distributive Property is used to distribute a coefficient across terms inside parentheses (e.g., \(a(b + c) = ab + ac\)). In this case, there's no coefficient to distribute.

2. **Step 2**: She then applies the Commutative Property, which states that the order of addition doesn't matter. However, since her prior step was incorrect, this step does not logically follow.

3. **Step 3**: The Associative Property allows for the regrouping of terms, but again, the prior steps have led to a misunderstanding, so this step is also flawed.

4. **Step 4**: Finally, she claims the expression simplifies to \(9 + x\), which is incorrect based on her previous steps.

The mistake lies primarily in **Step 1**. The expression \(2 + (x + 5)\) can be directly simplified to \(x + 7\) without needing to incorrectly apply the properties of algebra. The correct simplification should directly combine \(2\) and \(5\) as \(2 + 5 = 7\), leading to:

\[
2 + (x + 5) = x + 7
\]

So the error is in her initial assumption that she could distribute terms in that way.

Answer 5

Olivia's mistake occurs in **Step 1**. Let's break down each step to identify the error:

1. **Step 1**: Olivia claims to use the Distributive Property to rewrite \(2 + (x + 5)\) as \((2 + x) + (2 + 5)\). This is incorrect because the Distributive Property applies to expressions like \(a(b + c)\) but not in the way she is using it here. There is no distribution happening in the original expression; she should simply combine the constants. The correct way to simplify \(2 + (x + 5)\) is to add the constants directly, resulting in \(x + 7\).

2. **Step 2**: Olivia applies the Commutative Property, stating she can rearrange the expression. Although this property is valid, the previous step was incorrect, making the rearrangement not relevant.

3. **Step 3**: Olivia uses the Associative Property to group terms, but again this follows an incorrect foundation from Step 1.

4. **Step 4**: Finally, she combines the terms and concludes the expression is equivalent to \(9 + x\). This result is incorrect based on her initial mistake.

In summary, Olivia's reasoning is flawed starting from **Step 1**, where she misapplied the Distributive Property. The correct approach would simply combine \(2 + (x + 5)\) to yield \(x + 7\).