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?
1. i can use the distributive property to rewrite this expression as (2 + x) + (2 + 5)
2. then, i can use the commutative property to rewrite it again as (2 + 5) + (2 + x)
3. i can also use the associative property to write the expression as (2 + 5 + 2) + x
4. finally, i can simplify to show that the expression is equivalent to 9 + x
1. i can use the distributive property to rewrite this expression as (2 + x) + (2 + 5)
2. then, i can use the commutative property to rewrite it again as (2 + 5) + (2 + x)
3. i can also use the associative property to write the expression as (2 + 5 + 2) + x
4. finally, i can simplify to show that the expression is equivalent to 9 + x
Answers
GPT-4o mini
Olivia made an error in her first step when she applied the distributive property. The distributive property is used to multiply a term across a sum or difference, as in \( a(b + c) = ab + ac \). However, in her expression \( 2 + (x + 5) \), she is not able to use the distributive property because there is no multiplication occurring; instead, she should simply apply straightforward simplification.
Here’s where she went wrong step-by-step:
1. **Incorrect Step (Distributive Property)**: She should not have rewritten it as \( (2 + x) + (2 + 5) \). The expression \( 2 + (x + 5) \) can instead be simplified directly to \( 2 + x + 5 \).
2. **Correct Steps**: After correctly simplifying \( 2 + (x + 5) \) to \( 2 + x + 5 \), she could then use the commutative property to rearrange the terms, leading to \( (2 + 5) + x \).
3. **Correct Association**: Now it becomes \( 7 + x \).
4. **Final Simplification**: Therefore, the correct final simplified expression should be \( 7 + x\), not \( 9 + x\).
In conclusion, Olivia's final answer of \( 9 + x \) is incorrect because the distribution and manipulation of the expression were done incorrectly from the start. The correct simplification should lead to \( 7 + x \).
Here’s where she went wrong step-by-step:
1. **Incorrect Step (Distributive Property)**: She should not have rewritten it as \( (2 + x) + (2 + 5) \). The expression \( 2 + (x + 5) \) can instead be simplified directly to \( 2 + x + 5 \).
2. **Correct Steps**: After correctly simplifying \( 2 + (x + 5) \) to \( 2 + x + 5 \), she could then use the commutative property to rearrange the terms, leading to \( (2 + 5) + x \).
3. **Correct Association**: Now it becomes \( 7 + x \).
4. **Final Simplification**: Therefore, the correct final simplified expression should be \( 7 + x\), not \( 9 + x\).
In conclusion, Olivia's final answer of \( 9 + x \) is incorrect because the distribution and manipulation of the expression were done incorrectly from the start. The correct simplification should lead to \( 7 + x \).