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 the expression as (2 + x) + (2 + 5)
2. then I'll 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'll simplify it and show that the expression is equivalent to 9 + x
1. I can use the distributive property to rewrite the expression as (2 + x) + (2 + 5)
2. then I'll 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'll simplify it and show that the expression is equivalent to 9 + x
Answers
GPT-4o mini
Olivia made an error in her application of the properties of algebra. Let's break down where she went wrong in each step:
1. **Step 1**: Olivia states that she can use the distributive property to rewrite the expression as \((2 + x) + (2 + 5)\). However, this step is incorrect. The distributive property is used to distribute multiplication over addition (or vice versa), but in this case, she's not actually applying it correctly to this expression. The original expression \(2 + (x + 5)\) should simply simplify to \(2 + x + 5\) without applying the distributive property.
2. **Step 2**: She states that she can rewrite it again using the commutative property as \((2 + 5) + (2 + x)\). This is also incorrect because the commutative property changes the order of addition but does not change how the expression is structured. The expression \((2 + x) + (2 + 5)\) does not correspond to the original expression correctly.
3. **Step 3**: Olivia mentions using the associative property to write the expression as \((2 + 5 + 2) + x\). She's attempting to group the numbers differently, but since the previous steps were incorrect, this step is also flawed.
4. **Step 4**: Finally, Olivia simplifies to \(9 + x\). While \(2 + 5\) is indeed \(7\), her prior steps were incorrect, and she should have been handling the expression \(2 + x + 5\) directly instead of using incorrect applications of properties.
To correctly simplify the original expression \(2 + (x + 5)\), you would simply combine like terms:
1. Start with \(2 + (x + 5)\).
2. This becomes \(2 + x + 5\).
3. Then combine the numbers: \(2 + 5 = 7\).
4. The final simplified expression is \(7 + x\).
So the correct simplified expression is \(7 + x\) instead of \(9 + x\). Olivia's misunderstanding lies in the improper use of algebraic properties.
1. **Step 1**: Olivia states that she can use the distributive property to rewrite the expression as \((2 + x) + (2 + 5)\). However, this step is incorrect. The distributive property is used to distribute multiplication over addition (or vice versa), but in this case, she's not actually applying it correctly to this expression. The original expression \(2 + (x + 5)\) should simply simplify to \(2 + x + 5\) without applying the distributive property.
2. **Step 2**: She states that she can rewrite it again using the commutative property as \((2 + 5) + (2 + x)\). This is also incorrect because the commutative property changes the order of addition but does not change how the expression is structured. The expression \((2 + x) + (2 + 5)\) does not correspond to the original expression correctly.
3. **Step 3**: Olivia mentions using the associative property to write the expression as \((2 + 5 + 2) + x\). She's attempting to group the numbers differently, but since the previous steps were incorrect, this step is also flawed.
4. **Step 4**: Finally, Olivia simplifies to \(9 + x\). While \(2 + 5\) is indeed \(7\), her prior steps were incorrect, and she should have been handling the expression \(2 + x + 5\) directly instead of using incorrect applications of properties.
To correctly simplify the original expression \(2 + (x + 5)\), you would simply combine like terms:
1. Start with \(2 + (x + 5)\).
2. This becomes \(2 + x + 5\).
3. Then combine the numbers: \(2 + 5 = 7\).
4. The final simplified expression is \(7 + x\).
So the correct simplified expression is \(7 + x\) instead of \(9 + x\). Olivia's misunderstanding lies in the improper use of algebraic properties.