Answer :
To determine which property was applied in the transformation of the expression from \( 7x + (5 + 6x) \) to \((7x + 5) + 6x\), let's analyze the steps involved carefully.
### Original Expression
\( 7x + (5 + 6x) \)
### Transformed Expression
\( (7x + 5) + 6x \)
### Understanding the Properties Involved
1. Commutative Property:
The commutative property refers to the ability to change the order of the operands. For addition, \(a + b = b + a\).
2. Associative Property:
The associative property refers to the ability to change the grouping of the operands. For addition, \((a + b) + c = a + (b + c)\).
3. Distributive Property:
The distributive property involves distributing a common factor over addition or subtraction. For example, \(a(b + c) = ab + ac\).
### Identify the Property Used
In the transformation:
[tex]\[ 7x + (5 + 6x) \longrightarrow (7x + 5) + 6x \][/tex]
- We start with \(7x + (5 + 6x)\).
- The expression is regrouped to \((7x + 5) + 6x\).
### Analyzing the Steps
- We observe that the terms inside the parentheses have been rearranged in terms of their grouping but not in their sequence.
- From \( 7x + (5 + 6x) \) to \((7x + 5) + 6x\), the terms involved (7x, 5, and 6x) remain in their original order.
- The transformation involves changing the way the terms are grouped, specifically moving the parentheses from grouping 5 and \(6x\) together, to grouping \(7x\) and 5 together.
Given this transformation, the property used here is the associative property because it is changing the grouping of the addition without changing the actual order of the terms.
### Conclusion
Therefore, the property applied to create this equivalent expression is the associative property only.
The correct answer is:
[tex]\[ \boxed{2} \][/tex]
As [tex]\(2\)[/tex] corresponds to "the associative property only".
### Original Expression
\( 7x + (5 + 6x) \)
### Transformed Expression
\( (7x + 5) + 6x \)
### Understanding the Properties Involved
1. Commutative Property:
The commutative property refers to the ability to change the order of the operands. For addition, \(a + b = b + a\).
2. Associative Property:
The associative property refers to the ability to change the grouping of the operands. For addition, \((a + b) + c = a + (b + c)\).
3. Distributive Property:
The distributive property involves distributing a common factor over addition or subtraction. For example, \(a(b + c) = ab + ac\).
### Identify the Property Used
In the transformation:
[tex]\[ 7x + (5 + 6x) \longrightarrow (7x + 5) + 6x \][/tex]
- We start with \(7x + (5 + 6x)\).
- The expression is regrouped to \((7x + 5) + 6x\).
### Analyzing the Steps
- We observe that the terms inside the parentheses have been rearranged in terms of their grouping but not in their sequence.
- From \( 7x + (5 + 6x) \) to \((7x + 5) + 6x\), the terms involved (7x, 5, and 6x) remain in their original order.
- The transformation involves changing the way the terms are grouped, specifically moving the parentheses from grouping 5 and \(6x\) together, to grouping \(7x\) and 5 together.
Given this transformation, the property used here is the associative property because it is changing the grouping of the addition without changing the actual order of the terms.
### Conclusion
Therefore, the property applied to create this equivalent expression is the associative property only.
The correct answer is:
[tex]\[ \boxed{2} \][/tex]
As [tex]\(2\)[/tex] corresponds to "the associative property only".
