Which Property Was Applied To Create This Equivalent Expression?$\[ 7x + (5 + 6x) \longrightarrow (7x + 5) + 6x \\]A. The Commutative Property Only B. The Associative Property Only C. The Distributive Property Only D. Both The Commutative
In algebra, there are three main properties that help us simplify and manipulate expressions: the commutative property, the associative property, and the distributive property. These properties are essential in creating equivalent expressions, which is a crucial concept in mathematics. In this article, we will explore which property was applied to create the equivalent expression shown below.
The Given Expression
The given expression is:
To understand which property was applied, let's break down the expression and analyze the steps involved.
Step 1: Understanding the Distributive Property
The distributive property states that for any real numbers a, b, and c:
This property allows us to distribute a single term to multiple terms inside the parentheses.
Step 2: Applying the Distributive Property
In the given expression, we can see that the distributive property is applied to the term 7x. The expression (5 + 6x) is inside the parentheses, and 7x is distributed to both terms inside the parentheses.
Using the distributive property, we can rewrite the expression as:
Step 3: Understanding the Commutative Property
The commutative property states that for any real numbers a and b:
This property allows us to swap the order of the terms in an expression.
Step 4: Applying the Commutative Property
In the rewritten expression, we can see that the commutative property is not applied. The order of the terms is not swapped, and the expression remains the same.
Step 5: Understanding the Associative Property
The associative property states that for any real numbers a, b, and c:
This property allows us to regroup the terms in an expression.
Step 6: Applying the Associative Property
In the rewritten expression, we can see that the associative property is not applied. The terms are not regrouped, and the expression remains the same.
Conclusion
Based on the analysis above, we can conclude that the distributive property was applied to create the equivalent expression. The distributive property allowed us to distribute the term 7x to both terms inside the parentheses, resulting in the rewritten expression.
Answer
The correct answer is:
C. The distributive property only
Additional Tips and Examples
Here are some additional tips and examples to help you understand the properties of algebraic expressions:
- Commutative Property: The commutative property is applied when we swap the order of the terms in an expression. For example:
- Associative Property: The associative property is applied when we regroup the terms in an expression. For example:
- Distributive Property: The distributive property is applied when we distribute a single term to multiple terms inside the parentheses. For example:
By understanding and applying these properties, you can simplify and manipulate algebraic expressions with ease.
Final Thoughts
In this article, we will answer some frequently asked questions related to the properties of algebraic expressions.
Q: What is the commutative property?
A: The commutative property is a property of algebraic expressions that states that the order of the terms in an expression does not change the value of the expression. For example:
Q: What is the associative property?
A: The associative property is a property of algebraic expressions that states that the order in which we add or multiply terms does not change the value of the expression. For example:
Q: What is the distributive property?
A: The distributive property is a property of algebraic expressions that states that a single term can be distributed to multiple terms inside the parentheses. For example:
Q: How do I apply the distributive property?
A: To apply the distributive property, you need to multiply the single term by each term inside the parentheses. For example:
Q: Can I apply the distributive property to more than two terms?
A: Yes, you can apply the distributive property to more than two terms. For example:
Q: What is the difference between the commutative and associative properties?
A: The commutative property states that the order of the terms in an expression does not change the value of the expression, while the associative property states that the order in which we add or multiply terms does not change the value of the expression.
Q: Can I apply the commutative and associative properties together?
A: Yes, you can apply the commutative and associative properties together. For example:
Q: How do I know which property to apply?
A: To determine which property to apply, you need to look at the expression and identify the terms that need to be simplified or manipulated. Then, you can apply the appropriate property to simplify or manipulate the expression.
Q: Can I apply the properties of algebraic expressions to more complex expressions?
A: Yes, you can apply the properties of algebraic expressions to more complex expressions. For example:
By understanding and applying the properties of algebraic expressions, you can simplify and manipulate expressions with ease.
Conclusion
In conclusion, the properties of algebraic expressions are essential in simplifying and manipulating expressions. By understanding and applying the commutative, associative, and distributive properties, you can simplify and manipulate expressions with ease. Remember to apply the distributive property when distributing a single term to multiple terms inside the parentheses, and to apply the commutative and associative properties when swapping or regrouping terms. With practice and patience, you will become proficient in applying these properties and simplifying algebraic expressions.