Simplify The Expression: \[$(4x - 3) + (2x - 4)\$\]A. \[$6x + 7\$\] B. \[$6x - 7\$\] C. \[$6x + 1\$\] D. \[$6x - 1\$\]
Introduction
Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying the expression {(4x - 3) + (2x - 4)$}$, which is a common problem in algebra. We will break down the solution step by step, using clear and concise language to ensure that readers understand the process.
Understanding the Expression
The given expression is {(4x - 3) + (2x - 4)$}$. To simplify this expression, we need to apply the rules of algebra, which include the distributive property, the commutative property, and the associative property.
Step 1: Apply the Distributive Property
The distributive property states that for any numbers a, b, and c, the following equation holds:
a(b + c) = ab + ac
We can apply this property to the given expression by distributing the negative sign to the terms inside the parentheses:
[$(4x - 3) + (2x - 4)$ = 4x - 3 + 2x - 4
Step 2: Combine Like Terms
Like terms are terms that have the same variable raised to the same power. In this case, we have two like terms: 4x and 2x. We can combine these terms by adding their coefficients:
4x + 2x = 6x
So, the expression becomes:
[$6x - 3 - 4$
Step 3: Simplify the Constant Terms
The constant terms are the terms that do not have a variable. In this case, we have two constant terms: -3 and -4. We can combine these terms by adding them:
-3 + (-4) = -7
So, the expression becomes:
[$6x - 7$
Conclusion
In conclusion, the simplified expression is [$6x - 7$. This is the correct answer among the options provided.
Why is Simplifying Algebraic Expressions Important?
Simplifying algebraic expressions is an essential skill in mathematics because it helps us to:
- Understand the underlying structure of the expression: By simplifying an expression, we can identify the underlying structure and relationships between the variables and constants.
- Make calculations easier: Simplifying an expression can make it easier to perform calculations and solve problems.
- Identify patterns and relationships: Simplifying an expression can help us to identify patterns and relationships between the variables and constants.
Tips for Simplifying Algebraic Expressions
Here are some tips for simplifying algebraic expressions:
- Use the distributive property: The distributive property is a powerful tool for simplifying expressions. Use it to distribute the negative sign to the terms inside the parentheses.
- Combine like terms: Like terms are terms that have the same variable raised to the same power. Combine these terms by adding their coefficients.
- Simplify constant terms: Constant terms are terms that do not have a variable. Combine these terms by adding them.
- Use the commutative and associative properties: The commutative and associative properties can help us to rearrange the terms in an expression and make it easier to simplify.
Common Mistakes to Avoid
Here are some common mistakes to avoid when simplifying algebraic expressions:
- Not using the distributive property: Failing to use the distributive property can lead to incorrect simplifications.
- Not combining like terms: Failing to combine like terms can lead to incorrect simplifications.
- Not simplifying constant terms: Failing to simplify constant terms can lead to incorrect simplifications.
- Not using the commutative and associative properties: Failing to use the commutative and associative properties can lead to incorrect simplifications.
Conclusion
Introduction
In our previous article, we discussed the importance of simplifying algebraic expressions and provided a step-by-step guide on how to simplify the expression [$(4x - 3) + (2x - 4)$. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.
Q: What is the distributive property, and how is it used in simplifying algebraic expressions?
A: The distributive property is a fundamental concept in algebra that states that for any numbers a, b, and c, the following equation holds:
a(b + c) = ab + ac
The distributive property is used to simplify expressions by distributing the negative sign to the terms inside the parentheses. For example, in the expression [$(4x - 3) + (2x - 4)$, we can apply the distributive property to get:
4x - 3 + 2x - 4
Q: What are like terms, and how are they combined in simplifying algebraic expressions?
A: Like terms are terms that have the same variable raised to the same power. In the expression [$(4x - 3) + (2x - 4)$, we have two like terms: 4x and 2x. We can combine these terms by adding their coefficients:
4x + 2x = 6x
Q: How do I simplify constant terms in an algebraic expression?
A: Constant terms are terms that do not have a variable. In the expression [$(4x - 3) + (2x - 4)$, we have two constant terms: -3 and -4. We can combine these terms by adding them:
-3 + (-4) = -7
Q: What is the commutative property, and how is it used in simplifying algebraic expressions?
A: The commutative property states that the order of the terms in an expression does not change the value of the expression. For example, in the expression [$(4x - 3) + (2x - 4)$, we can rearrange the terms to get:
2x - 4 + 4x - 3
The commutative property is used to rearrange the terms in an expression and make it easier to simplify.
Q: What is the associative property, and how is it used in simplifying algebraic expressions?
A: The associative property states that the order in which we perform operations in an expression does not change the value of the expression. For example, in the expression [$(4x - 3) + (2x - 4)$, we can group the terms as follows:
(4x - 3) + (2x - 4) = (4x + 2x) + (-3 - 4)
The associative property is used to group the terms in an expression and make it easier to simplify.
Q: How do I know when to use the distributive property, the commutative property, and the associative property in simplifying algebraic expressions?
A: The distributive property, the commutative property, and the associative property are all used to simplify algebraic expressions. The distributive property is used to distribute the negative sign to the terms inside the parentheses. The commutative property is used to rearrange the terms in an expression. The associative property is used to group the terms in an expression.
Q: What are some common mistakes to avoid when simplifying algebraic expressions?
A: Some common mistakes to avoid when simplifying algebraic expressions include:
- Not using the distributive property
- Not combining like terms
- Not simplifying constant terms
- Not using the commutative and associative properties
Conclusion
In conclusion, simplifying algebraic expressions is an essential skill in mathematics. By understanding the distributive property, combining like terms, simplifying constant terms, and using the commutative and associative properties, we can simplify expressions and solve problems more effectively. By following the tips and avoiding common mistakes, we can become proficient in simplifying algebraic expressions and tackle complex problems with confidence.