Simplify The Expression:${ \frac{1}{5}(15 + 10x - 5) }$

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student should possess. In this article, we will focus on simplifying the given expression: $\frac{1}{5}(15 + 10x - 5)$. We will break down the problem into manageable steps, and by the end of this article, you will have a clear understanding of how to simplify complex algebraic expressions.

Understanding the Expression

Before we dive into simplifying the expression, let's take a closer look at what we're dealing with. The given expression is $\frac{1}{5}(15 + 10x - 5)$. This expression consists of a fraction, where the numerator is a polynomial expression, and the denominator is a constant. The polynomial expression inside the parentheses can be simplified by combining like terms.

Simplifying the Polynomial Expression

To simplify the polynomial expression inside the parentheses, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with the variable $x$: $10x$ and $-5$. However, we also have a constant term $15$ that is not associated with the variable $x$. To combine like terms, we need to add or subtract the coefficients of the like terms.

Combining Like Terms

Let's combine the like terms inside the parentheses:

$15 + 10x - 5$

We can start by combining the constant terms:

$15 - 5 = 10$

Now, we are left with the term $10x$. Since there are no other like terms with the variable $x$, we can leave it as is.

Simplifying the Expression

Now that we have simplified the polynomial expression inside the parentheses, we can rewrite the original expression with the simplified polynomial:

$\frac{1}{5}(10 + 10x)$

Distributing the Fraction

To simplify the expression further, we need to distribute the fraction to the terms inside the parentheses. This means that we need to multiply the fraction by each term inside the parentheses.

Distributing the Fraction to the Constant Term

Let's start by distributing the fraction to the constant term $10$:

$\frac{1}{5} \cdot 10 = 2$

Distributing the Fraction to the Variable Term

Now, let's distribute the fraction to the variable term $10x$:

$\frac{1}{5} \cdot 10x = 2x$

Simplifying the Expression

Now that we have distributed the fraction to each term inside the parentheses, we can rewrite the expression with the simplified terms:

$2 + 2x$

Conclusion

In this article, we simplified the given expression $\frac{1}{5}(15 + 10x - 5)$ by combining like terms, distributing the fraction, and simplifying the resulting expression. By following these steps, you can simplify complex algebraic expressions and become more confident in your ability to manipulate mathematical expressions.

Frequently Asked Questions

• Q: What is the simplified expression? A: The simplified expression is $2 + 2x$.
• Q: How do I simplify a complex algebraic expression? A: To simplify a complex algebraic expression, you need to combine like terms, distribute the fraction, and simplify the resulting expression.
• Q: What is the importance of simplifying algebraic expressions? A: Simplifying algebraic expressions is an essential skill that every student should possess. It helps you to understand the underlying structure of the expression and to solve mathematical problems more efficiently.

Final Thoughts

Simplifying algebraic expressions is a crucial aspect of mathematics, and it requires practice and patience to become proficient. By following the steps outlined in this article, you can simplify complex algebraic expressions and become more confident in your ability to manipulate mathematical expressions. Remember to always combine like terms, distribute the fraction, and simplify the resulting expression to get the final answer.

Introduction

In our previous article, we simplified the expression $\frac{1}{5}(15 + 10x - 5)$ by combining like terms, distributing the fraction, and simplifying the resulting expression. In this article, we will provide a Q&A guide to algebraic manipulation, covering common questions and topics related to simplifying algebraic expressions.

Q&A Guide

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to combine like terms. Like terms are terms that have the same variable raised to the same power.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have the expression $2x + 3x$, you can combine the like terms by adding the coefficients: $2x + 3x = 5x$.

Q: What is the difference between a coefficient and a variable?

A: A coefficient is a number that is multiplied by a variable, while a variable is a letter or symbol that represents a value. For example, in the expression $2x$, the coefficient is 2 and the variable is x.

Q: How do I distribute a fraction to a polynomial expression?

A: To distribute a fraction to a polynomial expression, you need to multiply the fraction by each term inside the parentheses. For example, if you have the expression $\frac{1}{2}(x + 3)$, you can distribute the fraction by multiplying it by each term inside the parentheses: $\frac{1}{2}x + \frac{3}{2}$.

Q: What is the importance of simplifying algebraic expressions?

A: Simplifying algebraic expressions is an essential skill that every student should possess. It helps you to understand the underlying structure of the expression and to solve mathematical problems more efficiently.

Q: How do I know if an expression is simplified?

A: An expression is simplified when there are no like terms that can be combined, and the fraction has been distributed to each term inside the parentheses.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

• Not combining like terms
• Not distributing the fraction to each term inside the parentheses
• Not simplifying the resulting expression

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through examples and exercises in your textbook or online resources. You can also try simplifying expressions on your own and then checking your work with a calculator or online tool.

Conclusion

Simplifying algebraic expressions is a crucial aspect of mathematics, and it requires practice and patience to become proficient. By following the steps outlined in this article and practicing regularly, you can become more confident in your ability to manipulate mathematical expressions.

Frequently Asked Questions

• Q: What is the difference between a coefficient and a variable? A: A coefficient is a number that is multiplied by a variable, while a variable is a letter or symbol that represents a value.
• Q: How do I distribute a fraction to a polynomial expression? A: To distribute a fraction to a polynomial expression, you need to multiply the fraction by each term inside the parentheses.
• Q: What is the importance of simplifying algebraic expressions? A: Simplifying algebraic expressions is an essential skill that every student should possess. It helps you to understand the underlying structure of the expression and to solve mathematical problems more efficiently.

Final Thoughts

Simplifying algebraic expressions is a crucial aspect of mathematics, and it requires practice and patience to become proficient. By following the steps outlined in this article and practicing regularly, you can become more confident in your ability to manipulate mathematical expressions. Remember to always combine like terms, distribute the fraction, and simplify the resulting expression to get the final answer.