Perform The Operation And Combine To One Fraction: 3 X − 1 X − 7 \frac{3}{x} - \frac{1}{x-7} X 3 − X − 7 1

Mar 11, 2025 by ADMIN 111 views

**Combining Fractions: A Step-by-Step Guide to Simplifying Complex Expressions**

Introduction

In mathematics, combining fractions is a crucial operation that involves adding or subtracting fractions with different denominators. This process can be challenging, especially when dealing with complex expressions. In this article, we will focus on combining two fractions with different denominators, specifically the expression $\frac{3}{x} - \frac{1}{x-7}$ . We will break down the steps involved in simplifying this expression and provide a clear understanding of the process.

Understanding the Problem

The given expression is $\frac{3}{x} - \frac{1}{x-7}$ . To combine these fractions, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the two denominators, which in this case is $x(x-7)$ .

Step 1: Find the Common Denominator

To find the common denominator, we need to multiply the two denominators together. In this case, we multiply $x$ and $x-7$ to get $x(x-7)$ .

x(x-7) = x^2 - 7x

Step 2: Rewrite the Fractions with the Common Denominator

Now that we have the common denominator, we can rewrite the fractions with the common denominator.

\frac{3}{x} = \frac{3(x-7)}{x(x-7)}
\frac{1}{x-7} = \frac{1(x)}{x(x-7)}

Step 3: Subtract the Fractions

Now that we have the fractions with the common denominator, we can subtract them.

\frac{3(x-7)}{x(x-7)} - \frac{1(x)}{x(x-7)} = \frac{3(x-7) - 1(x)}{x(x-7)}

Step 4: Simplify the Expression

Now that we have the expression with the common denominator, we can simplify it by combining like terms.

\frac{3(x-7) - 1(x)}{x(x-7)} = \frac{3x - 21 - x}{x(x-7)}
\frac{3x - 21 - x}{x(x-7)} = \frac{2x - 21}{x(x-7)}

Conclusion

Combining fractions is a crucial operation in mathematics that involves adding or subtracting fractions with different denominators. In this article, we focused on combining two fractions with different denominators, specifically the expression $\frac{3}{x} - \frac{1}{x-7}$ . We broke down the steps involved in simplifying this expression and provided a clear understanding of the process. By following these steps, you can simplify complex expressions and combine fractions with ease.

Common Mistakes to Avoid

When combining fractions, there are several common mistakes to avoid. Here are a few:

Not finding the common denominator: Failing to find the common denominator can lead to incorrect results.
Not rewriting the fractions with the common denominator: Failing to rewrite the fractions with the common denominator can lead to incorrect results.
Not simplifying the expression: Failing to simplify the expression can lead to incorrect results.

Tips and Tricks

Here are a few tips and tricks to help you combine fractions:

Use the least common multiple (LCM): The LCM is the smallest multiple that both denominators have in common.
Rewrite the fractions with the common denominator: This will help you to combine the fractions correctly.
Simplify the expression: This will help you to get the final answer.

Real-World Applications

Combining fractions has several real-world applications. Here are a few:

Finance: Combining fractions is used in finance to calculate interest rates and investment returns.
Science: Combining fractions is used in science to calculate chemical reactions and physical properties.
Engineering: Combining fractions is used in engineering to calculate stress and strain on materials.

Conclusion

Introduction

Combining fractions is a crucial operation in mathematics that involves adding or subtracting fractions with different denominators. In our previous article, we focused on combining two fractions with different denominators, specifically the expression $\frac{3}{x} - \frac{1}{x-7}$ . In this article, we will provide a Q&A guide to help you understand the process of combining fractions.

Q: What is the first step in combining fractions?

A: The first step in combining fractions is to find the common denominator. The common denominator is the least common multiple (LCM) of the two denominators.

Q: How do I find the common denominator?

A: To find the common denominator, you need to multiply the two denominators together. In the case of the expression $\frac{3}{x} - \frac{1}{x-7}$ , the common denominator is $x(x-7)$ .

Q: What is the next step in combining fractions?

A: The next step in combining fractions is to rewrite the fractions with the common denominator. This will help you to combine the fractions correctly.

Q: How do I rewrite the fractions with the common denominator?

A: To rewrite the fractions with the common denominator, you need to multiply the numerator and denominator of each fraction by the necessary factor. In the case of the expression $\frac{3}{x} - \frac{1}{x-7}$ , you would multiply the numerator and denominator of the first fraction by $(x-7)$ and the numerator and denominator of the second fraction by $x$ .

Q: What is the final step in combining fractions?

A: The final step in combining fractions is to simplify the expression. This will help you to get the final answer.

Q: How do I simplify the expression?

A: To simplify the expression, you need to combine like terms and cancel out any common factors. In the case of the expression $\frac{3}{x} - \frac{1}{x-7}$ , you would simplify the expression to $\frac{2x - 21}{x(x-7)}$ .

Q: What are some common mistakes to avoid when combining fractions?

A: Some common mistakes to avoid when combining fractions include:

Not finding the common denominator
Not rewriting the fractions with the common denominator
Not simplifying the expression

Q: What are some real-world applications of combining fractions?

A: Combining fractions has several real-world applications, including:

Finance: Combining fractions is used in finance to calculate interest rates and investment returns.
Science: Combining fractions is used in science to calculate chemical reactions and physical properties.
Engineering: Combining fractions is used in engineering to calculate stress and strain on materials.

Q: How can I practice combining fractions?

A: You can practice combining fractions by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.

Conclusion

Combining fractions is a crucial operation in mathematics that involves adding or subtracting fractions with different denominators. In this article, we provided a Q&A guide to help you understand the process of combining fractions. By following these steps and avoiding common mistakes, you can simplify complex expressions and combine fractions with ease.

Additional Resources

If you are looking for additional resources to help you practice combining fractions, here are a few suggestions:

Online practice tests: You can find online practice tests and quizzes to help you improve your skills.
Math textbooks: You can find math textbooks that include exercises and examples on combining fractions.
Math websites: You can find math websites that provide tutorials and examples on combining fractions.

Final Tips

Here are a few final tips to help you combine fractions:

Use the least common multiple (LCM): The LCM is the smallest multiple that both denominators have in common.
Rewrite the fractions with the common denominator: This will help you to combine the fractions correctly.
Simplify the expression: This will help you to get the final answer.

By following these tips and practicing combining fractions, you can improve your skills and become more confident in your ability to simplify complex expressions.