Simplifying the Final Expression

Oct 13, 2025 by recoveryshake.com 9 views

Introduction

In mathematics, simplifying complex fractions is an essential skill that can be applied to various problems in algebra, calculus, and other branches of mathematics. A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In this article, we will focus on simplifying a specific complex fraction, which is given as:

$\frac{x-y}{xy} - \frac{9-y}{9y}$

Understanding the Problem

The given expression is a complex fraction, which means it contains one or more fractions within it. To simplify this expression, we need to apply the rules of algebra and fraction manipulation. The goal is to simplify the expression to its simplest form, which can be done by combining the fractions and canceling out any common factors.

Step 1: Combine the Fractions

To simplify the given expression, we need to combine the two fractions. We can do this by finding a common denominator, which is the product of the denominators of the two fractions. In this case, the common denominator is $xy \times 9y = 9xy^2$ .

\frac{x-y}{xy} - \frac{9-y}{9y} = \frac{(x-y)(9y) - (9-y)(xy)}{9xy^2}

Step 2: Expand the Numerator

Now that we have combined the fractions, we need to expand the numerator. This involves multiplying out the terms in the numerator.

(x-y)(9y) - (9-y)(xy) = 9xy - 9y^2 - 9xy + xy^2

Step 3: Simplify the Numerator

We can simplify the numerator by combining like terms. In this case, we have two terms that are the same, $-9xy$ and $9xy$ , which cancel each other out.

9xy - 9y^2 - 9xy + xy^2 = -9y^2 + xy^2

Step 4: Write the Final Expression

Now that we have simplified the numerator, we can write the final expression. We can do this by dividing the numerator by the denominator.

\frac{-9y^2 + xy^2}{9xy^2} = \frac{xy^2 - 9y^2}{9xy^2}

Simplifying the Final Expression

We can simplify the final expression by factoring out a common factor from the numerator and denominator. In this case, we can factor out $y^2$ from the numerator and denominator.

\frac{xy^2 - 9y^2}{9xy^2} = \frac{y^2(x - 9)}{9xy^2}

Canceling Out Common Factors

We can simplify the expression further by canceling out any common factors between the numerator and denominator. In this case, we can cancel out $y^2$ from the numerator and denominator.

\frac{y^2(x - 9)}{9xy^2} = \frac{x - 9}{9x}

Conclusion

In this article, we have simplified a complex fraction using the rules of algebra and fraction manipulation. We started by combining the fractions, expanding the numerator, simplifying the numerator, and writing the final expression. We then simplified the final expression by factoring out a common factor and canceling out any common factors between the numerator and denominator. The final simplified expression is $\frac{x - 9}{9x}$ .

Final Answer

$The final answer is $\boxed{\frac{x - 9}{9x}}$.<br/> **Simplifying Complex Fractions: A Q&A Guide**$

Introduction

In our previous article, we simplified a complex fraction using the rules of algebra and fraction manipulation. In this article, we will provide a Q&A guide to help you understand the process of simplifying complex fractions.

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q: Why is it important to simplify complex fractions?

A: Simplifying complex fractions is important because it helps to make the expression easier to work with and understand. It also helps to avoid errors and make calculations more efficient.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to follow these steps:

Combine the fractions by finding a common denominator.
Expand the numerator by multiplying out the terms.
Simplify the numerator by combining like terms.
Write the final expression by dividing the numerator by the denominator.
Simplify the final expression by factoring out a common factor and canceling out any common factors between the numerator and denominator.

Q: What is the common denominator?

A: The common denominator is the product of the denominators of the two fractions.

Q: How do I find the common denominator?

A: To find the common denominator, you need to multiply the denominators of the two fractions together.

Q: What is the difference between a common factor and a common denominator?

A: A common factor is a factor that is common to both the numerator and denominator, while a common denominator is the product of the denominators of the two fractions.

Q: Can I simplify a complex fraction by canceling out common factors?

A: Yes, you can simplify a complex fraction by canceling out common factors between the numerator and denominator.

Q: What is the final simplified expression?

A: The final simplified expression is the result of simplifying the complex fraction by combining the fractions, expanding the numerator, simplifying the numerator, and canceling out any common factors between the numerator and denominator.

Q: How do I know if I have simplified the complex fraction correctly?

A: To check if you have simplified the complex fraction correctly, you need to make sure that the final simplified expression is in its simplest form.

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

A: Some common mistakes to avoid when simplifying complex fractions include:

Not finding the common denominator
Not expanding the numerator
Not simplifying the numerator
Not canceling out common factors between the numerator and denominator
Not checking if the final simplified expression is in its simplest form

Conclusion

In this article, we have provided a Q&A guide to help you understand the process of simplifying complex fractions. We have covered topics such as what a complex fraction is, why it is important to simplify complex fractions, and how to simplify a complex fraction. We have also provided some common mistakes to avoid when simplifying complex fractions.

Final Answer

The final answer is $\boxed{\frac{x - 9}{9x}}$ .