Simplify $\frac{x^2-y^2}{2y-y} \times \frac{2x^2}{xy-x^2}$ Under The Addition And Subtraction Of Algebraic Fractions.

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Introduction

Algebraic fractions are a fundamental concept in mathematics, and simplifying them is a crucial skill for any student or professional. In this article, we will focus on simplifying the expression x2−y22y−y×2x2xy−x2\frac{x^2-y^2}{2y-y} \times \frac{2x^2}{xy-x^2} using the rules of addition and subtraction of algebraic fractions. We will break down the problem step by step, and provide a clear and concise explanation of each step.

Understanding Algebraic Fractions

Before we dive into the problem, let's quickly review what algebraic fractions are. An algebraic fraction is a fraction that contains variables, such as x and y, in the numerator or denominator. Algebraic fractions can be added, subtracted, multiplied, and divided, just like regular fractions.

Properties of Algebraic Fractions

There are several properties of algebraic fractions that we need to be aware of:

  • Like terms: Like terms are terms that have the same variable and exponent. For example, 2x and 3x are like terms.
  • Unlike terms: Unlike terms are terms that have different variables or exponents. For example, 2x and 3y are unlike terms.
  • Common factors: Common factors are factors that are present in both the numerator and denominator of a fraction. For example, in the fraction 6x3x\frac{6x}{3x}, the common factor is 3x.

Simplifying the Expression

Now that we have a good understanding of algebraic fractions, let's focus on simplifying the expression x2−y22y−y×2x2xy−x2\frac{x^2-y^2}{2y-y} \times \frac{2x^2}{xy-x^2}.

Step 1: Factor the Numerator and Denominator

The first step in simplifying the expression is to factor the numerator and denominator of each fraction.

  • For the first fraction, x2−y22y−y\frac{x^2-y^2}{2y-y}, we can factor the numerator as (x+y)(x−y)(x+y)(x-y) and the denominator as 2y2y.
  • For the second fraction, 2x2xy−x2\frac{2x^2}{xy-x^2}, we can factor the numerator as 2x22x^2 and the denominator as x(y−x)x(y-x).

Step 2: Cancel Out Common Factors

Now that we have factored the numerator and denominator of each fraction, we can cancel out any common factors.

  • In the first fraction, (x+y)(x−y)2y\frac{(x+y)(x-y)}{2y}, we can cancel out the common factor of yy in the numerator and denominator.
  • In the second fraction, 2x2x(y−x)\frac{2x^2}{x(y-x)}, we can cancel out the common factor of xx in the numerator and denominator.

Step 3: Multiply the Fractions

Now that we have simplified each fraction, we can multiply them together.

  • The first fraction simplifies to x+y2\frac{x+y}{2}.
  • The second fraction simplifies to 2xy−x\frac{2x}{y-x}.

Step 4: Multiply the Numerators and Denominators

Now that we have the simplified fractions, we can multiply the numerators and denominators together.

  • The numerator of the product is (x+y)(2x)(x+y)(2x).
  • The denominator of the product is 2(y−x)2(y-x).

Step 5: Simplify the Product

Finally, we can simplify the product by combining like terms in the numerator and denominator.

  • The numerator simplifies to 2x2+2xy2x^2+2xy.
  • The denominator simplifies to 2y−2x2y-2x.

Conclusion

In this article, we simplified the expression x2−y22y−y×2x2xy−x2\frac{x^2-y^2}{2y-y} \times \frac{2x^2}{xy-x^2} using the rules of addition and subtraction of algebraic fractions. We broke down the problem step by step, and provided a clear and concise explanation of each step. By following these steps, we were able to simplify the expression and arrive at the final answer.

Final Answer

The final answer is x2+xyy−x\boxed{\frac{x^2+xy}{y-x}}.

Example Use Case

This problem can be used as an example in a mathematics textbook or online course to teach students how to simplify algebraic fractions. It can also be used as a practice problem for students who are struggling with simplifying algebraic fractions.

Future Work

In the future, we can explore other ways to simplify algebraic fractions, such as using the distributive property or combining like terms. We can also explore more complex algebraic fractions, such as those with multiple variables or exponents.

References

Acknowledgments

I would like to thank my editor and proofreader for their help in reviewing and editing this article. I would also like to thank my readers for their feedback and suggestions.

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Introduction

In our previous article, we simplified the expression x2−y22y−y×2x2xy−x2\frac{x^2-y^2}{2y-y} \times \frac{2x^2}{xy-x^2} using the rules of addition and subtraction of algebraic fractions. In this article, we will provide a Q&A section to help clarify any questions or doubts that readers may have.

Q&A

Q: What is the difference between an algebraic fraction and a regular fraction?

A: An algebraic fraction is a fraction that contains variables, such as x and y, in the numerator or denominator. A regular fraction, on the other hand, is a fraction that contains only numbers in the numerator and denominator.

Q: How do I simplify an algebraic fraction?

A: To simplify an algebraic fraction, you need to factor the numerator and denominator, cancel out any common factors, and then multiply the fractions together.

Q: What is the distributive property, and how do I use it to simplify algebraic fractions?

A: The distributive property is a rule that states that you can multiply a single term by multiple terms. To use the distributive property to simplify algebraic fractions, you need to multiply the numerator and denominator by the same term.

Q: How do I combine like terms in an algebraic fraction?

A: To combine like terms in an algebraic fraction, you need to add or subtract the coefficients of the like terms.

Q: What is the final answer to the expression x2−y22y−y×2x2xy−x2\frac{x^2-y^2}{2y-y} \times \frac{2x^2}{xy-x^2}?

A: The final answer to the expression x2−y22y−y×2x2xy−x2\frac{x^2-y^2}{2y-y} \times \frac{2x^2}{xy-x^2} is x2+xyy−x\boxed{\frac{x^2+xy}{y-x}}.

Q: Can I use this method to simplify other algebraic fractions?

A: Yes, you can use this method to simplify other algebraic fractions. However, you need to make sure that the numerator and denominator are factored correctly, and that you cancel out any common factors.

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

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

  • Not factoring the numerator and denominator correctly
  • Not canceling out common factors
  • Not multiplying the fractions together correctly
  • Not combining like terms correctly

Conclusion

In this article, we provided a Q&A section to help clarify any questions or doubts that readers may have about simplifying algebraic fractions. We covered topics such as the difference between algebraic fractions and regular fractions, how to simplify algebraic fractions, and common mistakes to avoid.

Final Answer

The final answer to the expression x2−y22y−y×2x2xy−x2\frac{x^2-y^2}{2y-y} \times \frac{2x^2}{xy-x^2} is x2+xyy−x\boxed{\frac{x^2+xy}{y-x}}.

Example Use Case

This Q&A section can be used as a resource for students who are struggling with simplifying algebraic fractions. It can also be used as a reference for teachers who are looking for ways to explain complex concepts to their students.

Future Work

In the future, we can explore other ways to simplify algebraic fractions, such as using the distributive property or combining like terms. We can also explore more complex algebraic fractions, such as those with multiple variables or exponents.

References

Acknowledgments

I would like to thank my editor and proofreader for their help in reviewing and editing this article. I would also like to thank my readers for their feedback and suggestions.