Simplify The Expression:$ \frac{5}{x^2 - X Y} - \frac{5}{y^2 - X Y} }$Choose The Correct Answer A. { \frac{5 {x Y}$}$B. { \frac{5(y-x)}{x Y (x-y)}$}$C. { \frac{5(x+y)}{x Y (x-y)}$}$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including fractions, variables, and algebraic manipulation. In this article, we will focus on simplifying a specific expression involving fractions and variables. We will break down the expression step by step, and by the end of this article, you will be able to simplify the expression with ease.

The Expression to Simplify

The expression we need to simplify is:

5x2−xy−5y2−xy\frac{5}{x^2 - xy} - \frac{5}{y^2 - xy}

Step 1: Factor the Denominators

To simplify the expression, we need to factor the denominators. The first denominator can be factored as follows:

x2−xy=x(x−y)x^2 - xy = x(x - y)

Similarly, the second denominator can be factored as follows:

y2−xy=y(y−x)y^2 - xy = y(y - x)

Step 2: Rewrite the Expression with Factored Denominators

Now that we have factored the denominators, we can rewrite the expression as follows:

5x(x−y)−5y(y−x)\frac{5}{x(x - y)} - \frac{5}{y(y - x)}

Step 3: Find a Common Denominator

To simplify the expression further, we need to find a common denominator. The common denominator of the two fractions is x(x−y)y(y−x)x(x - y)y(y - x).

Step 4: Rewrite the Expression with a Common Denominator

Now that we have found a common denominator, we can rewrite the expression as follows:

5y(y−x)x(x−y)y(y−x)−5x(x−y)x(x−y)y(y−x)\frac{5y(y - x)}{x(x - y)y(y - x)} - \frac{5x(x - y)}{x(x - y)y(y - x)}

Step 5: Simplify the Expression

Now that we have a common denominator, we can simplify the expression by combining the two fractions:

5y(y−x)−5x(x−y)x(x−y)y(y−x)\frac{5y(y - x) - 5x(x - y)}{x(x - y)y(y - x)}

Step 6: Expand and Simplify the Numerator

To simplify the expression further, we need to expand and simplify the numerator:

5y(y−x)−5x(x−y)=5y2−5xy−5x2+5xy5y(y - x) - 5x(x - y) = 5y^2 - 5xy - 5x^2 + 5xy

Step 7: Simplify the Numerator

Now that we have expanded and simplified the numerator, we can simplify the expression further:

5y2−5xy−5x2+5xy=5y2−5x25y^2 - 5xy - 5x^2 + 5xy = 5y^2 - 5x^2

Step 8: Simplify the Expression

Now that we have simplified the numerator, we can simplify the expression further:

5y2−5x2x(x−y)y(y−x)\frac{5y^2 - 5x^2}{x(x - y)y(y - x)}

Step 9: Factor the Numerator

To simplify the expression further, we need to factor the numerator:

5y2−5x2=5(y2−x2)5y^2 - 5x^2 = 5(y^2 - x^2)

Step 10: Simplify the Expression

Now that we have factored the numerator, we can simplify the expression further:

5(y2−x2)x(x−y)y(y−x)\frac{5(y^2 - x^2)}{x(x - y)y(y - x)}

Step 11: Simplify the Expression

Now that we have simplified the expression, we can simplify it further by factoring the difference of squares:

y2−x2=(y+x)(y−x)y^2 - x^2 = (y + x)(y - x)

Step 12: Simplify the Expression

Now that we have factored the difference of squares, we can simplify the expression further:

5(y+x)(y−x)x(x−y)y(y−x)\frac{5(y + x)(y - x)}{x(x - y)y(y - x)}

Step 13: Cancel Out Common Factors

To simplify the expression further, we need to cancel out common factors:

5(y+x)(y−x)x(x−y)y(y−x)=5(y+x)xy(x−y)\frac{5(y + x)(y - x)}{x(x - y)y(y - x)} = \frac{5(y + x)}{xy(x - y)}

Step 14: Simplify the Expression

Now that we have canceled out common factors, we can simplify the expression further:

5(y+x)xy(x−y)=5(y+x)xy(x−y)\frac{5(y + x)}{xy(x - y)} = \frac{5(y + x)}{xy(x - y)}

Conclusion

In this article, we have simplified the expression step by step. We have factored the denominators, found a common denominator, simplified the expression, factored the numerator, simplified the expression, and canceled out common factors. By following these steps, we have simplified the expression to:

5(y+x)xy(x−y)\frac{5(y + x)}{xy(x - y)}

This is the correct answer.

Final Answer

The final answer is:

\boxed{\frac{5(y + x)}{xy(x - y)}}$<br/> # Simplify the Expression: A Q&A Guide =====================================================

Introduction

In our previous article, we simplified the expression 5x2−xy−5y2−xy\frac{5}{x^2 - xy} - \frac{5}{y^2 - xy} step by step. In this article, we will answer some frequently asked questions related to simplifying the expression.

Q&A

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

A: The first step in simplifying the expression is to factor the denominators. We can factor the first denominator as x2−xy=x(x−y)x^2 - xy = x(x - y) and the second denominator as y2−xy=y(y−x)y^2 - xy = y(y - x).

Q: Why do we need to find a common denominator?

A: We need to find a common denominator to simplify the expression further. The common denominator of the two fractions is x(x−y)y(y−x)x(x - y)y(y - x).

Q: How do we simplify the numerator?

A: To simplify the numerator, we need to expand and simplify the expression 5y(y−x)−5x(x−y)5y(y - x) - 5x(x - y). This can be done by combining like terms and factoring out common factors.

Q: What is the final simplified expression?

A: The final simplified expression is 5(y+x)xy(x−y)\frac{5(y + x)}{xy(x - y)}.

Q: Why is it important to simplify expressions?

A: Simplifying expressions is important because it helps us to:

  • Reduce the complexity of the expression
  • Make it easier to work with
  • Identify patterns and relationships between variables
  • Solve problems more efficiently

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

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

  • Not factoring the denominators
  • Not finding a common denominator
  • Not expanding and simplifying the numerator
  • Not canceling out common factors

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by:

  • Working through examples and exercises
  • Using online resources and tools
  • Asking for help from a teacher or tutor
  • Practicing regularly to build your skills and confidence

Conclusion

Simplifying expressions is an important skill in mathematics, and it requires practice and patience to master. By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying expressions and solve problems more efficiently.

Final Tips

  • Always factor the denominators and find a common denominator
  • Expand and simplify the numerator carefully
  • Cancel out common factors to simplify the expression
  • Practice regularly to build your skills and confidence

By following these tips and practicing regularly, you can become a master of simplifying expressions and solve problems with ease.