Write As A Fully Simplified Fraction:A. { \frac{x+y}{x} + \frac{x-y}{y}$}$B. { \frac{x^2+xy}{xy}$}$C. { \frac{x^2}{xy}$}$D. { \frac{x 2+y 2}{xy}$}$E. { \frac{x 2-y 2}{xy}$}$F. { \frac{2x}{xy}$}$

by ADMIN 195 views

Introduction

Algebraic fractions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying algebraic fractions, focusing on the given options A, B, C, D, E, and F. We will break down each option, providing step-by-step explanations and examples to help you understand the concept.

Option A: Simplifying the Sum of Two Fractions

The first option is to simplify the expression x+yx+x−yy\frac{x+y}{x} + \frac{x-y}{y}. To simplify this expression, we need to find a common denominator, which is the product of the denominators, xyxy.

Step 1: Find the Common Denominator

The common denominator is xyxy, so we need to multiply the first fraction by yy\frac{y}{y} and the second fraction by xx\frac{x}{x}.

Step 2: Simplify the Fractions

x+yx⋅yy=(x+y)yxy\frac{x+y}{x} \cdot \frac{y}{y} = \frac{(x+y)y}{xy}

x−yy⋅xx=(x−y)xxy\frac{x-y}{y} \cdot \frac{x}{x} = \frac{(x-y)x}{xy}

Step 3: Add the Fractions

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

Step 4: Simplify the Expression

(x+y)y+(x−y)xxy=xy+y2+xy−xyxy\frac{(x+y)y + (x-y)x}{xy} = \frac{xy + y^2 + xy - xy}{xy}

xy+y2+xy−xyxy=y2+xyxy\frac{xy + y^2 + xy - xy}{xy} = \frac{y^2 + xy}{xy}

y2+xyxy=y(y+x)xy\frac{y^2 + xy}{xy} = \frac{y(y + x)}{xy}

y(y+x)xy=y+xy\frac{y(y + x)}{xy} = \frac{y + x}{y}

Option B: Simplifying the Fraction with a Quadratic Expression

The second option is to simplify the expression x2+xyxy\frac{x^2+xy}{xy}. To simplify this expression, we can factor out the common term xx from the numerator.

Step 1: Factor Out the Common Term

x2+xyxy=x(x+y)xy\frac{x^2+xy}{xy} = \frac{x(x+y)}{xy}

Step 2: Simplify the Fraction

x(x+y)xy=x+yy\frac{x(x+y)}{xy} = \frac{x+y}{y}

Option C: Simplifying the Fraction with a Quadratic Expression

The third option is to simplify the expression x2xy\frac{x^2}{xy}. To simplify this expression, we can factor out the common term xx from the numerator.

Step 1: Factor Out the Common Term

x2xy=x(x)xy\frac{x^2}{xy} = \frac{x(x)}{xy}

Step 2: Simplify the Fraction

x(x)xy=xy\frac{x(x)}{xy} = \frac{x}{y}

Option D: Simplifying the Fraction with a Quadratic Expression

The fourth option is to simplify the expression x2+y2xy\frac{x^2+y^2}{xy}. To simplify this expression, we can factor out the common term xx from the numerator.

Step 1: Factor Out the Common Term

x2+y2xy=x(x)+y(y)xy\frac{x^2+y^2}{xy} = \frac{x(x) + y(y)}{xy}

Step 2: Simplify the Fraction

x(x)+y(y)xy=x+yy\frac{x(x) + y(y)}{xy} = \frac{x+y}{y}

Option E: Simplifying the Fraction with a Quadratic Expression

The fifth option is to simplify the expression x2−y2xy\frac{x^2-y^2}{xy}. To simplify this expression, we can factor out the common term xx from the numerator.

Step 1: Factor Out the Common Term

x2−y2xy=x(x)−y(y)xy\frac{x^2-y^2}{xy} = \frac{x(x) - y(y)}{xy}

Step 2: Simplify the Fraction

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

Option F: Simplifying the Fraction with a Linear Expression

The sixth option is to simplify the expression 2xxy\frac{2x}{xy}. To simplify this expression, we can factor out the common term xx from the numerator.

Step 1: Factor Out the Common Term

2xxy=2(x)xy\frac{2x}{xy} = \frac{2(x)}{xy}

Step 2: Simplify the Fraction

2(x)xy=2y\frac{2(x)}{xy} = \frac{2}{y}

Conclusion

Simplifying algebraic fractions is a crucial skill to master in mathematics. By following the step-by-step guide provided in this article, you can simplify expressions with quadratic and linear terms. Remember to find the common denominator, factor out common terms, and simplify the expression to its simplest form. With practice and patience, you will become proficient in simplifying algebraic fractions.

Key Takeaways

  • To simplify an algebraic fraction, find the common denominator and factor out common terms.
  • Simplify the expression by canceling out common factors.
  • Use the distributive property to expand expressions and simplify.
  • Factor out common terms to simplify expressions with quadratic and linear terms.

Practice Problems

  1. Simplify the expression x2+xyxy\frac{x^2+xy}{xy}.
  2. Simplify the expression x2xy\frac{x^2}{xy}.
  3. Simplify the expression x2+y2xy\frac{x^2+y^2}{xy}.
  4. Simplify the expression x2−y2xy\frac{x^2-y^2}{xy}.
  5. Simplify the expression 2xxy\frac{2x}{xy}.

Answer Key

  1. x+yy\frac{x+y}{y}
  2. xy\frac{x}{y}
  3. x+yy\frac{x+y}{y}
  4. x−yy\frac{x-y}{y}
  5. 2y\frac{2}{y}
    Frequently Asked Questions: Simplifying Algebraic Fractions =============================================================

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

A: The first step in simplifying an algebraic fraction is to find the common denominator. The common denominator is the product of the denominators of the fractions.

Q: How do I find the common denominator?

A: To find the common denominator, multiply the denominators of the fractions together. For example, if you have the fractions xy\frac{x}{y} and yx\frac{y}{x}, the common denominator is xyxy.

Q: What is the next step after finding the common denominator?

A: After finding the common denominator, factor out common terms from the numerator and denominator. This will help you simplify the expression.

Q: How do I factor out common terms?

A: To factor out common terms, look for terms that are common to both the numerator and denominator. For example, if you have the expression x2+xyxy\frac{x^2+xy}{xy}, you can factor out the common term xx from the numerator.

Q: What is the final step in simplifying an algebraic fraction?

A: The final step in simplifying an algebraic fraction is to simplify the expression by canceling out common factors. This will give you the simplest form of the expression.

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

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

  • Not finding the common denominator
  • Not factoring out common terms
  • Not canceling out common factors
  • Not simplifying the expression to its simplest form

Q: How can I practice simplifying algebraic fractions?

A: You can practice simplifying algebraic fractions by working through practice problems. Start with simple expressions and gradually move on to more complex ones. You can also use online resources or math textbooks to find practice problems.

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

A: Simplifying algebraic fractions has many real-world applications, including:

  • Calculating rates and ratios
  • Solving equations and inequalities
  • Working with probability and statistics
  • Modeling real-world phenomena using mathematical equations

Q: Can I use a calculator to simplify algebraic fractions?

A: Yes, you can use a calculator to simplify algebraic fractions. However, it's always a good idea to check your work by hand to make sure you understand the process.

Q: How can I check my work when simplifying algebraic fractions?

A: To check your work when simplifying algebraic fractions, follow these steps:

  • Write out the original expression
  • Simplify the expression using the steps outlined above
  • Check that the simplified expression is equivalent to the original expression
  • Make sure you have not made any mistakes in the simplification process

Q: What are some common algebraic fractions that I should know how to simplify?

A: Some common algebraic fractions that you should know how to simplify include:

  • x2+xyxy\frac{x^2+xy}{xy}
  • x2xy\frac{x^2}{xy}
  • x2+y2xy\frac{x^2+y^2}{xy}
  • x2−y2xy\frac{x^2-y^2}{xy}
  • 2xxy\frac{2x}{xy}

Q: How can I use algebraic fractions in real-world applications?

A: Algebraic fractions can be used in a variety of real-world applications, including:

  • Calculating rates and ratios
  • Solving equations and inequalities
  • Working with probability and statistics
  • Modeling real-world phenomena using mathematical equations

Q: What are some tips for simplifying algebraic fractions?

A: Some tips for simplifying algebraic fractions include:

  • Start by finding the common denominator
  • Factor out common terms from the numerator and denominator
  • Simplify the expression by canceling out common factors
  • Check your work by hand to make sure you understand the process

Q: Can I use algebraic fractions to solve equations and inequalities?

A: Yes, you can use algebraic fractions to solve equations and inequalities. Algebraic fractions can be used to simplify complex expressions and solve equations and inequalities.

Q: How can I use algebraic fractions to model real-world phenomena?

A: Algebraic fractions can be used to model real-world phenomena by representing complex relationships between variables using mathematical equations. This can be used to solve problems in a variety of fields, including physics, engineering, and economics.