Which Expression Is Equivalent To The Following Complex Fraction?$\[ \frac{\frac{2}{x}-\frac{4}{y}}{\frac{-5}{y}+\frac{3}{x}} \\]A. \[$\frac{3y+5x}{2(y-2x)}\$\]B. \[$\frac{2(y-2x)}{3y-5x}\$\]C.

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Introduction

When dealing with complex fractions, it can be challenging to simplify them and find equivalent expressions. In this article, we will explore how to simplify the given complex fraction and find an equivalent expression. We will use algebraic manipulation and fraction simplification techniques to achieve this goal.

Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In the given complex fraction, we have two fractions in the numerator and two fractions in the denominator. To simplify this fraction, we need to find a common denominator for the fractions in the numerator and the fractions in the denominator.

Simplifying the Complex Fraction

To simplify the complex fraction, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expressions in the numerator and the denominator.
  2. Find a common denominator for the fractions in the numerator and the fractions in the denominator.
  3. Simplify the resulting fraction.

Let's start by evaluating the expressions in the numerator and the denominator:

2x−4y−5y+3x\frac{\frac{2}{x}-\frac{4}{y}}{\frac{-5}{y}+\frac{3}{x}}

We can rewrite the numerator and the denominator as:

2x−4y−5y+3x=2y−4xxy−5x+3yxy\frac{\frac{2}{x}-\frac{4}{y}}{\frac{-5}{y}+\frac{3}{x}} = \frac{\frac{2y-4x}{xy}}{\frac{-5x+3y}{xy}}

Now, we can simplify the resulting fraction by canceling out the common factors in the numerator and the denominator:

2y−4xxy−5x+3yxy=2y−4x−5x+3y\frac{\frac{2y-4x}{xy}}{\frac{-5x+3y}{xy}} = \frac{2y-4x}{-5x+3y}

Finding an Equivalent Expression

To find an equivalent expression, we need to manipulate the fraction to get it into a simpler form. We can do this by multiplying the numerator and the denominator by a suitable expression.

Let's multiply the numerator and the denominator by −1-1:

2y−4x−5x+3y=−(2y−4x)−(−5x+3y)\frac{2y-4x}{-5x+3y} = \frac{-(2y-4x)}{-( -5x+3y)}

Simplifying the resulting expression, we get:

−(2y−4x)−(−5x+3y)=4x−2y5x−3y\frac{-(2y-4x)}{-( -5x+3y)} = \frac{4x-2y}{5x-3y}

Comparing with the Options

Now, let's compare the resulting expression with the options:

A. 3y+5x2(y−2x)\frac{3y+5x}{2(y-2x)} B. 2(y−2x)3y−5x\frac{2(y-2x)}{3y-5x} C. 4x−2y5x−3y\frac{4x-2y}{5x-3y}

We can see that option C is equivalent to the resulting expression.

Conclusion

In this article, we simplified a complex fraction and found an equivalent expression. We used algebraic manipulation and fraction simplification techniques to achieve this goal. We also compared the resulting expression with the options and found that option C is equivalent to the resulting expression.

Frequently Asked Questions

  • What is a complex fraction? A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.
  • How do I simplify a complex fraction? To simplify a complex fraction, you need to follow the order of operations (PEMDAS) and find a common denominator for the fractions in the numerator and the fractions in the denominator.
  • What is an equivalent expression? An equivalent expression is a fraction that has the same value as the original fraction, but is expressed in a different form.

Final Answer

The final answer is option C: 4x−2y5x−3y\frac{4x-2y}{5x-3y}.

Introduction

Complex fractions can be a challenging topic in mathematics, but with the right approach, they can be simplified and understood. In this article, we will provide a comprehensive guide to simplifying complex fractions and finding equivalent expressions.

Frequently Asked Questions

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: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to follow the order of operations (PEMDAS) and find a common denominator for the fractions in the numerator and the fractions in the denominator.

Q: What is an equivalent expression?

A: An equivalent expression is a fraction that has the same value as the original fraction, but is expressed in a different form.

Q: How do I find an equivalent expression for a complex fraction?

A: To find an equivalent expression for a complex fraction, you need to manipulate the fraction to get it into a simpler form. You can do this by multiplying the numerator and the denominator by a suitable expression.

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

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

  • Not following the order of operations (PEMDAS)
  • Not finding a common denominator for the fractions in the numerator and the fractions in the denominator
  • Not simplifying the resulting fraction

Q: How do I know if a complex fraction is simplified?

A: A complex fraction is simplified when it has no fractions in the numerator or the denominator.

Q: Can a complex fraction have multiple equivalent expressions?

A: Yes, a complex fraction can have multiple equivalent expressions. This is because there are often multiple ways to simplify a complex fraction.

Advanced Topics

Q: What is the difference between a complex fraction and a nested fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. A nested fraction is a fraction that contains one or more fractions in its numerator or denominator, and the fractions are nested inside each other.

Q: How do I simplify a nested fraction?

A: To simplify a nested fraction, you need to follow the order of operations (PEMDAS) and find a common denominator for the fractions in the numerator and the fractions in the denominator.

Q: What is the difference between a complex fraction and a rational expression?

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

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to follow the order of operations (PEMDAS) and find a common denominator for the polynomials in the numerator and the polynomials in the denominator.

Conclusion

In this article, we have provided a comprehensive guide to simplifying complex fractions and finding equivalent expressions. We have also answered some frequently asked questions and covered some advanced topics.

Final Tips

  • Always follow the order of operations (PEMDAS) when simplifying complex fractions.
  • Find a common denominator for the fractions in the numerator and the fractions in the denominator.
  • Simplify the resulting fraction.
  • Be careful when multiplying the numerator and the denominator by a suitable expression.
  • Check your work to make sure that the complex fraction is simplified.

Final Answer

The final answer is that complex fractions can be simplified and understood with the right approach. By following the order of operations (PEMDAS) and finding a common denominator for the fractions in the numerator and the fractions in the denominator, you can simplify complex fractions and find equivalent expressions.