Simplify The Expression:$\[ \frac{\frac{2}{x}+\frac{3}{y}}{\frac{-5}{x}+\frac{7}{y}} \\]

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Introduction

Complex fractions can be a daunting task for many students and mathematicians alike. These fractions involve multiple layers of division and can be challenging to simplify. In this article, we will focus on simplifying the expression 2x+3y−5x+7y\frac{\frac{2}{x}+\frac{3}{y}}{\frac{-5}{x}+\frac{7}{y}}. We will break down the process into manageable steps and provide a clear explanation of each step.

Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In the given expression, we have a fraction in the numerator and a fraction in the denominator. To simplify this expression, we need to follow a specific order of operations.

Step 1: Simplify the Numerator

The numerator of the given expression is 2x+3y\frac{2}{x}+\frac{3}{y}. To simplify this expression, we need to find a common denominator for the two fractions. The common denominator is xyxy. We can rewrite the numerator as follows:

2x+3y=2yxy+3xxy=2y+3xxy\frac{2}{x}+\frac{3}{y} = \frac{2y}{xy} + \frac{3x}{xy} = \frac{2y+3x}{xy}

Step 2: Simplify the Denominator

The denominator of the given expression is −5x+7y\frac{-5}{x}+\frac{7}{y}. To simplify this expression, we need to find a common denominator for the two fractions. The common denominator is xyxy. We can rewrite the denominator as follows:

−5x+7y=−5yxy+7xxy=−5y+7xxy\frac{-5}{x}+\frac{7}{y} = \frac{-5y}{xy} + \frac{7x}{xy} = \frac{-5y+7x}{xy}

Step 3: Simplify the Complex Fraction

Now that we have simplified the numerator and denominator, we can simplify the complex fraction. We can rewrite the expression as follows:

2y+3xxy−5y+7xxy=2y+3xxy÷−5y+7xxy\frac{\frac{2y+3x}{xy}}{\frac{-5y+7x}{xy}} = \frac{2y+3x}{xy} \div \frac{-5y+7x}{xy}

To divide fractions, we need to multiply the numerator by the reciprocal of the denominator. The reciprocal of the denominator is xy−5y+7x\frac{xy}{-5y+7x}. We can rewrite the expression as follows:

2y+3xxy÷−5y+7xxy=2y+3xxy×xy−5y+7x\frac{2y+3x}{xy} \div \frac{-5y+7x}{xy} = \frac{2y+3x}{xy} \times \frac{xy}{-5y+7x}

Step 4: Simplify the Expression

Now that we have simplified the complex fraction, we can simplify the expression further. We can cancel out the common factors in the numerator and denominator. The common factors are xx and yy. We can rewrite the expression as follows:

2y+3xxy×xy−5y+7x=2y+3x−5y+7x\frac{2y+3x}{xy} \times \frac{xy}{-5y+7x} = \frac{2y+3x}{-5y+7x}

Conclusion

Simplifying complex fractions requires a step-by-step approach. We need to simplify the numerator and denominator separately and then combine them to simplify the complex fraction. In this article, we simplified the expression 2x+3y−5x+7y\frac{\frac{2}{x}+\frac{3}{y}}{\frac{-5}{x}+\frac{7}{y}} by following the order of operations and canceling out common factors. We hope this article has provided a clear explanation of how to simplify complex fractions.

Common Mistakes to Avoid

When simplifying complex fractions, there are several common mistakes to avoid. These include:

  • Not finding a common denominator: When simplifying the numerator and denominator, it is essential to find a common denominator. This ensures that the fractions are equivalent and can be combined.
  • Not canceling out common factors: When simplifying the complex fraction, it is essential to cancel out common factors in the numerator and denominator. This simplifies the expression and makes it easier to work with.
  • Not following the order of operations: When simplifying complex fractions, it is essential to follow the order of operations. This includes simplifying the numerator and denominator separately and then combining them to simplify the complex fraction.

Real-World Applications

Simplifying complex fractions has several real-world applications. These include:

  • Finance: Complex fractions are used in finance to calculate interest rates and investment returns.
  • Science: Complex fractions are used in science to calculate rates of change and solve equations.
  • Engineering: Complex fractions are used in engineering to calculate stress and strain on materials.

Conclusion

Introduction

Simplifying complex fractions can be a challenging task, but with the right guidance, it can be made easier. 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 makes the expression easier to work with and understand. It also helps to avoid errors and makes it easier to solve equations.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to follow the order of operations. This includes simplifying the numerator and denominator separately and then combining them to simplify the complex fraction.

Q: What is the order of operations for simplifying complex fractions?

A: The order of operations for simplifying complex fractions is as follows:

  1. Simplify the numerator and denominator separately.
  2. Find a common denominator for the numerator and denominator.
  3. Combine the numerator and denominator to simplify the complex fraction.
  4. Cancel out common factors in the numerator and denominator.

Q: How do I find a common denominator for the numerator and denominator?

A: To find a common denominator for the numerator and denominator, you need to identify the least common multiple (LCM) of the denominators. The LCM is the smallest number that both denominators can divide into evenly.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest number that both denominators can divide into evenly. It is used to find a common denominator for the numerator and denominator.

Q: How do I cancel out common factors in the numerator and denominator?

A: To cancel out common factors in the numerator and denominator, you need to identify the common factors and divide them out. This will simplify the expression and make it easier to work with.

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 a common denominator
  • Not canceling out common factors
  • Not following the order of operations

Q: How do I apply simplifying complex fractions in real-world scenarios?

A: Simplifying complex fractions has several real-world applications, including finance, science, and engineering. It is used to calculate interest rates, investment returns, rates of change, and stress and strain on materials.

Q: What are some examples of simplifying complex fractions?

A: Some examples of simplifying complex fractions include:

  • 2x+3y−5x+7y\frac{\frac{2}{x}+\frac{3}{y}}{\frac{-5}{x}+\frac{7}{y}}
  • 4x+6y−2x+3y\frac{\frac{4}{x}+\frac{6}{y}}{\frac{-2}{x}+\frac{3}{y}}
  • 1x+2y−1x+1y\frac{\frac{1}{x}+\frac{2}{y}}{\frac{-1}{x}+\frac{1}{y}}

Conclusion

Simplifying complex fractions is a crucial skill in mathematics and has several real-world applications. By following the order of operations and canceling out common factors, we can simplify complex fractions and make them easier to work with. We hope this Q&A guide has provided a clear explanation of how to simplify complex fractions and has helped you to avoid common mistakes.