Simplify The Expression:$\[ \frac{y^2 + 4y - 5}{y^2} \div \frac{y^2 + 6y + 3}{y^2} \\]

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities more efficiently. One of the most common ways to simplify an expression is by dividing one fraction by another. In this article, we will focus on simplifying the given expression: ${ \frac{y^2 + 4y - 5}{y^2} \div \frac{y^2 + 6y + 3}{y^2} }$. We will break down the steps involved in simplifying this expression and provide a clear explanation of each step.

Understanding the Expression

The given expression is a division of two fractions. To simplify it, we need to first understand the concept of dividing fractions. When we divide one fraction by another, we can multiply the first fraction by the reciprocal of the second fraction. In other words, ${ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} }$.

Step 1: Rewrite the Expression as a Multiplication of Fractions

Using the concept of dividing fractions, we can rewrite the given expression as a multiplication of fractions: ${ \frac{y^2 + 4y - 5}{y^2} \times \frac{y2}{y2 + 6y + 3} }$.

Step 2: Simplify the Numerators and Denominators

Now that we have rewritten the expression as a multiplication of fractions, we can simplify the numerators and denominators separately. The numerator of the first fraction is y2+4y−5y^2 + 4y - 5, and the denominator is y2y^2. The numerator of the second fraction is y2y^2, and the denominator is y2+6y+3y^2 + 6y + 3.

Step 3: Factor the Numerators and Denominators

To simplify the expression further, we can factor the numerators and denominators. The numerator y2+4y−5y^2 + 4y - 5 can be factored as (y+5)(y−1)(y + 5)(y - 1). The denominator y2+6y+3y^2 + 6y + 3 can be factored as (y+1)(y+3)(y + 1)(y + 3).

Step 4: Cancel Out Common Factors

Now that we have factored the numerators and denominators, we can cancel out common factors. The expression becomes: ${ \frac{(y + 5)(y - 1)}{y^2} \times \frac{y^2}{(y + 1)(y + 3)} }$.

Step 5: Simplify the Expression

We can simplify the expression further by canceling out common factors. The y2y^2 terms cancel out, leaving us with: ${ \frac{(y + 5)(y - 1)}{(y + 1)(y + 3)} }$.

Step 6: Final Simplification

The expression is now simplified, and we can write the final answer.

Conclusion

Simplifying expressions is an essential skill in algebra that helps us solve equations and inequalities more efficiently. In this article, we have walked through the steps involved in simplifying the given expression: ${ \frac{y^2 + 4y - 5}{y^2} \div \frac{y^2 + 6y + 3}{y^2} }$. We have broken down the steps involved in simplifying this expression and provided a clear explanation of each step. By following these steps, you can simplify any expression involving fractions and algebraic manipulations.

Frequently Asked Questions

  • Q: What is the concept of dividing fractions? A: When we divide one fraction by another, we can multiply the first fraction by the reciprocal of the second fraction.
  • Q: How do we simplify the expression ${ \frac{y^2 + 4y - 5}{y^2} \div \frac{y^2 + 6y + 3}{y^2} }$? A: We can rewrite the expression as a multiplication of fractions, simplify the numerators and denominators, factor the numerators and denominators, cancel out common factors, and finally simplify the expression.
  • Q: What is the final simplified expression? A: The final simplified expression is ${ \frac{(y + 5)(y - 1)}{(y + 1)(y + 3)} }$.

Additional Resources

  • For more information on simplifying expressions, check out the following resources:
  • Khan Academy: Simplifying Expressions
  • Mathway: Simplifying Expressions
  • Wolfram Alpha: Simplifying Expressions

Final Answer

The final answer is ${ \frac{(y + 5)(y - 1)}{(y + 1)(y + 3)} }$.

Introduction

In our previous article, we walked through the steps involved in simplifying the expression: ${ \frac{y^2 + 4y - 5}{y^2} \div \frac{y^2 + 6y + 3}{y^2} }$. In this article, we will provide a Q&A guide to help you understand the concept of simplifying expressions and algebraic manipulations.

Q&A Guide

Q: What is the concept of dividing fractions?

A: When we divide one fraction by another, we can multiply the first fraction by the reciprocal of the second fraction. In other words, ${ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} }$.

Q: How do we simplify the expression ${

\frac{y^2 + 4y - 5}{y^2} \div \frac{y^2 + 6y + 3}{y^2} }$?

A: We can rewrite the expression as a multiplication of fractions, simplify the numerators and denominators, factor the numerators and denominators, cancel out common factors, and finally simplify the expression.

Q: What is the difference between simplifying expressions and solving equations?

A: Simplifying expressions involves reducing the complexity of an expression by combining like terms, canceling out common factors, and rearranging the terms. Solving equations, on the other hand, involves finding the value of a variable that makes the equation true.

Q: How do we know when to simplify an expression?

A: We should simplify an expression when it is necessary to make the expression more manageable or to make it easier to solve an equation. Simplifying expressions can also help us identify patterns and relationships between variables.

Q: Can we simplify expressions with variables in the denominator?

A: Yes, we can simplify expressions with variables in the denominator. However, we need to be careful when canceling out common factors to avoid dividing by zero.

Q: What is the final simplified expression for the given problem?

A: The final simplified expression is ${ \frac{(y + 5)(y - 1)}{(y + 1)(y + 3)} }$.

Q: How do we check our work when simplifying expressions?

A: We can check our work by plugging in values for the variables and verifying that the simplified expression is true. We can also use algebraic manipulations to verify that the simplified expression is equivalent to the original expression.

Additional Tips and Tricks

  • Always start by simplifying the expression by combining like terms and canceling out common factors.
  • Use algebraic manipulations to verify that the simplified expression is equivalent to the original expression.
  • Be careful when canceling out common factors to avoid dividing by zero.
  • Use variables in the denominator to simplify expressions, but be careful when canceling out common factors.

Conclusion

Simplifying expressions is an essential skill in algebra that helps us solve equations and inequalities more efficiently. In this article, we have provided a Q&A guide to help you understand the concept of simplifying expressions and algebraic manipulations. By following these tips and tricks, you can simplify any expression involving fractions and algebraic manipulations.

Frequently Asked Questions

  • Q: What is the concept of dividing fractions? A: When we divide one fraction by another, we can multiply the first fraction by the reciprocal of the second fraction.
  • Q: How do we simplify the expression ${ \frac{y^2 + 4y - 5}{y^2} \div \frac{y^2 + 6y + 3}{y^2} }$? A: We can rewrite the expression as a multiplication of fractions, simplify the numerators and denominators, factor the numerators and denominators, cancel out common factors, and finally simplify the expression.
  • Q: What is the final simplified expression? A: The final simplified expression is ${ \frac{(y + 5)(y - 1)}{(y + 1)(y + 3)} }$.

Additional Resources

  • For more information on simplifying expressions, check out the following resources:
  • Khan Academy: Simplifying Expressions
  • Mathway: Simplifying Expressions
  • Wolfram Alpha: Simplifying Expressions

Final Answer

The final answer is ${ \frac{(y + 5)(y - 1)}{(y + 1)(y + 3)} }$.