Simplify The Expression: Y X 2 − 1 ⋅ X − 1 2 Y \frac{y}{x^2-1} \cdot \frac{x-1}{2y} X 2 − 1 Y ​ ⋅ 2 Y X − 1 ​

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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 techniques used to simplify expressions is factoring. In this article, we will focus on simplifying the given expression: yx2−1⋅x−12y\frac{y}{x^2-1} \cdot \frac{x-1}{2y}. 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 product of two fractions: yx2−1\frac{y}{x^2-1} and x−12y\frac{x-1}{2y}. To simplify this expression, we need to first understand the properties of fractions and how they can be manipulated.

Properties of Fractions

Fractions are a way of representing a part of a whole. They consist of a numerator (the top number) and a denominator (the bottom number). When we multiply fractions, we multiply the numerators and denominators separately.

Multiplying Fractions

To multiply fractions, we follow these steps:

  1. Multiply the numerators: y⋅(x−1)y \cdot (x-1)
  2. Multiply the denominators: (x2−1)⋅2y(x^2-1) \cdot 2y

Simplifying the Expression

Now that we have understood the properties of fractions and how to multiply them, we can simplify the given expression.

Step 1: Factor the Denominator

The denominator of the first fraction is x2−1x^2-1. We can factor this expression as (x+1)(x−1)(x+1)(x-1).

import sympy as sp

x = sp.symbols('x')



denominator = x**2 - 1
factored_denominator = sp.factor(denominator)


print(factored_denominator)

Step 2: Cancel Common Factors

Now that we have factored the denominator, we can cancel common factors between the numerator and denominator.

import sympy as sp


x = sp.symbols('x')
y = sp.symbols('y')



numerator = y * (x - 1)
denominator = (x + 1) * (x - 1) * 2 * y



simplified_expression = sp.cancel(numerator / denominator)


print(simplified_expression)

Final Answer

After simplifying the expression, we get:

12(x+1)\frac{1}{2(x+1)}

This is the final answer to the problem.

Conclusion

Simplifying expressions is an essential skill in algebra that helps us solve equations and inequalities more efficiently. In this article, we have broken down the steps involved in simplifying the given expression: yx2−1⋅x−12y\frac{y}{x^2-1} \cdot \frac{x-1}{2y}. We have used factoring and canceling common factors to simplify the expression. By following these steps, we have arrived at the final answer: 12(x+1)\frac{1}{2(x+1)}.

Introduction

In our previous article, we simplified the expression: yx2−1⋅x−12y\frac{y}{x^2-1} \cdot \frac{x-1}{2y}. We broke down the steps involved in simplifying this expression and provided a clear explanation of each step. In this article, we will answer some of the most frequently asked questions related to simplifying expressions.

Q&A

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

A: The first step in simplifying an expression is to understand the properties of fractions and how they can be manipulated. We need to identify the numerator and denominator of each fraction and determine how they can be combined.

Q: How do I factor the denominator of a fraction?

A: To factor the denominator of a fraction, we need to look for common factors between the numerator and denominator. We can use the distributive property to factor the denominator.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that a single term can be distributed to multiple terms. For example, a(b+c)=ab+aca(b+c) = ab + ac.

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

A: To cancel common factors between the numerator and denominator, we need to identify the common factors and divide them out. We can use the cancel function in a computer algebra system (CAS) to cancel common factors.

Q: What is a computer algebra system (CAS)?

A: A computer algebra system (CAS) is a software program that can perform mathematical operations, such as simplifying expressions, solving equations, and graphing functions.

Q: How do I use a CAS to simplify an expression?

A: To use a CAS to simplify an expression, we need to enter the expression into the CAS and use the simplify function. The CAS will then simplify the expression and provide the result.

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

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

  • Not factoring the denominator
  • Not canceling common factors
  • Not using the distributive property
  • Not using a CAS to simplify the expression

Q: How do I know if an expression is simplified?

A: An expression is simplified when there are no common factors between the numerator and denominator that can be canceled out. We can use a CAS to check if an expression is simplified.

Conclusion

Simplifying expressions is an essential skill in algebra that helps us solve equations and inequalities more efficiently. In this article, we have answered some of the most frequently asked questions related to simplifying expressions. By following these steps and avoiding common mistakes, we can simplify expressions and arrive at the final answer.

Additional Resources

Final Answer

After simplifying the expression: yx2−1⋅x−12y\frac{y}{x^2-1} \cdot \frac{x-1}{2y}, we get:

12(x+1)\frac{1}{2(x+1)}

This is the final answer to the problem.

Frequently Asked Questions

  • Q: What is the first step in simplifying an expression? A: The first step in simplifying an expression is to understand the properties of fractions and how they can be manipulated.
  • Q: How do I factor the denominator of a fraction? A: To factor the denominator of a fraction, we need to look for common factors between the numerator and denominator.
  • Q: What is the distributive property? A: The distributive property is a mathematical property that states that a single term can be distributed to multiple terms.
  • Q: How do I cancel common factors between the numerator and denominator? A: To cancel common factors between the numerator and denominator, we need to identify the common factors and divide them out.
  • Q: What is a computer algebra system (CAS)? A: A computer algebra system (CAS) is a software program that can perform mathematical operations, such as simplifying expressions, solving equations, and graphing functions.
  • Q: How do I use a CAS to simplify an expression? A: To use a CAS to simplify an expression, we need to enter the expression into the CAS and use the simplify function.
  • Q: What are some common mistakes to avoid when simplifying expressions? A: Some common mistakes to avoid when simplifying expressions include not factoring the denominator, not canceling common factors, not using the distributive property, and not using a CAS to simplify the expression.
  • Q: How do I know if an expression is simplified? A: An expression is simplified when there are no common factors between the numerator and denominator that can be canceled out.