Simplifying (x-y)/(xy) Divided By (x^2-y^2)/(3x^2y)

In algebra, simplifying expressions is a fundamental skill. This article focuses on dividing and simplifying algebraic fractions, specifically addressing the expression: $\frac{x-y}{xy} \div \frac{x^2-y^2}{3x^2y}$. Understanding how to manipulate these fractions is crucial for solving more complex algebraic problems. We will explore the steps involved, from converting division to multiplication by the reciprocal, to factoring and canceling common terms. This comprehensive guide aims to provide a clear and concise explanation, making the process accessible to learners of all levels. By mastering these techniques, you will be well-equipped to tackle a wide range of algebraic expressions and equations. Let's dive into the step-by-step process of simplifying this expression.

Understanding Algebraic Fractions

Before diving into the problem, let's clarify what algebraic fractions are and why they matter. Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions. These expressions can involve variables, constants, and mathematical operations. The ability to manipulate these fractions is essential in various areas of mathematics, including calculus, trigonometry, and advanced algebra. Simplifying algebraic fractions often involves techniques such as factoring, canceling common factors, and combining fractions using common denominators.

The expression $\frac{x-y}{xy} \div \frac{x^2-y^2}{3x^2y}$ is a classic example of an algebraic fraction problem. It requires a combination of skills, including understanding the rules of fraction division and the ability to factor algebraic expressions. The beauty of algebra lies in its ability to generalize arithmetic operations to variables and expressions. This means that the same principles that apply to numerical fractions also apply to algebraic fractions. For instance, dividing one fraction by another is the same as multiplying by the reciprocal of the second fraction. This simple yet powerful rule is the cornerstone of simplifying the expression at hand. Moreover, the concept of factoring, which involves breaking down complex expressions into simpler factors, plays a crucial role in identifying and canceling common terms. By mastering these foundational skills, you can confidently tackle a wide array of algebraic challenges.

Step-by-Step Solution

Step 1: Convert Division to Multiplication

The first step in simplifying the expression $\frac{x-y}{xy} \div \frac{x^2-y^2}{3x^2y}$ is to convert the division operation into multiplication. To do this, we multiply the first fraction by the reciprocal of the second fraction. This is a fundamental rule of fraction division: $a/b \div c/d = a/b \times d/c$. Applying this rule, we get:

$\frac{x-y}{xy} \div \frac{x^2-y^2}{3x^2y} = \frac{x-y}{xy} \times \frac{3x^2y}{x^2-y^2}$

This transformation is crucial because multiplication is often easier to handle than division, especially when dealing with algebraic fractions. By converting division to multiplication, we set the stage for further simplification by allowing us to combine the numerators and denominators into single expressions. This step also highlights the importance of understanding the inverse relationship between multiplication and division. The reciprocal of a fraction is simply flipping the numerator and the denominator. This seemingly simple operation is a powerful tool in algebra, enabling us to manipulate complex expressions more easily. The next step involves factoring the expressions, which will allow us to identify and cancel common factors, leading to the simplified form.

Step 2: Factor the Expressions

The next crucial step involves factoring the algebraic expressions in the numerators and denominators. Factoring simplifies the terms, making it easier to identify common factors that can be canceled out. In our expression, $\frac{x-y}{xy} \times \frac{3x^2y}{x^2-y^2}$, we need to focus on factoring $x^2 - y^2$. This is a classic difference of squares, which can be factored as $(x-y)(x+y)$. Recognizing and applying factoring patterns like the difference of squares is a key skill in algebra. The ability to quickly factor expressions significantly speeds up the simplification process.

So, the expression becomes:

$\frac{x-y}{xy} \times \frac{3x^2y}{(x-y)(x+y)}$

Now that we have factored the expressions, we can see common factors emerging in the numerator and the denominator. Factoring is like dissecting a complex expression into its building blocks, revealing the underlying structure and relationships. In this case, factoring $x^2 - y^2$ into $(x-y)(x+y)$ allows us to see the $(x-y)$ term that is present in both the numerator and the denominator. This is a critical observation, as it sets the stage for the next step: canceling common factors. The art of factoring extends beyond simple patterns like the difference of squares; it includes techniques such as factoring by grouping, factoring trinomials, and more. Mastering these techniques is essential for simplifying a wide range of algebraic expressions.

Step 3: Cancel Common Factors

Now that we have the expression in the form $\frac{x-y}{xy} \times \frac{3x^2y}{(x-y)(x+y)}$, we can cancel out common factors present in both the numerator and the denominator. Canceling common factors is akin to dividing both the numerator and the denominator by the same quantity, which doesn't change the value of the fraction but simplifies its form. We can see that $(x-y)$ appears in both the numerator and the denominator, so we can cancel it out. Also, $xy$ in the denominator of the first fraction and $x^2y$ in the numerator of the second fraction share common factors of $x$ and $y$. Specifically, $x^2y$ can be written as $x \cdot xy$, so we can cancel $xy$ from both the numerator and the denominator.

After canceling $(x-y)$ and $xy$, we are left with:

$\frac{1}{1} \times \frac{3x}{x+y}$

This simplification step is where the power of factoring truly shines. By factoring and canceling common terms, we reduce the complexity of the expression, making it easier to understand and work with. The cancellation process highlights the importance of identifying factors, which are multiplicative components of an expression. By removing common factors, we are essentially stripping away the redundant parts of the expression, revealing its core structure. This skill is crucial not only in simplifying fractions but also in solving equations, finding limits in calculus, and many other mathematical contexts. The ability to spot and cancel common factors is a hallmark of algebraic proficiency.

Step 4: Simplify the Expression

After canceling the common factors, we have the expression $\frac{1}{1} \times \frac{3x}{x+y}$. The final step is to multiply the remaining terms to obtain the simplified form. Multiplying the fractions, we get:

$\frac{1 \times 3x}{1 \times (x+y)} = \frac{3x}{x+y}$

This is the simplest form of the original expression. The simplified expression is not only more concise but also easier to interpret and use in further calculations. The process of simplification has removed the complexity and revealed the underlying essence of the expression. The ability to simplify algebraic expressions is a fundamental skill that underpins much of advanced mathematics. It allows us to manipulate equations, solve problems, and gain deeper insights into mathematical relationships. In this case, the journey from the initial expression to the simplified form $\frac{3x}{x+y}$ showcases the power of algebraic techniques such as converting division to multiplication, factoring, and canceling common factors. This final form is not only more elegant but also more practical for various applications.

Conclusion

In conclusion, we have successfully simplified the algebraic expression $\frac{x-y}{xy} \div \frac{x^2-y^2}{3x^2y}$ to $\frac{3x}{x+y}$. This process involved several key steps: converting division to multiplication by the reciprocal, factoring algebraic expressions, canceling common factors, and simplifying the result. Each of these steps is a fundamental technique in algebra, and mastering them is crucial for success in more advanced mathematical topics.

The ability to divide and simplify algebraic fractions is not just an abstract mathematical skill; it has practical applications in various fields, including engineering, physics, and computer science. These skills are essential for modeling real-world phenomena, solving complex problems, and making informed decisions. The journey through this simplification process highlights the interconnectedness of algebraic concepts and the importance of a systematic approach to problem-solving. By understanding the underlying principles and practicing these techniques, you can build a strong foundation in algebra and unlock the door to a world of mathematical possibilities. Remember, the key to success in mathematics is not just memorizing formulas but also understanding the logic and reasoning behind them. By embracing this approach, you can become a confident and proficient problem-solver.