Simplifying Algebraic Expressions A Step By Step Guide

In the realm of mathematics, simplifying complex algebraic expressions is a fundamental skill. These expressions often appear daunting at first glance, but with a systematic approach and a firm grasp of algebraic principles, they can be tamed and reduced to their most basic form. In this article, we will delve into the process of simplifying a specific expression, providing a step-by-step guide and highlighting key concepts along the way. Our focus will be on the expression (3/x - 15/2y) ÷ (6/xy), which involves fractions, variables, and division. By carefully dissecting each step, we aim to not only simplify this particular expression but also equip you with the tools and understanding necessary to tackle similar challenges in the future. The journey of simplification is not just about arriving at a final answer; it's about understanding the underlying structure of the expression and the transformations that preserve its value. So, let's embark on this mathematical adventure and unravel the complexities of this expression together. Remember, mathematics is a journey of exploration, and each simplification is a step forward in our understanding of the elegant world of algebra. Mastering these techniques is crucial for success in higher-level mathematics and various fields that rely on mathematical modeling and analysis. So, let's dive in and begin the process of simplification.

Step 1: Combining Fractions within the Parentheses

The initial step in simplifying the expression (3/x - 15/2y) ÷ (6/xy) involves dealing with the terms within the parentheses. We have a subtraction of two fractions, 3/x and 15/2y. To combine these fractions, we need to find a common denominator. The least common denominator (LCD) for x and 2y is 2xy. This is because 2xy is the smallest expression that both x and 2y can divide into evenly. To get the first fraction, 3/x, to have a denominator of 2xy, we multiply both the numerator and the denominator by 2y. This gives us (3 * 2y) / (x * 2y) = 6y / 2xy. Similarly, for the second fraction, 15/2y, we multiply both the numerator and the denominator by x to get a denominator of 2xy. This results in (15 * x) / (2y * x) = 15x / 2xy. Now that both fractions have the same denominator, we can subtract them: 6y / 2xy - 15x / 2xy. This subtraction involves combining the numerators while keeping the common denominator. So, we have (6y - 15x) / 2xy. This step is crucial as it consolidates the two fractions into a single fraction, making the subsequent division step more manageable. Understanding how to find and use the least common denominator is a fundamental skill in algebra, and it's essential for simplifying expressions involving fractions. By mastering this technique, you'll be able to confidently tackle more complex expressions in the future. This foundational step sets the stage for the rest of the simplification process, ensuring a clear and efficient path to the final answer.

Step 2: Converting Division to Multiplication

Now that we have simplified the expression within the parentheses to (6y - 15x) / 2xy, the next step is to address the division operation. We are dividing this fraction by 6/xy. A fundamental rule in algebra states that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by simply swapping the numerator and the denominator. Therefore, the reciprocal of 6/xy is xy/6. So, the division operation becomes a multiplication: [(6y - 15x) / 2xy] ÷ (6/xy) is equivalent to [(6y - 15x) / 2xy] * (xy/6). This conversion is a cornerstone of algebraic manipulation, allowing us to transform a division problem into a multiplication problem, which is often easier to handle. Understanding this principle is crucial for simplifying various algebraic expressions and solving equations. By changing division to multiplication, we open the door to further simplification through cancellation of common factors, as we will see in the next step. This step highlights the interconnectedness of mathematical operations and the power of applying fundamental rules to simplify complex problems. Mastering this conversion technique is essential for anyone seeking to excel in algebra and related fields. It's a testament to the elegance and efficiency of mathematical methods that allow us to transform seemingly complex problems into simpler, more manageable forms.

Step 3: Factoring and Cancelling Common Factors

With the expression now in the form [(6y - 15x) / 2xy] * (xy/6), we can proceed to simplify it further by factoring and cancelling common factors. The numerator of the first fraction, 6y - 15x, has a common factor of 3. Factoring out the 3, we get 3(2y - 5x). This factorization is a crucial step as it reveals common factors that can be cancelled out, leading to a simpler expression. Now the expression looks like this: [3(2y - 5x) / 2xy] * (xy/6). Next, we can observe that there are common factors in the numerators and denominators of the two fractions. We have 'xy' in both the denominator of the first fraction and the numerator of the second fraction. These factors can be cancelled out, as they appear in both the numerator and the denominator of the overall expression. Similarly, the numbers 3 and 6 have a common factor of 3. We can divide both 3 and 6 by 3, which gives us 1 and 2, respectively. After cancelling these common factors, the expression simplifies significantly. The 'xy' terms are completely eliminated, and the numerical factors are reduced. This process of cancelling common factors is a fundamental technique in simplifying algebraic expressions. It allows us to reduce the complexity of the expression by removing terms that contribute to both the numerator and the denominator. By mastering this technique, you can significantly simplify complex expressions and make them easier to work with. This step demonstrates the power of factorization and cancellation in algebraic simplification, highlighting the importance of recognizing and utilizing common factors. The result of this step sets us up for the final simplification, where we will arrive at the most reduced form of the expression.

Step 4: Arriving at the Simplified Expression

After factoring and cancelling common factors, the expression [3(2y - 5x) / 2xy] * (xy/6) has been transformed. We cancelled out the 'xy' terms from the numerator and denominator, and we simplified the numerical factors by dividing both 3 and 6 by their common factor, 3. This left us with a much simpler expression. Now, let's put together what we have. The 'xy' terms are gone, and the 3 in the numerator of the first fraction has been divided out, leaving 1. The 6 in the denominator of the second fraction has been divided by 3, leaving 2. So, the expression now looks like [(2y - 5x) / 2] * (1/2). Multiplying these fractions, we get (2y - 5x) / (2 * 2), which simplifies to (2y - 5x) / 4. This is the simplified form of the original expression. We have successfully reduced the complex expression (3/x - 15/2y) ÷ (6/xy) to its most basic form: (2y - 5x) / 4. This final step showcases the culmination of all the previous steps. By systematically applying algebraic principles, we have navigated through the complexities of the expression and arrived at a concise and simplified result. This process underscores the importance of each step in the simplification journey, from finding common denominators to factoring and cancelling common factors. The simplified expression (2y - 5x) / 4 is much easier to understand and work with than the original expression. This simplification is not just an exercise in algebraic manipulation; it's a powerful tool for solving equations, analyzing relationships between variables, and building mathematical models.

In conclusion, simplifying algebraic expressions is a fundamental skill in mathematics that requires a systematic approach and a strong understanding of algebraic principles. Throughout this article, we have demonstrated the step-by-step process of simplifying the expression (3/x - 15/2y) ÷ (6/xy), starting from combining fractions within parentheses, converting division to multiplication, factoring and cancelling common factors, and finally arriving at the simplified expression (2y - 5x) / 4. Each step in this process is crucial, and mastering these techniques is essential for success in algebra and related fields. The ability to simplify complex expressions not only makes mathematical problems more manageable but also provides a deeper understanding of the underlying relationships between variables and operations. This skill is invaluable in various applications, from solving equations to building mathematical models for real-world phenomena. The journey of simplification is a testament to the elegance and efficiency of mathematical methods. By applying these methods, we can transform seemingly complex problems into simpler, more understandable forms. This simplification process is not just about finding the answer; it's about developing a strong foundation in algebraic thinking and problem-solving. As you continue your mathematical journey, remember that practice is key. The more you work with algebraic expressions, the more comfortable and confident you will become in simplifying them. So, embrace the challenge, and continue to explore the fascinating world of mathematics. The skills you develop in simplifying expressions will serve you well in your academic pursuits and beyond, opening doors to new opportunities and deeper understanding.