Simplifying The Expression (3/x - 15/2y) ÷ (6/xy) A Step-by-Step Guide

by Admin 71 views

In mathematics, simplifying expressions is a fundamental skill. When dealing with rational expressions, which are fractions containing polynomials, the process can sometimes seem daunting. However, by understanding the underlying principles and applying a systematic approach, we can simplify even the most complex expressions. This article focuses on simplifying a specific rational expression, providing a step-by-step guide and highlighting key concepts along the way.

The expression we aim to simplify is (3/x - 15/2y) ÷ (6/xy). This expression involves fractions within fractions, making it a complex rational expression. To simplify it effectively, we'll need to address the subtraction within the parentheses first, then handle the division.

Step 1: Finding a Common Denominator

The first step in simplifying the expression is to address the subtraction within the parentheses: (3/x - 15/2y). To subtract these fractions, they must have a common denominator. The least common multiple (LCM) of the denominators x and 2y is 2xy. Therefore, we need to rewrite each fraction with 2xy as the denominator.

To rewrite 3/x with a denominator of 2xy, we multiply both the numerator and denominator by 2y:

(3/x) * (2y/2y) = 6y/2xy

Similarly, to rewrite 15/2y with a denominator of 2xy, we multiply both the numerator and denominator by x:

(15/2y) * (x/x) = 15x/2xy

Now we can rewrite the expression within the parentheses with the common denominator:

3/x - 15/2y = 6y/2xy - 15x/2xy

With the common denominator in place, we can subtract the numerators:

6y/2xy - 15x/2xy = (6y - 15x) / 2xy

This simplifies the expression within the parentheses. We now have a single fraction that represents the result of the subtraction.

Step 2: Division as Multiplication by the Reciprocal

Now that we've simplified the expression within the parentheses, we can address the division: ((6y - 15x) / 2xy) ÷ (6/xy). Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 6/xy is xy/6. Therefore, we can rewrite the division as multiplication:

((6y - 15x) / 2xy) ÷ (6/xy) = ((6y - 15x) / 2xy) * (xy/6)

This transformation is crucial because it allows us to simplify the expression further by canceling common factors.

Step 3: Multiplying and Simplifying

Now we multiply the fractions: ((6y - 15x) / 2xy) * (xy/6). When multiplying fractions, we multiply the numerators and the denominators:

((6y - 15x) * xy) / (2xy * 6)

This gives us:

(6xy^2 - 15x^2y) / 12xy

Now we look for common factors in the numerator and denominator that can be canceled. Both the numerator and denominator share common factors of 3, x, and y. Let's factor out 3xy from the numerator:

3xy(2y - 5x) / 12xy

Now we can cancel the common factor of 3xy from the numerator and denominator:

(3xy(2y - 5x)) / (12xy) = (2y - 5x) / 4

Step 4: Factoring and Cancelling Common Factors

After multiplying the fractions, we obtained the expression (6xy^2 - 15x^2y) / 12xy. The next step is to factor out the greatest common factor (GCF) from both the numerator and the denominator. This will allow us to identify any common factors that can be cancelled, further simplifying the expression.

Looking at the numerator, 6xy^2 - 15x^2y, we can see that both terms have a factor of 3xy. Factoring out 3xy, we get:

3xy(2y - 5x)

In the denominator, 12xy, the factors are 2 * 2 * 3 * x * y. The GCF of the numerator and denominator is 3xy. Now we can rewrite the expression as:

[3xy(2y - 5x)] / (12xy)

Now we can cancel the common factor of 3xy from the numerator and denominator. This leaves us with:

(2y - 5x) / 4

Step 5: The Final Simplified Expression

After performing all the steps of finding a common denominator, multiplying by the reciprocal, and cancelling common factors, we arrive at the simplified expression:

(2y - 5x) / 4

This is the simplest form of the original expression, (3/x - 15/2y) ÷ (6/xy). It's crucial to present the final answer in its most simplified form, which often involves factoring and canceling common factors.

Key Concepts and Considerations

Throughout this simplification process, several key mathematical concepts have been applied. Understanding these concepts is crucial for simplifying other rational expressions:

  • Common Denominators: Fractions can only be added or subtracted if they have the same denominator. Finding the least common multiple (LCM) of the denominators is key to finding the common denominator efficiently.
  • Reciprocal: Dividing by a fraction is equivalent to multiplying by its reciprocal. This transformation is essential for simplifying expressions involving division of fractions.
  • Factoring: Factoring out common factors from the numerator and denominator allows for cancellation, which is a crucial step in simplifying rational expressions.
  • Cancellation: Only common factors can be canceled. It's important to ensure that you're canceling factors and not terms.

Common Mistakes to Avoid

When simplifying rational expressions, several common mistakes can occur. Being aware of these pitfalls can help you avoid them:

  • Incorrectly finding the common denominator: Make sure to find the least common multiple (LCM) of the denominators. Using a common multiple that is not the LCM can lead to more complex calculations.
  • Forgetting to distribute negative signs: When subtracting fractions, remember to distribute the negative sign to all terms in the numerator of the second fraction.
  • Incorrectly canceling terms: Only common factors can be canceled, not terms. For example, in the expression (2y - 5x) / 4, you cannot cancel the 2 in 2y with the 4 in the denominator because 2y is a term, not a factor.
  • Skipping steps: It's crucial to show all the steps in the simplification process. This helps to avoid errors and makes it easier to track your work.

Conclusion

Simplifying rational expressions requires a systematic approach and a solid understanding of fundamental mathematical concepts. By following the steps outlined in this article, you can confidently simplify even complex expressions. Remember to find common denominators, rewrite division as multiplication by the reciprocal, factor out common factors, and cancel where possible. By avoiding common mistakes and practicing regularly, you can master the art of simplifying rational expressions.

The key to simplifying complex rational expressions lies in breaking down the problem into manageable steps. By mastering these steps and understanding the underlying principles, you can approach any rational expression with confidence and achieve the correct simplified form.