Subtracting Rational Expressions A Step-by-Step Guide

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In the realm of algebra, manipulating rational expressions is a fundamental skill. Rational expressions, which are essentially fractions with polynomials in the numerator and denominator, often require simplification through various operations, including subtraction. This article delves into the process of subtracting rational expressions, focusing on the specific example of ${\frac{4y+9}{3y-27} - \frac{y+6}{y-9}}$. We will break down each step, ensuring a clear understanding of the underlying principles and techniques involved. Mastering these techniques is crucial for success in algebra and calculus, where rational expressions frequently appear.

Understanding Rational Expressions

Before diving into the subtraction process, it's crucial to understand what rational expressions are and the basic operations that can be performed on them. A rational expression is a fraction where the numerator and the denominator are polynomials. Polynomials, in turn, are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. For instance, ${4y+9}$ and ${3y-27}$ are polynomials, making ${\frac{4y+9}{3y-27}}$ a rational expression. Similarly, ${\frac{y+6}{y-9}}$ is another rational expression.

When dealing with rational expressions, it's often necessary to simplify them. Simplification can involve factoring, canceling common factors, and performing algebraic operations such as addition, subtraction, multiplication, and division. These operations follow similar rules to those for numerical fractions, but with the added complexity of polynomial expressions. The goal is typically to reduce the expression to its simplest form, making it easier to work with in further calculations or analyses. This foundation is vital for tackling the subtraction problem at hand, ensuring a solid grasp of the initial steps required for simplification.

Step-by-Step Subtraction of ${\frac{4y+9}{3y-27} - \frac{y+6}{y-9}}$

The given problem requires us to subtract two rational expressions: ${\frac{4y+9}{3y-27} - \frac{y+6}{y-9}}$. To subtract these expressions, we need to follow a series of steps, which include factoring, finding a common denominator, adjusting the numerators, performing the subtraction, and simplifying the result. Each of these steps is critical to arriving at the correct solution and understanding the broader principles of rational expression manipulation.

Step 1: Factoring the Denominators

The first step in subtracting rational expressions is to factor the denominators. Factoring helps us identify common factors, which are essential for finding a common denominator. Looking at the expression ${\frac{4y+9}{3y-27} - \frac{y+6}{y-9}}$, we can see that the denominator ${3y-27}$ can be factored. We can factor out a ${3}$ from ${3y-27}$:

${ 3y - 27 = 3(y - 9) }$

Now our expression looks like this:

{ rac{4y+9}{3(y-9)} - rac{y+6}{y-9} }

The denominator ${y-9}$ in the second term is already in its simplest form and cannot be factored further. Factoring is a crucial step because it simplifies the process of finding a common denominator, which is the next key step in subtracting rational expressions. This step highlights the importance of recognizing opportunities for simplification before proceeding with more complex operations.

Step 2: Finding the Least Common Denominator (LCD)

To subtract rational expressions, we need a common denominator. The least common denominator (LCD) is the smallest expression that is a multiple of both denominators. In our case, the denominators are ${3(y-9)}$ and ${y-9}$. To find the LCD, we need to identify all unique factors present in the denominators and take the highest power of each factor. The factors are ${3}$ and ${(y-9)}$.

The LCD is the product of these factors:

${ LCD = 3(y - 9) }$

This means that the first fraction already has the common denominator, while the second fraction needs to be adjusted. Finding the LCD is a fundamental step in adding or subtracting fractions, whether they are numerical or algebraic. The LCD ensures that we are working with equivalent fractions, which allows us to combine the numerators correctly.

Step 3: Adjusting the Numerators

Now that we have the LCD, we need to adjust the numerators so that each fraction has the common denominator. The first fraction, ${\frac{4y+9}{3(y-9)}}$, already has the LCD, so we don't need to change its numerator. The second fraction, ${\frac{y+6}{y-9}}$, needs to be multiplied by ${\frac{3}{3}}$ to get the LCD in the denominator:

{ rac{y+6}{y-9} imes rac{3}{3} = rac{3(y+6)}{3(y-9)} = rac{3y+18}{3(y-9)} }

Now our expression looks like this:

{ rac{4y+9}{3(y-9)} - rac{3y+18}{3(y-9)} }

Adjusting the numerators is a critical step because it ensures that we are subtracting equivalent fractions. By multiplying the numerator and denominator of the second fraction by the appropriate factor, we maintain the fraction's value while achieving a common denominator. This step sets the stage for the actual subtraction operation.

Step 4: Subtracting the Numerators

With both fractions now having the same denominator, we can subtract the numerators. We subtract the second numerator from the first:

{ rac{4y+9}{3(y-9)} - rac{3y+18}{3(y-9)} = rac{(4y+9) - (3y+18)}{3(y-9)} }

Distribute the negative sign to both terms in the second numerator:

{ rac{4y+9 - 3y - 18}{3(y-9)} }

Combine like terms in the numerator:

{ rac{(4y - 3y) + (9 - 18)}{3(y-9)} = rac{y - 9}{3(y-9)} }

This step involves careful application of algebraic principles, particularly the distribution of the negative sign and the combination of like terms. Accurate execution of this step is vital for arriving at the correct simplified expression.

Step 5: Simplifying the Result

Finally, we simplify the resulting rational expression. We have:

{ rac{y-9}{3(y-9)} }

Notice that ${(y-9)}$ appears in both the numerator and the denominator. We can cancel this common factor, provided that (y eq 9) (since division by zero is undefined):

{ rac{y-9}{3(y-9)} = rac{1}{3} }

Thus, the simplified form of the expression is ${\frac{1}{3}}$. Simplification is the final touch that presents the expression in its most concise and manageable form. It often involves canceling common factors, which not only simplifies the expression but also reveals its fundamental structure.

Final Answer

Therefore, the simplified result of subtracting the given rational expressions is:

{ rac{4y+9}{3y-27} - rac{y+6}{y-9} = rac{1}{3} }

This final answer represents the culmination of all the steps, from factoring and finding a common denominator to subtracting numerators and simplifying. The process demonstrates a systematic approach to handling rational expressions, highlighting the importance of each step in achieving the correct result.

Common Mistakes to Avoid

When subtracting rational expressions, several common mistakes can occur. Being aware of these pitfalls can help you avoid them and ensure accurate solutions. Here are some frequent errors:

1. Forgetting to Distribute the Negative Sign: A common mistake is not distributing the negative sign correctly when subtracting numerators. Remember that you are subtracting the entire numerator, so each term in the second numerator must have its sign changed. For instance, in our example, it’s crucial to subtract both ${3y}$ and ${18}$ from the first numerator.

2. Not Finding a Common Denominator: Subtracting fractions requires a common denominator. Failing to find and establish a common denominator is a fundamental error that invalidates the subtraction process. Make sure to identify the least common denominator (LCD) before combining the numerators.

3. Incorrect Factoring: Factoring the denominators correctly is crucial for finding the LCD. Mistakes in factoring can lead to an incorrect LCD and, consequently, an incorrect final answer. Always double-check your factoring to ensure accuracy.

4. Canceling Terms Incorrectly: Simplification involves canceling common factors, not terms. You can only cancel factors that are multiplied by the entire numerator and denominator. Canceling terms that are added or subtracted is a common error. For example, you cannot cancel the ${y}$ in ${\frac{y-9}{3(y-9)}}$ until you’ve factored out the common factor of ${(y-9)}$.

5. Not Simplifying the Final Answer: Always simplify your final answer as much as possible. This often involves canceling common factors or combining like terms. Failing to simplify can result in a correct but unnecessarily complex answer. Simplification is a critical step in presenting the solution in its most understandable form.

By being mindful of these common mistakes and double-checking your work, you can improve your accuracy and confidence in subtracting rational expressions. The process of subtraction requires attention to detail and a systematic approach, but with practice, these skills can be mastered.

Practice Problems

To solidify your understanding of subtracting rational expressions, working through practice problems is essential. Here are a few problems you can try:

1. {\frac{5x}{x^2 - 4} - rac{2}{x - 2}}
2. {\frac{3}{y + 1} - rac{2}{y - 1}}
3. {\frac{4a}{a^2 + 2a + 1} - rac{1}{a + 1}}

Working through these problems will give you hands-on experience in applying the steps we’ve discussed. Remember to factor the denominators, find the LCD, adjust the numerators, subtract, and simplify the result. Each problem provides an opportunity to reinforce your skills and identify areas where you may need additional practice.

Conclusion

Subtracting rational expressions is a key skill in algebra, requiring a systematic approach that includes factoring, finding a common denominator, adjusting numerators, subtracting, and simplifying. By following these steps carefully and avoiding common mistakes, you can confidently tackle these problems. The specific example of ${\frac{4y+9}{3y-27} - \frac{y+6}{y-9}}$ illustrates the entire process, from initial simplification to the final answer of ${\frac{1}{3}}$. Practice is essential for mastering these techniques, so be sure to work through additional problems and apply what you’ve learned. With a solid understanding of these principles, you'll be well-equipped to handle more advanced algebraic concepts that build upon these foundational skills. The ability to manipulate rational expressions is not only crucial for success in mathematics but also for various fields that rely on mathematical modeling and analysis.