Simplify Algebraic Fractions: Step-by-Step Guide

Mathematics is a vast and fascinating field, and one of the fundamental skills within it is the ability to simplify algebraic fractions. This process is crucial for solving more complex equations and understanding various mathematical concepts. Today, we're going to break down how to find the sum of algebraic fractions, focusing on a specific example to make it crystal clear. We'll guide you through each step, highlighting the values you need to determine to complete the process. So, grab your pencils, and let's dive into the wonderful world of algebraic manipulation!

Understanding the Problem: Adding Algebraic Fractions

Our journey begins with a common challenge: adding two algebraic fractions that, at first glance, might seem a bit tricky. The expression we're working with is:

$$\frac{3 x}{2 x-6}+\frac{9}{6-2 x}$$

Notice that the denominators, ${2x-6}$ and ${6-2x}$, are not identical. However, they are closely related! One is the negative of the other. This is a key observation that will help us simplify the problem significantly. Our goal is to find a common denominator so we can add the numerators. We'll be filling in some blanks, represented by variables ${a}$, ${b}$, and ${c}$, as we go through the steps. This isn't just about getting an answer; it's about understanding the why behind each step in simplifying algebraic fractions.

Step 1: Making the Denominators Consistent

The first crucial step in adding algebraic fractions is to ensure that both fractions have the same denominator. Looking at our expression, we have ${2x-6}$ and ${6-2x}$. We can rewrite the second denominator, ${6-2x}$, by factoring out a -1:

$$6-2x = -1(2x-6)$$

Now, let's substitute this back into our original expression:

$$\frac{3 x}{2 x-6}+\frac{9}{-1(2 x-6)}$$

To make the denominator of the second fraction match the first, we can move the negative sign from the factored -1 to the numerator. This means multiplying both the numerator and the denominator of the second fraction by -1 (effectively multiplying by 1, which doesn't change the value of the fraction):

$$\frac{3 x}{2 x-6}+\frac{9 \times (-1)}{-1(2 x-6) \times (-1)} = \frac{3 x}{2 x-6}+\frac{-9}{2 x-6}$$

Now, both fractions share the same denominator, ${2x-6}$. This is a pivotal moment in simplifying algebraic fractions. The problem statement guides us here, showing the first step as:

$$\frac{3 x}{2 x-6}+\frac{9}{a(2 x-6)}$$

Comparing this to our work, we can see that ${a}$ must be -1. This value of ${a}$ is essential for proceeding correctly. Remember, the goal is always to create a common ground, and in adding algebraic fractions, that common ground is the denominator.

Step 2: Combining the Numerators

With a common denominator established, the next logical step in adding algebraic fractions is to combine the numerators. Since both fractions now have the denominator ${2x-6}$, we can simply add their numerators:

$$\frac{3 x + (-9)}{2 x-6}$$

This simplifies to:

$$\frac{3 x-9}{2 x-6}$$

This step directly corresponds to the next part of the problem statement:

$$=\frac{3 x-b}{2 x-6}$$

By comparing ${\frac{3x-9}{2x-6}}$ with ${\frac{3x-b}{2x-6}}$, we can clearly identify the value of ${b}$. The numerator is ${3x - 9}$, and in the provided format, it's ${3x - b}$. Therefore, ${b}$ must be 9. It's important to note that while we added ${-9}$ in our calculation, the format given ${3x-b}$ implies that ${b}$ itself is the positive value that is being subtracted. This attention to detail is key when working with simplifying algebraic fractions.

Step 3: Factoring and Final Simplification

We've successfully combined the fractions, resulting in ${\frac{3x-9}{2x-6}}$. Now, the final step in simplifying algebraic fractions is to see if the resulting fraction can be further reduced. To do this, we look for common factors in the numerator and the denominator.

Let's factor the numerator, ${3x-9}$. The greatest common factor here is 3. So, we can rewrite the numerator as:

$$3x-9 = 3(x-3)$$

Now, let's factor the denominator, ${2x-6}$. The greatest common factor here is 2. So, we can rewrite the denominator as:

$$2x-6 = 2(x-3)$$

Substituting these factored forms back into our fraction, we get:

$$\frac{3(x-3)}{2(x-3)}$$

We can see that ${(x-3)}$ is a common factor in both the numerator and the denominator. As long as ${x \neq 3}$ (because division by zero is undefined), we can cancel out this common factor:

\frac{3${x-3}$}{2${x-3}$} = \frac{3}{2}

This is the simplified sum of the original algebraic fractions. Now, let's relate this back to the final step in the problem description:

$$\frac{3 x-c}{2 x-6}$$

Wait, there seems to be a slight discrepancy between our calculated numerator ${3x-9}$ and the format ${3x-c}$ in the original problem description, which was meant to be before factoring the numerator. Let's re-examine the steps. The problem intended for us to reach ${\frac{3x-9}{2x-6}}$ and then potentially simplify. However, the provided structure seems to imply the step after combining numerators but before canceling common factors. If we interpret the step ${\frac{3 x}{2 x-6}+\frac{b}{2 x-6} =\frac{3 x-c}{2 x-6}}$ as the direct result of combining ${\frac{3x}{2x-6}}$ and ${\frac{b}{2x-6}}$, then our combined numerator was ${3x + (-b)}$ if ${b}$ was the numerator we added, or ${3x - b}$ if ${b}$ was meant to be added as a positive value.

Let's follow the structure exactly as given.

Step 1 leads to ${\frac{3 x}{2 x-6}+\frac{9}{a(2 x-6)}}$ where ${a=-1}$. This becomes ${\frac{3 x}{2 x-6}+\frac{-9}{2 x-6}}$.

Step 2 is ${\frac{3 x}{2 x-6}+\frac{b}{2 x-6}}$. This implies that the second fraction's numerator was adjusted to ${b}$. From our Step 1, the second fraction became ${\frac{-9}{2x-6}}$. So, in this step, ${b = -9}$. This is the value that gets added to ${3x}$.

Step 3 is ${\frac{3 x-c}{2 x-6}}$. This represents the combined numerator. If we combine ${3x}$ and ${b}$ (which is -9), we get ${3x + b = 3x + (-9) = 3x - 9}$. Therefore, comparing ${3x - 9}$ to ${3x - c}$, we find that ${c = 9}$.

So, the values are: ${a = -1}$, ${b = -9}$, and ${c = 9}$.

Let's re-evaluate the problem statement's presentation of the steps to ensure clarity. The initial form suggests a direct addition: ${\frac{3 x}{2 x-6}+\frac{9}{6-2 x}}$.

We established ${\frac{9}{6-2 x} = \frac{9}{-1(2x-6)} = \frac{-9}{2x-6}}$.

So the sum is ${\frac{3 x}{2 x-6} + \frac{-9}{2 x-6}}$.

The problem shows ${\frac{3 x}{2 x-6}+\frac{9}{a(2 x-6)}}$ as the first transformation. This means ${\frac{9}{6-2x} = \frac{9}{a(2x-6)}}$. For this to be true, ${6-2x = a(2x-6)}$. Factoring ${6-2x = -1(2x-6)}$, we get ${-1(2x-6) = a(2x-6)}$. Thus, ${a=-1}$.

Next, the problem shows ${\frac{3 x}{2 x-6}+\frac{b}{2 x-6}}$. This means that ${\frac{9}{a(2 x-6)}}$ was transformed into ${\frac{b}{2 x-6}}$. Since ${a=-1}$, this is ${\frac{9}{-1(2 x-6)} = \frac{-9}{2 x-6}}$. So, ${b = -9}$.

Finally, the problem shows ${\frac{3 x-c}{2 x-6}}$. This is the result of adding the numerators: ${\frac{3x + b}{2x-6} = \frac{3x + (-9)}{2x-6} = \frac{3x-9}{2x-6}}$. Comparing this to ${\frac{3 x-c}{2 x-6}}$, we see that ${c = 9}$.

Therefore, the values are ${a=-1}$, ${b=-9}$, and ${c=9}$.

This detailed breakdown ensures that every step in simplifying algebraic fractions is understood, from finding common denominators to combining numerators and performing the final, crucial simplification. It's a process that builds confidence and mathematical skill.

The Importance of Simplifying Algebraic Fractions

Simplifying algebraic fractions is not just an academic exercise; it's a fundamental skill that underpins much of higher mathematics. Whether you're dealing with calculus, trigonometry, or advanced algebra, the ability to manipulate and simplify these expressions efficiently can make complex problems manageable. It's like learning to tie your shoelaces before running a marathon; you need the basic skills to tackle the bigger challenges. The process we've just walked through – finding common denominators, combining terms, and canceling out common factors – are techniques that will serve you well throughout your mathematical journey. Adding algebraic fractions correctly involves careful attention to signs and factors, ensuring that no errors creep in. Remember that the goal is always to express a mathematical statement in its simplest, most elegant form, making it easier to understand and use.

Conclusion: Mastering Algebraic Simplification

We've successfully navigated the steps to find the sum of algebraic fractions, demystifying the process by filling in the required variables. We found that ${a = -1}$, ${b = -9}$, and ${c = 9}$, leading us to the simplified expression ${\frac{3}{2}}$. The key takeaways here are the importance of finding a common denominator, the careful handling of negative signs, and the power of factoring to reveal common factors for cancellation. Simplifying algebraic fractions is a skill that, with practice, becomes second nature. It allows us to reduce complex expressions to their simplest forms, which is invaluable in problem-solving. Keep practicing these techniques, and you'll find that more advanced mathematical concepts become much more accessible. Remember, every mathematician started with the basics, and mastering these algebraic manipulations is a significant step forward.

For further exploration into algebraic concepts and practice problems, you might find the resources at Khan Academy incredibly helpful. They offer a wide range of lessons and exercises on algebra, including detailed explanations of fractions and algebraic manipulation.