Simplifying $\frac{4 W}{w-2}+\frac{3 W}{w-3}$: A Step-by-Step Guide

Hey guys! Today, we're diving into simplifying rational expressions, which might sound intimidating, but it's totally manageable once you break it down. We're going to tackle a specific example: simplifying the expression $\frac{4w}{w-2} + \frac{3w}{w-3}$. This type of problem is common in algebra, and mastering it will definitely boost your math skills. So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into the solution, let's understand what makes this a rational expression. A rational expression is basically a fraction where the numerator and the denominator are polynomials. In our case, we have two rational expressions being added together. Our goal is to combine these into a single, simplified fraction. Think of it like adding regular fractions – you need a common denominator first!

Now, why is simplifying important? Simplifying expressions makes them easier to work with in further calculations. It's like tidying up your workspace before starting a project; a clean and organized expression is much easier to manipulate and understand. Plus, in many real-world applications, simplified forms are essential for getting accurate results. For example, in physics or engineering, you might encounter similar expressions when dealing with rates, ratios, or proportions. Simplifying these expressions can help you model and solve complex problems more efficiently.

To simplify this expression, we'll need to find a common denominator, combine the fractions, and then see if we can simplify the result further. This involves a few key algebraic techniques, like finding the least common multiple (LCM) of the denominators and distributing terms. Don't worry if these terms sound unfamiliar right now; we'll walk through each step together. The key is to take it one step at a time and understand the logic behind each operation. Remember, math is like building blocks – you need a solid foundation to move on to more complex concepts. So, let's lay that foundation by understanding the basic principles of adding rational expressions.

Finding a Common Denominator

Okay, so the first major step in simplifying our expression, $\frac{4w}{w-2} + \frac{3w}{w-3}$, is to find a common denominator. Remember how when you add fractions like $\frac{1}{2} + \frac{1}{3}$, you need to find a common denominator (which would be 6 in that case)? It's the same idea here, just with algebraic expressions. The common denominator allows us to combine the two fractions into one.

So, how do we find this common denominator? In our expression, the denominators are (w - 2) and (w - 3). Since these expressions don't share any common factors, the least common denominator (LCD) is simply their product. This is similar to finding the LCM for numbers – you multiply them together if they don't have any common factors. Therefore, our common denominator is (w - 2)(w - 3).

Now that we have our common denominator, we need to rewrite each fraction with this new denominator. To do this, we'll multiply the numerator and denominator of each fraction by the missing factor from the common denominator. For the first fraction, $\frac{4w}{w-2}$, we'll multiply both the numerator and denominator by (w - 3). This gives us $\frac{4w(w-3)}{(w-2)(w-3)}$. For the second fraction, $\frac{3w}{w-3}$, we'll multiply both the numerator and denominator by (w - 2). This gives us $\frac{3w(w-2)}{(w-2)(w-3)}$.

Why do we multiply both the numerator and denominator? This is a crucial point! When you multiply both the top and bottom of a fraction by the same expression, you're essentially multiplying by 1. This doesn't change the value of the fraction, but it does change its form, allowing us to combine them. It's like renaming a fraction without altering its amount. Now that both fractions have the same denominator, we're one step closer to simplifying the whole expression. Next, we'll focus on combining the numerators and simplifying the resulting expression. So, let's move on to the next step and see how we can put these fractions together!

Combining the Fractions

Alright, we've successfully found our common denominator, (w - 2)(w - 3), and rewritten our fractions. Now comes the fun part: combining these fractions! We're taking the expression $\frac{4w(w-3)}{(w-2)(w-3)} + \frac{3w(w-2)}{(w-2)(w-3)}$ and merging it into a single fraction. This step is pretty straightforward once you have the common denominator sorted out.

Since both fractions now have the same denominator, we can simply add the numerators. This means we'll be adding 4w(w - 3) and 3w(w - 2). So, our expression becomes $\frac{4w(w-3) + 3w(w-2)}{(w-2)(w-3)}$. See how we've placed the sum of the numerators over the common denominator? That's the key to combining fractions!

But we're not done yet! The next step is to simplify the numerator by distributing and combining like terms. This is where our algebraic skills come into play. First, let's distribute: 4w(w - 3) becomes 4w² - 12w, and 3w(w - 2) becomes 3w² - 6w. So, our numerator now looks like 4w² - 12w + 3w² - 6w.

Now, we can combine the like terms. We have 4w² and 3w², which add up to 7w². We also have -12w and -6w, which combine to -18w. So, our simplified numerator is 7w² - 18w. This gives us the combined fraction $\frac{7w^2 - 18w}{(w-2)(w-3)}$. We're getting closer to our simplest form! We've combined the fractions and simplified the numerator. The last step is to simplify the denominator and see if we can reduce the entire fraction further. Let's head to the next section to tackle that!

Simplifying the Numerator and Denominator

Okay, we've reached a crucial stage in simplifying our expression. We've combined the fractions and arrived at $\frac{7w^2 - 18w}{(w-2)(w-3)}$. Now, we need to simplify both the numerator and the denominator as much as possible. This involves factoring, expanding, and looking for any common factors that we can cancel out.

Let's start with the numerator, 7w² - 18w. Can we factor this? Absolutely! Both terms have a common factor of 'w'. We can factor out 'w' to get w(7w - 18). Factoring is super useful because it allows us to see if there are any terms that can be canceled with the denominator.

Now, let's move on to the denominator, (w - 2)(w - 3). This is already in factored form, which is great! However, to see if anything cancels with the numerator after factoring, it's beneficial to expand it. Expanding means multiplying out the terms. So, (w - 2)(w - 3) becomes w² - 3w - 2w + 6, which simplifies to w² - 5w + 6.

So, our expression now looks like $\frac{w(7w - 18)}{w^2 - 5w + 6}$. At this point, it's tempting to try and cancel terms, but we need to be careful! We can only cancel factors that are common to both the numerator and the denominator. Looking at our expression, we don't see any obvious factors that match between w(7w - 18) and w² - 5w + 6. This means we've simplified as much as we can by factoring and expanding.

Now we need to consider the available answer options. Since we couldn't cancel any factors, we'll keep the denominator in its expanded form, w² - 5w + 6, and the numerator as 7w² - 18w (after distributing the 'w' back in). This is a common strategy in simplifying rational expressions: simplify as much as possible, and then compare your result to the given choices.

Choosing the Correct Answer

We've done the hard work of simplifying the expression! We started with $\frac{4w}{w-2} + \frac{3w}{w-3}$, found a common denominator, combined the fractions, and simplified the numerator and denominator. Our simplified expression is $\frac{7w^2 - 18w}{w^2 - 5w + 6}$. Now, it's time to choose the correct answer from the options provided.

Let's quickly recap the options we had:

A. $\frac{7w}{2w-5}$ B. $\frac{7w^2-5}{w^2-5w+6}$ C. $\frac{7w^2-18w}{w^2-5w+6}$ D. $\frac{7w}{w^2+6}$

By comparing our simplified expression, $\frac{7w^2 - 18w}{w^2 - 5w + 6}$, with the given options, we can clearly see that it matches option C. The numerator, 7w² - 18w, is exactly what we got after combining like terms, and the denominator, w² - 5w + 6, is the expanded form of (w - 2)(w - 3).

Why is it important to double-check your answer against the options? Well, sometimes the simplified form you arrive at might look slightly different from the options due to different factoring or expanding choices. So, comparing your result with the options ensures you select the most accurate answer.

We can confidently say that option C, $\frac{7w^2-18w}{w^2-5w+6}$, is the correct simplified form of the given expression. We did it! We successfully navigated through the steps of simplifying rational expressions. Now, let's take a moment to reflect on what we've learned and how these skills can be applied to other problems.

Conclusion: Mastering Rational Expressions

Awesome job, guys! We've reached the end of our journey in simplifying the rational expression $\frac{4w}{w-2} + \frac{3w}{w-3}$. We tackled each step methodically, from finding a common denominator to combining fractions and simplifying the result. We successfully identified the correct answer as option C, $\frac{7w^2-18w}{w^2-5w+6}$.

What are the key takeaways from this problem? First, remember the importance of finding a common denominator when adding or subtracting rational expressions. This is the foundation for combining the fractions. Second, practice your factoring and expanding skills. These are essential for simplifying numerators and denominators and identifying common factors that can be canceled. Finally, always double-check your answer against the given options to ensure you've chosen the correct simplified form.

Simplifying rational expressions might seem challenging at first, but with practice, it becomes second nature. The steps we followed today can be applied to a wide range of similar problems. Think of this as another tool in your math toolbox! These skills are not just useful in algebra; they also pop up in calculus, physics, engineering, and other fields where mathematical modeling is crucial.

So, how can you continue to improve your skills? The best way is to practice! Try working through more examples of simplifying rational expressions. You can find plenty of practice problems in textbooks, online resources, or even create your own. The more you practice, the more confident you'll become in your ability to tackle these types of problems. And remember, if you ever get stuck, don't hesitate to review the steps we covered today or seek help from a teacher, tutor, or online community. Keep up the great work, and happy simplifying!