Adding Fractions With Variables: Step-by-Step Solution

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Hey guys! Today, we're diving into the world of algebra to tackle a common problem: adding fractions that contain variables. Specifically, we're going to break down how to solve the expression: $\frac{3}{x+y} + \frac{4}{x-y} = ?$ This might seem intimidating at first, but don't worry, we'll go through it together, step by step, so you can master this skill. Understanding how to add these types of fractions is crucial for more advanced algebraic manipulations, and it's a fundamental concept in many areas of mathematics and science. So, let's get started and make sure you're confident in handling these problems!

Understanding the Basics of Fraction Addition

Before we jump into the problem, let's quickly refresh the basic principles of adding fractions. You see, adding fractions with different denominators requires a crucial first step: finding a common denominator. This common denominator allows us to express both fractions with the same "size of pieces," making it possible to combine them. Think of it like trying to add apples and oranges – you need a common unit (like β€œfruit”) to make the addition meaningful. So, to effectively add these fractions, especially when variables are involved, it's important to remember the fundamental steps. Essentially, we need to transform the fractions so they share the same denominator, which then lets us simply add the numerators. This process ensures that we're adding comparable quantities, leading us to the correct and simplified result. Once we've found that common ground, we can then add the numerators while keeping the denominator the same. This gives us a single fraction that represents the sum of the original two. But what happens when variables enter the picture? Well, the same principles apply, but we need to be a little more careful with our algebraic manipulations. The key is to identify the common denominator, which will often involve expressions with variables, and then adjust the numerators accordingly. This might sound complex, but we'll break it down in this example so it becomes clear and manageable for you. Remember, mastering this skill is essential for tackling more advanced algebraic concepts, so let's dive in and make sure we understand each step thoroughly!

Finding the Common Denominator

Our fractions are 3x+y\frac{3}{x+y} and 4xβˆ’y\frac{4}{x-y}. Notice that the denominators are (x+y) and (x-y). To add these fractions, we need a common denominator. The least common denominator (LCD) here is simply the product of these two denominators: (x+y)(x-y). Think of it this way: if you were adding fractions like 1/2 and 1/3, the common denominator would be 2 * 3 = 6. We're doing the same thing here, just with algebraic expressions. Identifying the correct common denominator is a critical step because it sets the stage for correctly adding the fractions. If we don't find the LCD, we won't be able to combine the numerators in a meaningful way. In this particular case, the denominators (x+y) and (x-y) don't share any common factors, so their product is indeed the simplest common denominator we can use. This is a common scenario when dealing with algebraic fractions, especially those involving sums and differences of variables. Once we've established the common denominator, our next task is to rewrite each fraction so that it has this denominator. This involves multiplying both the numerator and the denominator of each fraction by the appropriate factor, which we'll cover in the next section. Keep in mind that finding the common denominator is half the battle in adding fractions, so make sure you're comfortable with this step before moving on. It's a foundational concept that will help you tackle more complex algebraic problems with confidence. So, with our LCD identified, we're well-prepared to proceed to the next step in solving this problem!

Rewriting the Fractions

Now that we've identified our common denominator as (x+y)(x-y), we need to rewrite each fraction with this new denominator. Let's start with the first fraction, 3x+y\frac{3}{x+y}. To get the denominator (x+y) to become (x+y)(x-y), we need to multiply it by (x-y). But remember, to keep the fraction equivalent, we must also multiply the numerator by the same factor. So, we multiply both the numerator and denominator by (x-y):

3x+yβˆ—xβˆ’yxβˆ’y=3(xβˆ’y)(x+y)(xβˆ’y)\frac{3}{x+y} * \frac{x-y}{x-y} = \frac{3(x-y)}{(x+y)(x-y)}

Next, let's do the same for the second fraction, 4xβˆ’y\frac{4}{x-y}. To get the denominator (x-y) to become (x+y)(x-y), we need to multiply it by (x+y). Again, we multiply both the numerator and denominator by (x+y):

4xβˆ’yβˆ—x+yx+y=4(x+y)(x+y)(xβˆ’y)\frac{4}{x-y} * \frac{x+y}{x+y} = \frac{4(x+y)}{(x+y)(x-y)}

Now, both fractions have the same denominator! Rewriting the fractions in this way is absolutely crucial because it allows us to combine them in the next step. Without a common denominator, we simply can't add the numerators together meaningfully. This process ensures that we're adding like terms, which is a fundamental principle in algebra. You see, by multiplying both the numerator and the denominator by the same factor, we're essentially multiplying the fraction by 1, which doesn't change its value. We're just changing the way it looks. This technique is a cornerstone of working with fractions, and it's something you'll use again and again in algebra and beyond. So, make sure you grasp the concept of rewriting fractions with a common denominator – it's a skill that will serve you well. With our fractions now sharing a common denominator, we're perfectly set up to move on to the next phase: adding the numerators!

Adding the Numerators

Okay, now we have our fractions rewritten with the common denominator (x+y)(x-y). They look like this:

3(xβˆ’y)(x+y)(xβˆ’y)+4(x+y)(x+y)(xβˆ’y)\frac{3(x-y)}{(x+y)(x-y)} + \frac{4(x+y)}{(x+y)(x-y)}

Since the denominators are the same, we can now add the numerators. This is the exciting part where we actually combine the fractions into one! We simply add the expressions in the numerators together, keeping the common denominator:

3(xβˆ’y)+4(x+y)(x+y)(xβˆ’y)\frac{3(x-y) + 4(x+y)}{(x+y)(x-y)}

Now, let's simplify the numerator by distributing and combining like terms. First, we distribute the 3 and the 4:

3xβˆ’3y+4x+4y(x+y)(xβˆ’y)\frac{3x - 3y + 4x + 4y}{(x+y)(x-y)}

Next, we combine the like terms (the x terms and the y terms):

(3x+4x)+(βˆ’3y+4y)(x+y)(xβˆ’y)=7x+y(x+y)(xβˆ’y)\frac{(3x + 4x) + (-3y + 4y)}{(x+y)(x-y)} = \frac{7x + y}{(x+y)(x-y)}

Adding the numerators is a key step in simplifying the expression, and it directly follows the principle of adding fractions with a common denominator. The careful distribution and combination of like terms in the numerator is essential to arriving at the correct simplified form. This process shows how important it is to pay attention to details and algebraic rules when working with fractions. By correctly adding the numerators, we've transformed the two separate fractions into a single fraction that represents their sum. Now, our next step is to see if we can simplify this fraction any further. This might involve factoring, canceling common factors, or other algebraic manipulations. So, we're not quite finished yet, but we've made excellent progress. Let's move on to the final step of simplifying the result to its simplest form!

Simplifying the Result

We've arrived at the fraction 7x+y(x+y)(xβˆ’y)\frac{7x + y}{(x+y)(x-y)}. Now, we need to see if we can simplify this any further. Look at the numerator, 7x + y. There aren't any common factors we can pull out, and it doesn't seem to factor in any obvious way. Now let’s consider the denominator, (x+y)(x-y). This looks familiar, doesn't it? It's actually a difference of squares pattern! We know that (a+b)(a-b) = a^2 - b^2. So, we can rewrite the denominator as x^2 - y^2.

Our fraction now looks like this:

7x+yx2βˆ’y2\frac{7x + y}{x^2 - y^2}

Now, we need to ask ourselves: can we simplify this fraction any further? Are there any common factors between the numerator (7x + y) and the denominator (x^2 - y^2)? In this case, the answer is no. The numerator and denominator don't share any common factors, so we can't simplify the fraction any further. This means we've reached the simplest form of our answer! Simplifying the result is crucial because it ensures that we've expressed our answer in the most concise and understandable way possible. Recognizing patterns, like the difference of squares in the denominator, is a powerful tool in simplification. It's also important to check whether any further simplification is possible by looking for common factors between the numerator and denominator. In this case, since we can't find any, we can confidently say that our final answer is in its simplest form. So, after all the steps we've taken – finding the common denominator, rewriting the fractions, adding the numerators, and simplifying – we've successfully added the fractions and arrived at the final solution. Give yourself a pat on the back!

The Final Answer

Therefore, the simplified sum of the fractions is:

7x+yx2βˆ’y2\frac{7x + y}{x^2 - y^2}

Or, equivalently:

7x+y(x+y)(xβˆ’y)\frac{7x + y}{(x+y)(x-y)}

Both forms are correct, and the second form can be useful because it shows the factored form of the denominator. So, there you have it! We've successfully added the fractions 3x+y\frac{3}{x+y} and 4xβˆ’y\frac{4}{x-y}. You can see how breaking down the problem into smaller, manageable steps makes it much easier to solve. Remember the key steps: find the common denominator, rewrite the fractions, add the numerators, and simplify the result. By mastering these steps, you'll be well-equipped to tackle all sorts of fraction addition problems in algebra. And that’s a big win! This is the final step, and it's always satisfying to arrive at the solution. Reviewing the steps we took can reinforce your understanding and help you apply these techniques to future problems. Remember, practice makes perfect, so try working through similar examples to build your confidence and skills. You've got this! Whether you present the answer with the denominator factored or expanded, the important thing is that you understand the process and can apply it effectively. So, keep practicing and exploring algebraic concepts – you'll be amazed at how much you can achieve! And that's it for today's lesson on adding fractions with variables. I hope you found this guide helpful and that you're now more confident in your ability to solve these types of problems. Keep up the great work, and I'll see you next time for more algebra adventures!