Solving Fraction Algebra: A Step-by-Step Guide

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Hey guys! Let's dive into a common algebra problem that often pops up: solving the sum of two fractions with variables. Specifically, we're going to break down how to find the result of the expression: (2x - 1) / 4 + (4x + 2) / 3. Don't worry, it might seem a bit tricky at first, but I'll guide you through it step by step, making it super easy to understand. This is a fundamental concept, so getting a solid grasp of it will really help you with future math problems. We'll find the correct answer from the provided options, showing exactly how to get there. Ready? Let's get started!

Understanding the Problem: The Core of the Math

Okay, so the core of our problem is to add two algebraic fractions. Before we jump into the calculation, let's make sure we're all on the same page. Adding fractions, in general, requires a common denominator. This is the foundation for our operation. Remember, fractions are just parts of a whole, and to add them, the 'wholes' must be divided into the same number of parts. In our case, the denominators are 4 and 3. So, before we can add the numerators, we need to convert both fractions to have the same denominator. This part is crucial, so pay close attention. It involves finding the Least Common Multiple (LCM) of the denominators, which in our case is pretty straightforward. Once we have the common denominator, we'll adjust the numerators accordingly, and then it's smooth sailing. The whole point here is to make sure we're combining apples with apples – or in our case, fourths with fourths, and thirds with thirds.

Finding the Least Common Multiple (LCM)

To add fractions, we first need to find the Least Common Multiple (LCM) of the denominators, which are 4 and 3. The LCM is the smallest number that both denominators can divide into evenly.

  • For the number 4, the multiples are: 4, 8, 12, 16, …
  • For the number 3, the multiples are: 3, 6, 9, 12, 15, …

The smallest number that appears in both lists is 12. Therefore, the LCM of 4 and 3 is 12. This means that 12 will be our common denominator.

Converting the Fractions

Now, we'll convert each fraction so that it has a denominator of 12. Let's start with the first fraction, (2x - 1) / 4.

To get a denominator of 12, we multiply both the numerator and the denominator by 3:

  • ((2x - 1) / 4) * (3 / 3) = (6x - 3) / 12

Next, let's convert the second fraction, (4x + 2) / 3.

To get a denominator of 12, we multiply both the numerator and the denominator by 4:

  • ((4x + 2) / 3) * (4 / 4) = (16x + 8) / 12

So, now our problem looks like this: (6x - 3) / 12 + (16x + 8) / 12. We've done the hardest part, believe me!

Step-by-Step Calculation: Making it Simple

Alright, now that we have our fractions with a common denominator, we can add them. This part is pretty straightforward. You just need to add the numerators while keeping the common denominator the same. Think of it like this: you're just adding the top parts of the fractions, and the bottom part (the denominator) stays put. This keeps everything consistent and correct. This step is all about combining like terms in the numerators. Remember that we're dealing with algebraic expressions, which means we'll have variables (like x) and constants (just numbers). We need to combine the 'x' terms together and the constant terms together. It's like sorting things – 'x' with 'x', and numbers with numbers. By the end of this step, we'll have a single, simplified fraction representing the sum of our original fractions. And then, we're almost there! Let's get to it!

Adding the Numerators

Now, we add the numerators of the fractions we converted: (6x - 3) / 12 + (16x + 8) / 12.

When you add fractions with the same denominator, you add the numerators and keep the denominator.

So, (6x - 3) + (16x + 8) = 6x - 3 + 16x + 8.

Combining Like Terms

Next, we'll combine like terms in the numerator. Like terms are terms that have the same variable raised to the same power (or are both constants).

  • Combine the 'x' terms: 6x + 16x = 22x
  • Combine the constant terms: -3 + 8 = 5

So, our numerator becomes 22x + 5.

The Final Fraction

Now, we put it all together. The sum of the two fractions is (22x + 5) / 12.

Selecting the Correct Answer: Identifying the Solution

Okay, guys, now that we've crunched all the numbers, it's time to see where our answer fits in with the choices given. This is where we verify if our calculations are spot-on. Carefully compare our final answer with the provided options (A, B, C, and D) to find the perfect match. This step is all about confirming our hard work. Making sure we select the correct answer helps solidify our understanding of the whole process. There are plenty of opportunities to make a small error, so double-checking the answer is always a good idea. This helps make sure you understood every part of the question. You can use this as a chance to reflect on your problem-solving skills and see where you can improve for similar math problems. Let’s do this!

Comparing with the Options

  • We found the result to be (22x + 5) / 12.

Now, let's compare this with the options provided:

  • A. (22x + 5) / 12
  • B. (22x - 5) / 12
  • C. (-22x - 5) / 12
  • D. (-22x + 5) / 12

The correct answer is A. (22x + 5) / 12. We did it, guys! We successfully added the fractions and found the right answer from the options.

Conclusion: Mastering Fraction Addition

Congratulations, guys! We've successfully navigated the process of adding algebraic fractions. We started by finding the common denominator, converting the fractions, adding the numerators, and simplifying the expression. Remember, this is a fundamental skill in algebra, and with practice, you'll become more confident in solving these types of problems. Remember to keep practicing and always double-check your work, particularly when dealing with negative signs and combining like terms. You are now equipped with the tools to tackle similar problems. Keep up the awesome work!

Key Takeaways

  • Finding the Common Denominator: The first and most crucial step is to identify the common denominator. This sets the stage for the rest of the calculation.
  • Converting Fractions: Ensure you convert the fractions correctly, so they all share the same denominator.
  • Combining Like Terms: This step ensures that the expressions are simplified and easy to understand.
  • Double-Check Your Work: Always review your steps to make sure that there are no mistakes. Even the smallest of details can make the difference between a correct and an incorrect answer.

Keep practicing, and you'll be acing these problems in no time. See ya!