Solving For 'k': A Step-by-Step Guide

Hey everyone! Today, we're diving into a classic algebra problem: solving for k in the equation $\frac{1}{3} + \frac{2}{k} = \frac{13}{3k}$. Don't worry if it looks a little intimidating at first; we'll break it down step by step, making it super easy to understand. This is a fundamental concept in mathematics, so understanding how to solve for a variable in an equation is a super valuable skill, like knowing how to ride a bike – you'll use it everywhere! We'll go through each step carefully, explaining the 'why' behind each move, so you'll not only get the answer but also truly grasp the underlying principles. Get ready to flex those math muscles – let's do this!

Understanding the Problem: The Basics

Alright, guys, before we jump into the nitty-gritty, let's make sure we're all on the same page. The equation $\frac{1}{3} + \frac{2}{k} = \frac{13}{3k}$ is an algebraic equation. Our goal is to find the value of k that makes this equation true. Think of k as a mystery number we're trying to unveil. The key here is to isolate k on one side of the equation. We do this by performing mathematical operations (like adding, subtracting, multiplying, and dividing) on both sides of the equation. The golden rule is: whatever you do to one side, you must do to the other to keep things balanced. This is kinda like a seesaw; if you add weight to one side, you have to add the same weight to the other to keep it level. Also, notice that we have fractions, so we need to be careful with how we handle them. The presence of fractions means we'll likely deal with common denominators, which we will cover in our next section! Remember, patience is key. Take your time, and you'll nail it.

Now, let's recap the important things about solving an equation for 'k'. First, we need to know that 'k' is a variable, an unknown value that we aim to find. Second, to find the value of 'k', we will perform mathematical operations like addition, subtraction, multiplication, and division to both sides of the equation. This is a fundamental concept in algebra. Third, our goal is to isolate 'k' on one side of the equation. Fourth, always remember that whatever operation you perform on one side, you must perform on the other to maintain the equation's balance. Finally, understanding these basics will lay a strong foundation for tackling more complex algebraic problems. Ready? Let's proceed to the next step.

Step-by-Step Solution: Finding the Value of k

Okay, guys, let's get down to business and solve for k! We'll break down the equation $\frac{1}{3} + \frac{2}{k} = \frac{13}{3k}$ into manageable steps. This will make it easier to follow and understand. The first thing we want to do is eliminate those pesky fractions. A common strategy here is to multiply both sides of the equation by the least common denominator (LCD). The denominators we have are 3 and k. So, what is the LCD of 3 and k? It's simply 3k. Now, let's multiply both sides of the equation by 3k. Remember, we have to do this to every term!

So, the equation is $\frac{1}{3} + \frac{2}{k} = \frac{13}{3k}$. Multiplying both sides by 3k, we get: 3k * (\frac1}{3} + \frac{2}{k}) = 3k * (\frac{13}{3k}). Distribute the 3k on the left side (3k * $\frac{1{3}$) + (3k * $\frac{2}{k}$) = 3k * $\frac{13}{3k}$. Simplifying each term: k + 6 = 13. Great job, guys! We've successfully removed the fractions. Now, we are left with a much simpler equation. This step is all about making the equation easier to handle by getting rid of fractions. The next step involves isolating k by performing the appropriate operations. We are getting closer to finding the value of k!

Now, to isolate k, we need to get rid of the +6 on the left side. We can do this by subtracting 6 from both sides of the equation. This is where that 'balancing the equation' rule comes into play. So, we have k + 6 - 6 = 13 - 6. Simplifying this, we get k = 7. Voila! We've found the solution. This is awesome! k equals 7! We’ve successfully solved the equation by systematically applying mathematical operations while keeping the equation balanced. The best way to check that you are right is to plug it back into the original equation! Let’s do it!

Verifying the Solution: Checking Our Work

Alright, we've got our answer: k = 7. But are we absolutely sure? It's always a good practice to check your answer, just to make sure you didn't make any silly mistakes along the way. Think of it as double-checking your work before you hand in the test. To verify, we're going to plug our value of k (which is 7) back into the original equation: $\frac{1}{3} + \frac{2}{k} = \frac{13}{3k}$.

Substitute k with 7: $\frac{1}{3} + \frac{2}{7} = \frac{13}{3*7}$. Now let’s simplify: $\frac{1}{3} + \frac{2}{7} = \frac{13}{21}$. To add $\frac{1}{3}$ and $\frac{2}{7}$, we need a common denominator, which is 21. Rewrite $\frac{1}{3}$ as $\frac{7}{21}$ and $\frac{2}{7}$ as $\frac{6}{21}$. Now our equation is $\frac{7}{21} + \frac{6}{21} = \frac{13}{21}$.

Adding the fractions on the left side, we get $\frac{13}{21} = \frac{13}{21}$. Wow! The left side of the equation equals the right side. This means that our solution, k = 7, is correct. Nice work, everyone! Verifying the solution is super important because it helps you build confidence in your ability to solve the problems. It’s also a fantastic way to catch any sneaky arithmetic errors. Now you know how to solve for k and check your answer. Keep practicing, and you'll become a pro in no time! Remember, practice makes perfect. Keep solving different types of equations, and you'll become more and more comfortable with the process. Keep up the awesome work!

Common Mistakes and How to Avoid Them

Alright, guys, let's talk about some common pitfalls that people run into when solving this type of equation. Recognizing these mistakes is half the battle! One of the most common mistakes is messing up the distribution when multiplying by the least common denominator. Remember, you have to multiply every term on both sides of the equation. Make sure you don't forget any terms! Another mistake is making arithmetic errors, especially with fractions. Take your time, double-check your calculations, and use a calculator if you need to (but make sure you understand the steps!).

Sometimes, students might get confused about when to add, subtract, multiply, or divide. Always remember the goal: to isolate the variable (k in our case). You do this by performing the opposite operation to what’s being done to the variable. For example, if a number is being added to k, you subtract it from both sides. If a number is multiplying k, you divide both sides by that number. Also, don’t forget the negative signs! A common sign error can completely change your answer. Pay very close attention to whether the terms are positive or negative.

Finally, don't be afraid to take your time and break the problem down into smaller steps. Solving for 'k', or any variable, takes practice. The more you work through problems, the better you'll get at recognizing patterns and avoiding these mistakes. Always, always check your answer! This is your safety net. If your answer doesn't make the equation true, then you know you've made a mistake somewhere along the line. Learning from your mistakes is one of the best ways to improve, so embrace them and keep going! You got this!

Conclusion: Mastering the Art of Solving for k

Awesome work, everyone! We've successfully navigated the process of solving for k in the equation $\frac{1}{3} + \frac{2}{k} = \frac{13}{3k}$. You've learned how to isolate the variable, handle fractions, and check your answer. Remember, the key takeaways are: always balance the equation, follow the order of operations, and check your work. These principles apply to all sorts of algebraic equations, not just this one!

Now you're equipped with the skills and knowledge to tackle similar problems with confidence. Keep practicing! The more you solve these types of equations, the more comfortable and confident you'll become. Each problem you solve is a building block in your math journey. Don't be afraid to ask for help if you get stuck. Your teachers, classmates, and online resources are all valuable tools. Keep challenging yourself, and remember, mathematics is a skill that improves with practice. Keep learning, keep growing, and keep exploring the amazing world of mathematics!

I hope this guide was helpful. If you have any questions or want to try another example, just let me know. Happy solving, and keep up the great work, everyone!