Solving For K: A Step-by-Step Guide

Hey math enthusiasts! Let's dive into the world of algebra and tackle a common problem: solving for a variable. Specifically, we're going to solve for k in the equation: $2 - k - \frac{2}{5} = 3 + \frac{3}{2} k - \frac{1}{2} k$. Don't worry if it looks a little daunting at first – we'll break it down into easy-to-follow steps. This guide is designed to help you understand the process, build your confidence, and master the art of equation solving. Get ready to flex those math muscles!

Understanding the Basics: Equations and Variables

Before we jump into the equation, let's make sure we're all on the same page. An equation is a mathematical statement that asserts the equality of two expressions. It's like a balanced scale; what's on one side is equal to what's on the other. Our equation, $2 - k - \frac{2}{5} = 3 + \frac{3}{2} k - \frac{1}{2} k$, has two sides separated by an equals sign (=). Our goal is to find the value of the variable k that makes the equation true. The variable k represents an unknown number, and our mission is to uncover its hidden value. The basic principle is to isolate k on one side of the equation and numbers on the other side. This is achieved by performing the same operations on both sides of the equation to maintain the balance.

Think of it this way: the equals sign is the fulcrum of our balance. To keep the scale balanced, any change we make on one side must be mirrored on the other. This principle of balance is fundamental to solving equations. In our case, the variable k appears on both sides of the equation, so we will need to rearrange the equation to have all the terms with k on one side and the constant terms (numbers without k) on the other. We use the properties of equality—addition, subtraction, multiplication, and division—to manipulate the equation while preserving its validity. We can add or subtract the same value from both sides, multiply or divide both sides by the same non-zero value, and the equation will remain true. These are the tools of the trade when it comes to equation solving.

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

Now, let's get down to business and solve for k. We'll break down the process into clear, manageable steps to ensure you follow along with ease. This is the heart of the matter – the actual process of finding the value of k. Each step is crucial, and understanding each one will help build your problem-solving skills and mathematical intuition. Ready? Let's go!

Step 1: Simplify Both Sides

First, let's simplify each side of the equation separately. On the left side, we have $2 - k - \frac{2}{5}$. We can rewrite 2 as $\frac{10}{5}$ to have a common denominator with the fraction. So, the left side becomes $\frac{10}{5} - k - \frac{2}{5}$. Combining the fractions, we get $\frac{8}{5} - k$. On the right side, we have $3 + \frac{3}{2} k - \frac{1}{2} k$. We can combine the k terms: $\frac{3}{2} k - \frac{1}{2} k = \frac{2}{2} k = k$. Thus, the right side becomes $3 + k$. Now our simplified equation is $\frac{8}{5} - k = 3 + k$. This step is all about making the equation cleaner and easier to work with. Remember, the goal is to make the equation simpler to manage. Always simplify each side of the equation before moving on to the next step. By carefully simplifying both sides, we reduce the complexity and minimize the chances of making a mistake later on.

Step 2: Isolate the k Terms

Next, we need to get all the k terms on one side of the equation. Let's add k to both sides. This gives us $\frac{8}{5} - k + k = 3 + k + k$. Simplifying, we get $\frac{8}{5} = 3 + 2k$. Now we have successfully moved all the k terms to the right side of the equation. This is a crucial step towards isolating k. This step is crucial because it consolidates all the terms with k together, setting the stage for the final isolation of the variable. We add k to both sides of the equation. This ensures that the balance of the equation is maintained. By applying this operation, the k term on the left side cancels out, leaving us with a cleaner equation. This step is about grouping like terms. By strategically manipulating the equation, we bring the k terms together, inching closer to our goal.

Step 3: Isolate the Constant Terms

Now, let's isolate the constant terms on one side. Subtract 3 from both sides. This gives us $\frac{8}{5} - 3 = 3 + 2k - 3$. Rewrite 3 as $\frac{15}{5}$ so we can subtract it from $\frac{8}{5}$. Thus, $\frac{8}{5} - \frac{15}{5} = 2k$. Simplifying the left side, we get $-\frac{7}{5} = 2k$. We are inching closer to our solution. We subtract 3 from both sides of the equation. This isolates the terms without k on the left side and keeps the k terms on the right side. This step brings the numbers together. We perform arithmetic operations to simplify the equation. This manipulation puts the constant terms on one side of the equation. This is where we bring together all the numbers on one side of the equation. It's a step-by-step process, and each step helps move closer to the solution.

Step 4: Solve for k

Finally, we solve for k. Divide both sides of the equation by 2. This gives us $-\frac{7}{5} \div 2 = 2k \div 2$. So, $k = -\frac{7}{10}$. And there you have it! We have successfully solved for k! This is the grand finale. We divide both sides of the equation by 2. We isolate the variable k and unveil its value. This is where the solution emerges. We find the value of k by performing this final operation. This step unveils the numerical value of k. We perform a simple division to unveil the solution. The goal is to get k by itself. We have isolated k and have our solution. This final step reveals the answer. With a final division, the value of k is revealed. This is where the culmination of all previous steps becomes clear, and the value of k is revealed.

Conclusion: The Answer Revealed

And there you have it! The solution to the equation $2 - k - \frac{2}{5} = 3 + \frac{3}{2} k - \frac{1}{2} k$ is $k = -\frac{7}{10}$. By following these steps, you've successfully navigated the process of solving for a variable. Remember, practice makes perfect. Keep working through problems, and you'll become more confident in your ability to solve equations. Embrace the challenge, enjoy the process, and celebrate your successes! Solving for a variable is a fundamental skill in mathematics. This guide breaks down the process step by step, making it easy to understand. By following these steps, you can confidently solve for any variable in an equation. With practice, you'll become a pro at solving equations. Now that you've solved for k, consider tackling more complex equations. Keep practicing, and your skills will improve over time. Congratulations, you’ve done it!

Why is Solving for k Important?

Solving for a variable is not just an abstract mathematical exercise; it's a fundamental skill that underpins many areas of life. Whether you are calculating the cost of a purchase, figuring out distances, or analyzing data in science and engineering, the ability to manipulate and solve equations is essential. It's a key skill for various careers in science, technology, engineering, and mathematics (STEM) fields, and it enhances logical thinking and problem-solving abilities applicable in everyday situations. Understanding how to solve for a variable helps in problem-solving in many different fields.

Tips for Success

Here are a few additional tips to help you succeed in solving equations:

• Practice Regularly: The more you practice, the better you'll become. Work through different types of problems to build your skills. Work through various problems to reinforce your skills. Practice improves your understanding and speed.
• Show Your Work: Write down each step. This helps you avoid mistakes and makes it easier to find errors. Writing down steps helps avoid mistakes. Clear steps make it easier to find and correct errors.
• Check Your Answer: Substitute your solution back into the original equation to verify that it's correct. Check your answer to make sure it's right. Verify the result by substituting the solution back into the original equation.
• Stay Organized: Keep your work neat and organized. This reduces the chances of making errors and makes it easier to review your work. Organize your work for clarity. Organized work reduces errors and makes reviewing easier.
• Don't Be Afraid to Ask for Help: If you're stuck, ask your teacher, a friend, or use online resources for help. Seek help when needed. Asking for help is important when struggling.

We hope this guide has helped you understand how to solve for k. Keep practicing, and you'll become a math wizard in no time! Remember to practice regularly to sharpen your skills. With perseverance and practice, you will master equation-solving.

For further information and practice, please visit Khan Academy's Algebra Section. This is an excellent resource for additional practice and explanations.