Solve (1/2) - (X/3) = X/6: A Step-by-Step Guide

Hey guys! Ever get those math problems that look like a jumbled mess of fractions and variables? Don't worry, we've all been there. Today, we're going to break down a common type of equation that involves fractions and a single variable (that's our 'X'!). We're tackling the equation (1/2) - (X/3) = X/6, and I promise, by the end of this guide, you'll be solving it like a pro. So, grab your pencils, your favorite notebook, and let's dive into the world of algebraic fractions! We will explore every detail, making sure you understand each step. Get ready to transform from a math newbie to a confident equation solver. The best part? This skill isn't just for textbooks; it’s super useful in everyday life, from splitting bills to planning projects. Let’s get started and make math a little less intimidating and a lot more fun!

Understanding the Equation: Setting the Stage

Before we jump into the solution, let's make sure we understand what the equation (1/2) - (X/3) = X/6 is actually telling us. Think of an equation like a balanced scale. The equals sign (=) is the center, and whatever is on one side must be equal to whatever is on the other side. In our case, we have two expressions: (1/2) - (X/3) on the left and X/6 on the right. Our mission is to find the value of 'X' that makes this scale perfectly balanced. In simpler terms, we need to figure out what number 'X' represents so that when we subtract it (divided by 3) from 1/2, we get the same result as when we divide 'X' by 6. Now, let's talk about why this matters. Equations like this pop up everywhere! Imagine you're trying to figure out how to divide a pizza fairly among friends, or maybe you're calculating the discounts at your favorite store. Understanding how to solve equations is a key skill for problem-solving in all sorts of situations. Plus, mastering these basics will set you up for more advanced math concepts later on. We're not just learning to solve one equation here; we're building a foundation for all sorts of mathematical adventures! So, with this foundational understanding, we’re ready to roll up our sleeves and get into the nitty-gritty of solving (1/2) - (X/3) = X/6. Remember, each step we take is a step towards mastering this crucial mathematical skill. Let's do this!

Step 1: Finding the Least Common Multiple (LCM)

The first thing we need to do when dealing with fractions in an equation is to get rid of them! And the best way to do that is by finding the Least Common Multiple (LCM) of the denominators. Denominators, remember, are the bottom numbers of our fractions (2, 3, and 6 in this case). The LCM is the smallest number that all the denominators can divide into evenly. So, why do we need the LCM? Think of it like this: we want to multiply every term in our equation by the same number so that we can cancel out the denominators. The LCM is the perfect number for this job. It ensures that when we multiply, we'll get whole numbers, making our equation much easier to handle. Now, how do we find the LCM of 2, 3, and 6? There are a couple of ways to do it. One way is to list out the multiples of each number until you find a common one: Multiples of 2: 2, 4, 6, 8, 10... Multiples of 3: 3, 6, 9, 12... Multiples of 6: 6, 12, 18... See that? 6 is the smallest number that appears in all three lists. Another method is to use prime factorization, but for these smaller numbers, listing multiples is usually quicker. So, we've found our LCM: it's 6! This is the magic number that will help us clear those fractions and make our equation much friendlier. Now that we have our LCM, we're ready to move on to the next step: multiplying both sides of the equation by this LCM. Get ready, because things are about to get even clearer! By mastering this step, you are not only solving this particular equation, but you are also equipping yourself with a fundamental skill that is applicable across a wide array of mathematical problems. Fantastic!

Step 2: Multiplying Both Sides by the LCM

Alright, now that we've found our LCM (which is 6, remember?), it's time to put it to work! We're going to multiply every term on both sides of the equation by 6. This is a crucial step, and it’s important to do it carefully. Remember that balanced scale we talked about earlier? If we do something to one side, we must do the same thing to the other side to keep it balanced. So, multiplying by 6 on both sides keeps the equation true. Let's break it down. Our equation is (1/2) - (X/3) = X/6. We're going to multiply each term by 6: 6 * (1/2) - 6 * (X/3) = 6 * (X/6) Now, let's simplify each term. 6 * (1/2) is the same as 6 divided by 2, which equals 3. 6 * (X/3) is the same as 6X divided by 3, which simplifies to 2X. And finally, 6 * (X/6) is the same as 6X divided by 6, which simplifies to just X. See how the fractions are disappearing? That's the power of the LCM! After multiplying and simplifying, our equation now looks like this: 3 - 2X = X. Much cleaner, right? We've transformed our equation from a fraction-filled puzzle into a simpler, more manageable form. This step is super important because it makes the rest of the solving process much smoother. We've essentially cleared the path for the final steps, where we'll isolate 'X' and find its value. So, by multiplying by the LCM, we've not only eliminated fractions but also simplified our equation significantly. Now, we're perfectly set up to move on to the next phase of solving for 'X'. You're doing great – keep the momentum going! This is how you conquer those mathematical challenges, one step at a time. Awesome job!

Step 3: Isolating the Variable (X)

Okay, we're making great progress! We've cleared the fractions, and our equation now looks like 3 - 2X = X. The next step is to isolate our variable, 'X'. This means we want to get all the 'X' terms on one side of the equation and all the constant terms (the numbers without 'X') on the other side. Think of it like sorting laundry: we want to group similar items together. In our equation, we have -2X on the left side and X on the right side. Let's move the -2X to the right side. To do this, we'll add 2X to both sides of the equation. Remember, we have to keep the equation balanced! Adding 2X to both sides gives us: 3 - 2X + 2X = X + 2X. On the left side, -2X and +2X cancel each other out, leaving us with just 3. On the right side, X + 2X combines to give us 3X. So, our equation now looks like this: 3 = 3X. We're almost there! We have all the 'X' terms on one side and the constant on the other. Now, we just need to get 'X' completely by itself. To do this, we'll divide both sides of the equation by 3 (the number that's multiplying 'X'). Dividing both sides by 3 gives us: 3 / 3 = (3X) / 3. On the left side, 3 divided by 3 is 1. On the right side, (3X) divided by 3 simplifies to just X. So, we're left with: 1 = X. We did it! We've isolated 'X' and found its value. This step of isolating the variable is a fundamental skill in algebra. It's like being a detective, carefully moving pieces around until you reveal the hidden answer. By mastering this technique, you'll be able to solve a wide range of equations with confidence. Fantastic work!

Step 4: Solving for X

We've reached the exciting part – actually solving for X! After all the hard work of clearing fractions and isolating the variable, we're now at the finish line. In the previous step, we transformed our equation into 1 = X. Guess what? That's it! We've already solved for X. This equation tells us that the value of X is 1. Sometimes, the solution is this straightforward. It's like finding the missing puzzle piece and realizing it fits perfectly. However, it's always a good idea to double-check our answer to make sure it's correct. This is especially important in math, where a small mistake early on can lead to a wrong final answer. To check our solution, we'll substitute X = 1 back into the original equation: (1/2) - (X/3) = X/6. Replacing X with 1, we get: (1/2) - (1/3) = 1/6. Now, we need to see if this statement is true. To subtract fractions, we need a common denominator. The common denominator for 2 and 3 is 6. So, let's rewrite the fractions: (3/6) - (2/6) = 1/6. Now we can subtract: (3/6) - (2/6) = 1/6. So, we have 1/6 = 1/6. This is a true statement! Our solution checks out. We've confirmed that X = 1 is indeed the correct answer. This step of verifying our solution is crucial for building confidence and ensuring accuracy in math. It's like proofreading a document before submitting it – a final check to catch any errors. By solving for X and verifying our solution, we've demonstrated a solid understanding of algebraic equation solving. Congratulations! You've conquered this equation, and you're well on your way to mastering algebra. Keep up the great work!

Step 5: Checking Your Answer (Verification)

We touched on checking our answer in the previous step, but it's so important that it deserves its own dedicated discussion! Checking your answer is like the secret weapon of successful problem solvers. It's the final step that ensures you've not only found a solution but that your solution is correct. Think of it as quality control for your math work. We've solved for X and found that X = 1. But how do we know for sure that this is the right answer? That's where verification comes in. The process is simple: we take our solution (X = 1) and plug it back into the original equation. This is super important – we use the original equation to avoid carrying over any mistakes we might have made along the way. Our original equation is (1/2) - (X/3) = X/6. Substituting X = 1, we get: (1/2) - (1/3) = 1/6. Now, we need to simplify both sides of the equation and see if they are equal. Let's start with the left side: (1/2) - (1/3). To subtract these fractions, we need a common denominator, which is 6. So, we rewrite the fractions: (3/6) - (2/6). Now we can subtract: (3/6) - (2/6) = 1/6. So, the left side simplifies to 1/6. The right side of the equation is already 1/6. So, we have: 1/6 = 1/6. This is a true statement! This confirms that our solution, X = 1, is correct. If the two sides of the equation didn't match after substituting our solution, it would mean we made a mistake somewhere along the way. We would then need to go back and carefully review each step to find the error. Checking your answer is not just about getting the right answer; it's about building confidence in your problem-solving skills. It's about knowing that you've done your work thoroughly and accurately. Make it a habit, and you'll become a math whiz in no time!

Conclusion: Mastering Algebraic Equations

Wow, guys! We've come a long way. We started with a seemingly complicated equation, (1/2) - (X/3) = X/6, and we've successfully solved it, step by step. We found that X = 1, and we even verified our answer to make sure it was correct. But more importantly, we've learned a valuable process for tackling algebraic equations that involve fractions. We've covered finding the Least Common Multiple (LCM), multiplying both sides of the equation by the LCM to clear fractions, isolating the variable, solving for X, and the crucial step of checking our answer. These steps aren't just for this one equation; they're a powerful toolkit that you can use to solve all sorts of similar problems. Think about it: you can now confidently approach equations that used to seem intimidating. You understand the logic behind each step, and you know how to verify your solutions. This is the essence of mathematical mastery – not just memorizing formulas, but truly understanding the process. Remember, math isn't about being perfect; it's about learning and growing. Every equation you solve, every problem you tackle, makes you a stronger and more confident problem solver. So, keep practicing, keep exploring, and keep challenging yourself. The world of mathematics is vast and fascinating, and you've just unlocked another door to explore it further. And remember, if you ever get stuck, don't hesitate to break the problem down into smaller steps, just like we did here. You've got this! This journey through solving (1/2) - (X/3) = X/6 is just the beginning. With the skills you've gained, you're ready to take on new mathematical adventures. So, go out there and conquer those equations! You've earned it!