Solving X/3 + X/6 = 3 A Step-by-Step Guide

Hey guys! Let's dive into solving a common algebraic equation: x/3 + x/6 = 3. Math can seem daunting, but breaking it down step-by-step makes it super manageable. In this guide, we’ll walk through each stage of solving this equation, making sure you understand not just the how, but also the why behind each step. Whether you're a student tackling homework or just brushing up on your algebra skills, this guide will help you master this type of problem.

Understanding the Basics

Before we jump into the solution, let’s quickly recap some fundamental algebraic principles. When you're faced with an equation like x/3 + x/6 = 3, the main goal is to isolate the variable x on one side of the equation. This means we want to manipulate the equation in a way that we end up with x = some number. To do this, we'll use inverse operations—operations that undo each other. For example, addition and subtraction are inverse operations, as are multiplication and division. Keeping this in mind will help you navigate through the steps more confidently.

The equation x/3 + x/6 = 3 involves fractions, and dealing with fractions might seem tricky at first. But don't worry, we've got a strategy for that! The key to handling fractions in equations is to find a common denominator. A common denominator is a number that each of the denominators (the bottom numbers in the fractions) can divide into evenly. Once we have a common denominator, we can combine the fractions and simplify the equation. Think of it like this: you can't easily add apples and oranges, but if you convert them both to pieces of fruit, you can add them up. Similarly, we need a common denominator to add the fractions involving x.

Remember, whatever operation you perform on one side of the equation, you must perform the same operation on the other side to maintain balance. It’s like a see-saw; if you add weight to one side, you need to add the same weight to the other side to keep it level. This principle is crucial in algebra because it ensures that the equality remains true throughout the solving process. So, with these basics in mind, let’s get started on solving our equation!

Step 1: Finding the Common Denominator

Okay, so our equation is x/3 + x/6 = 3. The first thing we need to tackle is those fractions. To combine them, we need a common denominator. Look at the denominators we have: 3 and 6. What's the smallest number that both 3 and 6 can divide into evenly? That’s right, it’s 6! So, 6 will be our common denominator. Finding the least common multiple (LCM) is essential here because it simplifies the process and keeps the numbers manageable. If we chose a larger common multiple, like 12, we'd still arrive at the correct answer, but the numbers we'd be working with would be bigger and potentially more complex.

Now that we know our common denominator is 6, we need to convert each fraction to have this denominator. The fraction x/6 already has a denominator of 6, so we can leave it as is. But what about x/3? To change the denominator from 3 to 6, we need to multiply it by 2. Remember, though, that we can't just multiply the denominator; we also have to multiply the numerator (the top number) by the same amount to keep the fraction equivalent. So, we multiply both the numerator and the denominator of x/3 by 2, which gives us (x * 2) / (3 * 2) = 2x/6. Now, both fractions have the same denominator, and we’re one step closer to solving the equation!

This step is crucial because it sets the stage for combining the fractions. Without a common denominator, we can't simply add the numerators. Think of it like trying to add different units; you can't add meters and centimeters directly until you convert them to the same unit. Similarly, we need the fractions to have the same denominator before we can add them. So, with x/3 converted to 2x/6, our equation now looks like 2x/6 + x/6 = 3. We’re ready to move on to the next step, where we’ll combine these fractions and simplify the equation further. Great job so far, guys! You're nailing it!

Step 2: Combining the Fractions

Alright, we've got our common denominator sorted out, and our equation now reads 2x/6 + x/6 = 3. The next step is to combine these fractions. Since they both have the same denominator (which is 6), we can simply add the numerators. Think of it as adding like terms – if you have two of something and you add one more, you have three of that something. In this case, we have 2x and we're adding x to it.

So, 2x + x equals 3x. This means we can rewrite the left side of the equation as 3x/6. The equation now looks like 3x/6 = 3. See how much simpler it's becoming? By finding the common denominator and combining the fractions, we’ve reduced the equation to a more manageable form. This is a key strategy in algebra: simplification. The simpler the equation, the easier it is to solve.

But we're not done simplifying yet! We can actually reduce the fraction 3x/6 further. Notice that both the numerator (3x) and the denominator (6) have a common factor, which is 3. We can divide both the numerator and the denominator by 3 to simplify the fraction. When we divide 3x by 3, we get x. And when we divide 6 by 3, we get 2. So, 3x/6 simplifies to x/2. Now our equation looks even cleaner: x/2 = 3. We're on the home stretch now! Simplifying fractions is a crucial skill in algebra because it not only makes the equation easier to work with but also ensures that we get the solution in its simplest form. In the next step, we’ll isolate x and find the final answer. Keep up the great work; you’re doing fantastic!

Step 3: Isolating the Variable

Okay, guys, we're getting super close to the finish line! Our equation is now x/2 = 3. Our goal, as always, is to isolate x, meaning we want to get x by itself on one side of the equation. Right now, x is being divided by 2. To undo this division, we need to perform the inverse operation, which is multiplication. Remember, what we do to one side of the equation, we must also do to the other side to keep things balanced.

So, we're going to multiply both sides of the equation by 2. On the left side, we have (x/2) * 2. Multiplying x/2 by 2 cancels out the division by 2, leaving us with just x. This is exactly what we wanted! On the right side, we have 3 * 2, which equals 6. So, after multiplying both sides by 2, our equation becomes x = 6. Ta-da! We’ve isolated x and found our solution.

This step highlights the power of using inverse operations to solve equations. By identifying the operation that’s affecting the variable (in this case, division) and performing its inverse (multiplication), we can systematically peel away the layers and reveal the value of x. Isolating the variable is a fundamental technique in algebra, and mastering it will help you solve a wide range of equations. Now that we’ve found our solution, it’s always a good idea to check our work to make sure we haven’t made any mistakes. In the next step, we’ll do just that. You’re doing an awesome job, so let’s keep the momentum going!

Step 4: Checking the Solution

Alright, we've solved the equation and found that x = 6. But before we declare victory, it's super important to check our solution. This step is like the quality control of algebra – it ensures that our answer is correct and that we haven't made any sneaky errors along the way. Checking our solution involves plugging the value we found for x back into the original equation and seeing if it holds true. If it does, we know we've nailed it; if not, we need to go back and look for any mistakes.

Our original equation was x/3 + x/6 = 3. Now, let's substitute x = 6 into this equation. We get 6/3 + 6/6 = 3. Let’s simplify each fraction. 6/3 is equal to 2, and 6/6 is equal to 1. So, the equation now looks like 2 + 1 = 3. And guess what? 2 + 1 does indeed equal 3! This means our solution, x = 6, is correct. High five!

Checking your solution is a crucial habit to develop in algebra. It not only confirms that your answer is right but also helps you catch any errors you might have made in the process. Think of it as a safety net – it’s there to prevent you from confidently submitting a wrong answer. Plus, the act of checking reinforces your understanding of the equation and the steps you took to solve it. So, always make time to check your solution, guys. It’s a small step that makes a big difference. And with that, we’ve successfully solved and checked our equation! You’ve done an amazing job following along, and you’re now one step closer to mastering algebra. Great work!

Conclusion

So, there you have it! We’ve successfully solved the equation x/3 + x/6 = 3, step by step. We started by finding a common denominator, then combined the fractions, simplified the equation, isolated the variable, and finally, checked our solution. Remember, the key to solving algebraic equations is to break them down into manageable steps and understand the principles behind each step. Math might seem intimidating, but with a methodical approach and a bit of practice, you can tackle any equation that comes your way.

We hope this guide has been helpful and has given you a clearer understanding of how to solve equations involving fractions. Keep practicing, keep asking questions, and most importantly, keep believing in yourself. You’ve got this! Whether you're acing your math class or just expanding your problem-solving skills, remember that every step you take is a step forward. Thanks for joining us on this mathematical journey, and we’ll see you next time with more exciting math adventures! Keep up the fantastic work, guys!