Solving Proportions: X - 1/4 = 2x + 1/3 - Step-by-Step Guide

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Hey guys! Today, we're diving into the exciting world of proportions and tackling a common problem: solving equations where you've got variables mixed with fractions. Specifically, we’re going to break down how to solve the equation x - 1/4 = 2x + 1/3. This type of problem might seem a little intimidating at first, but trust me, with a few simple steps, you’ll be solving these like a pro. We’ll go through each stage methodically, ensuring you understand not just the how, but also the why behind each step. Plus, we'll consider the given options (A) x = 1/2, (B) x = 1, (C) x = 3/4, and (D) x = 2, to figure out which one is the actual solution. Ready to get started? Let's jump right in!

Understanding the Problem

Before we dive into the solution, let’s make sure we understand what we're dealing with. The equation x - 1/4 = 2x + 1/3 is a proportion problem, but it’s also a linear equation because the highest power of our variable x is 1. The key to solving these types of equations is to isolate x on one side of the equation. This means we need to get all the terms with x on one side and all the constant terms (the numbers) on the other. Now, why is this important? Isolating x allows us to clearly see what value of x makes the equation true. It's like finding the missing piece of a puzzle! This involves several algebraic manipulations, like adding and subtracting terms, and sometimes multiplying or dividing both sides of the equation by the same number. These operations maintain the balance of the equation, ensuring that the left side remains equal to the right side throughout the solving process. Think of it like a seesaw – if you add or remove weight from one side, you need to do the same on the other side to keep it balanced. By understanding this fundamental principle, we can confidently move forward in solving the equation.

Step 1: Eliminate the Fractions

Fractions can sometimes make equations look more complicated than they actually are. So, our first step is to get rid of them! To do this, we'll find the least common multiple (LCM) of the denominators, which in our case are 4 and 3. The LCM of 4 and 3 is 12. Why the LCM? Well, multiplying both sides of the equation by the LCM will allow us to cancel out the denominators, effectively eliminating the fractions. It’s a neat trick that simplifies the equation and makes it much easier to work with. Now, let's multiply both sides of the equation x - 1/4 = 2x + 1/3 by 12. This gives us: 12 * (x - 1/4) = 12 * (2x + 1/3). Next, we distribute the 12 to each term inside the parentheses on both sides: 12 * x - 12 * (1/4) = 12 * 2x + 12 * (1/3). This simplifies to 12x - 3 = 24x + 4. See how much cleaner the equation looks now? No more fractions! This step is crucial because it transforms the equation into a form that is easier to manipulate and solve. We’re one step closer to finding the value of x!

Step 2: Group Like Terms

Now that we've eliminated the fractions, the next step is to group the like terms together. This means we want to get all the x terms on one side of the equation and all the constant terms on the other side. This is like sorting your laundry – putting all the socks together, all the shirts together, and so on. In our equation, 12x - 3 = 24x + 4, we have x terms (12x and 24x) and constant terms (-3 and 4). To group these, we can subtract 12x from both sides of the equation. This will move the x term from the left side to the right side. Doing so, we get: 12x - 3 - 12x = 24x + 4 - 12x, which simplifies to -3 = 12x + 4. Next, we want to move the constant term 4 from the right side to the left side. We can do this by subtracting 4 from both sides: -3 - 4 = 12x + 4 - 4, which simplifies to -7 = 12x. By grouping like terms, we’ve simplified the equation and made it much clearer how to isolate x. This step is all about organization and setting ourselves up for the final step of solving for x.

Step 3: Isolate x

We're almost there! The final step is to isolate x. Remember, our goal is to get x by itself on one side of the equation. Right now, we have -7 = 12x. This means 12 is multiplying x. To undo this multiplication and isolate x, we need to do the opposite operation, which is division. We'll divide both sides of the equation by 12. This gives us: -7 / 12 = (12x) / 12. On the right side, the 12s cancel each other out, leaving us with just x. So, we have: x = -7/12. And there you have it! We’ve successfully isolated x and found the solution to the equation. This final step is the culmination of all our hard work, bringing us to the answer we were looking for. By isolating the variable, we've revealed its value and solved the equation.

Step 4: Check the Solution

Before we declare victory, it’s always a good idea to check our solution. This is like proofreading your work before submitting it. It helps catch any mistakes and ensures that our answer is correct. To check our solution, we substitute x = -7/12 back into the original equation, x - 1/4 = 2x + 1/3. This means we replace every instance of x in the equation with -7/12. So, the equation becomes: (-7/12) - 1/4 = 2*(-7/12) + 1/3. Now, we simplify both sides of the equation separately. On the left side, we need a common denominator to subtract the fractions. The LCM of 12 and 4 is 12, so we rewrite 1/4 as 3/12. The left side becomes: -7/12 - 3/12 = -10/12, which simplifies to -5/6. On the right side, we first multiply 2 by -7/12, which gives us -14/12. Then, we need a common denominator to add the fractions. The LCM of 12 and 3 is 12, so we rewrite 1/3 as 4/12. The right side becomes: -14/12 + 4/12 = -10/12, which simplifies to -5/6. Now, we compare both sides of the equation. We have -5/6 = -5/6. Since both sides are equal, our solution x = -7/12 is correct! Checking our solution is a crucial step because it confirms that we haven’t made any errors along the way. It gives us confidence in our answer and ensures that we’ve truly solved the problem.

Step 5: Determine the Correct Alternative

Now that we've found the solution x = -7/12, let's look back at the given alternatives: (A) x = 1/2 (B) x = 1 (C) x = 3/4 (D) x = 2. Clearly, our solution x = -7/12 does not match any of these options. This is an important observation! It tells us that the correct answer is not among the provided choices. In a real-world scenario, this might indicate a mistake in the options or a need to re-evaluate the problem statement. Sometimes, multiple-choice questions might include distractors, which are incorrect answers designed to mislead you. However, in this case, our systematic approach has led us to the correct solution, even though it's not listed. This highlights the importance of understanding the process rather than simply trying to match an answer to the options. By solving the equation ourselves, we've gained a deeper understanding of the problem and can confidently say that none of the given alternatives are correct. This reinforces the value of critical thinking and problem-solving skills in mathematics.

Conclusion

So, to wrap things up, we’ve successfully solved the equation x - 1/4 = 2x + 1/3 by following a step-by-step approach. We eliminated the fractions, grouped like terms, isolated x, and checked our solution. We found that x = -7/12, which wasn't among the provided alternatives. Remember, guys, the key to mastering these types of problems is practice and understanding the underlying principles. Don’t be afraid to break down complex equations into smaller, manageable steps. Each step we took – from eliminating fractions to isolating x – was a building block towards the final solution. And remember, checking your answer is just as important as solving the equation itself! It’s like the final polish on a masterpiece. Keep practicing, and you’ll become a proportion-solving pro in no time! If you have any more questions or want to tackle another equation, just let me know. Happy solving!