Solving For X: A Step-by-Step Guide

Hey guys! Ever stumbled upon an equation that looks a bit intimidating but is actually quite simple once you break it down? Today, we're going to tackle one such equation: 3/2 + 1/3 = 1/x. This falls squarely into the realm of mathematics, specifically algebraic equations involving fractions. Don't worry if fractions make you sweat a little – we'll go through this step-by-step, making it super clear and easy to follow. This isn't just about getting the right answer; it’s about understanding the process so you can confidently solve similar problems in the future. We'll cover everything from finding a common denominator to isolating the variable, ensuring you grasp each concept along the way. So, grab your pencils, your thinking caps, and let's dive in! Understanding the fundamentals of fraction manipulation and equation solving is crucial, not just for math class, but for numerous real-life applications as well. Think about scaling recipes, calculating proportions, or even understanding financial ratios – these all rely on the basic principles we'll be exploring today. Let's turn that initial intimidation into a feeling of accomplishment and mastery!

1. Finding a Common Denominator

The very first hurdle in solving our equation, 3/2 + 1/3 = 1/x, is dealing with those pesky fractions on the left side. We can't simply add them as they are because they have different denominators (the bottom numbers). Remember, fractions represent parts of a whole, and we can only add parts if they're based on the same whole – hence the need for a common denominator. So, what exactly is a common denominator? It's simply a number that both denominators can divide into evenly. In our case, we have denominators of 2 and 3. The easiest way to find a common denominator is often to multiply the two denominators together: 2 * 3 = 6. Six works perfectly because both 2 and 3 divide into it without leaving a remainder. Now, we need to rewrite each fraction with this new denominator. This is where the magic happens! To convert 3/2 to an equivalent fraction with a denominator of 6, we need to figure out what we multiplied the original denominator (2) by to get 6. The answer is 3 (2 * 3 = 6). So, we also multiply the numerator (3) by 3: 3 * 3 = 9. This gives us the equivalent fraction 9/6. Similarly, for 1/3, we multiplied the denominator (3) by 2 to get 6 (3 * 2 = 6). Therefore, we multiply the numerator (1) by 2 as well: 1 * 2 = 2. This gives us the equivalent fraction 2/6. Now our equation looks like this: 9/6 + 2/6 = 1/x. See how we’ve transformed the equation into something much more manageable? We've successfully found a common denominator and rewritten our fractions, setting the stage for the next step in solving for x. This skill of finding common denominators is fundamental in fraction arithmetic, and mastering it will make your life so much easier when dealing with equations like this. So, take a deep breath, pat yourself on the back for getting this far, and let's move on!

2. Adding the Fractions

Now that we've successfully navigated the common denominator waters, we're ready to add those fractions on the left side of our equation: 9/6 + 2/6 = 1/x. Remember, the beauty of having a common denominator is that we can now directly add the numerators (the top numbers) while keeping the denominator the same. It's like adding slices of the same-sized pie – if you have 9 slices and then get 2 more, you have a total of 11 slices! So, 9 + 2 = 11. Therefore, 9/6 + 2/6 equals 11/6. Our equation now looks even simpler: 11/6 = 1/x. We've effectively combined the two fractions on the left side into a single, manageable fraction. This is a significant step forward in isolating x and finding its value. Think of it like simplifying a complex puzzle – we're gradually reducing the number of pieces, making the solution clearer and clearer. The ability to add fractions with a common denominator is another cornerstone of fraction arithmetic, and it's essential for solving a wide variety of mathematical problems. This step demonstrates the power of simplification – by combining terms, we move closer to our goal. Aren't you feeling more confident already? We're making great progress, guys! Let's keep that momentum going and tackle the next step.

3. Isolating x Using Reciprocals

We're getting closer to cracking the code! Our equation currently stands at 11/6 = 1/x. The goal, as always, is to isolate x, meaning we want to get x all by itself on one side of the equation. This might seem a bit tricky since x is currently in the denominator of a fraction. But don't worry, we have a powerful tool at our disposal: reciprocals. The reciprocal of a fraction is simply the fraction flipped upside down. For example, the reciprocal of 2/3 is 3/2. The magic of reciprocals comes into play when we multiply a fraction by its reciprocal – the result is always 1. And that's exactly what we're going to use to our advantage. To isolate x, we'll take the reciprocal of both sides of the equation. This is crucial – we must perform the same operation on both sides to maintain the balance of the equation. It's like a seesaw; if you add weight to one side, you need to add the same weight to the other side to keep it level. The reciprocal of 11/6 is 6/11. The reciprocal of 1/x is x/1, which is simply x. So, taking the reciprocal of both sides gives us: 6/11 = x. Boom! We've done it! We've isolated x and found its value. This step highlights a key principle in algebra: performing the same operation on both sides of an equation maintains its equality. And reciprocals are a fantastic tool for dealing with fractions and isolating variables. You've successfully navigated this crucial step, guys. Give yourselves a round of applause!

4. The Solution

After all our hard work, we've arrived at the solution! By finding a common denominator, adding the fractions, and using reciprocals to isolate x, we've determined that x = 6/11. That's it! We've successfully solved the equation 3/2 + 1/3 = 1/x. But before we celebrate too wildly, it's always a good idea to double-check our answer. This is a crucial step in problem-solving – it ensures we haven't made any silly mistakes along the way. To check our answer, we'll substitute x = 6/11 back into the original equation and see if it holds true. So, we have: 3/2 + 1/3 = 1 / (6/11). Dividing by a fraction is the same as multiplying by its reciprocal, so 1 / (6/11) becomes 1 * (11/6), which equals 11/6. Now, let's revisit the left side of the equation: 3/2 + 1/3. We already know from our earlier work that this equals 11/6. Therefore, we have 11/6 = 11/6. The equation holds true! Our solution is correct. We can confidently say that the value of x that satisfies the equation 3/2 + 1/3 = 1/x is indeed 6/11. This verification step underscores the importance of accuracy and attention to detail in mathematics. It's not enough to just get an answer; we need to be sure it's the right answer. And we've done just that! You've conquered this equation, guys. You've demonstrated your understanding of fractions, reciprocals, and equation solving. This is a fantastic achievement!

5. Key Takeaways and Practice

So, what have we learned on this mathematical adventure? We've successfully solved the equation 3/2 + 1/3 = 1/x, and more importantly, we've reinforced some fundamental concepts in algebra and fraction arithmetic. Let's recap the key steps we took:

1. Finding a Common Denominator: We learned how to find a common denominator to add fractions with different denominators. This involved identifying a common multiple of the denominators and rewriting each fraction with that new denominator.
2. Adding the Fractions: Once we had a common denominator, we could simply add the numerators while keeping the denominator the same. This simplified the left side of the equation.
3. Isolating x Using Reciprocals: We discovered the power of reciprocals in isolating variables. By taking the reciprocal of both sides of the equation, we effectively flipped the fractions and brought x out of the denominator.
4. Verification: We emphasized the importance of checking our solution by substituting it back into the original equation. This ensured the accuracy of our answer.

These steps are not just specific to this equation; they represent a general approach to solving algebraic equations involving fractions. By mastering these techniques, you'll be well-equipped to tackle a wide range of mathematical challenges. Now, the best way to solidify your understanding is through practice. Try solving similar equations on your own. For example:

• Solve for x: 1/4 + 2/5 = 1/x
• Solve for y: 2/3 + 1/6 = 1/y
• Solve for z: 5/8 + 1/2 = 1/z

Work through these problems step-by-step, applying the techniques we've discussed. Don't be afraid to make mistakes – they're part of the learning process! The more you practice, the more confident you'll become in your problem-solving abilities. Remember, guys, mathematics is not just about memorizing formulas; it's about understanding concepts and developing logical thinking skills. You've taken a significant step in that direction today. Keep up the great work, and happy solving!