Solving Direct Variation Find Y When X Is 5

Hey guys! Let's dive into a cool math problem that involves direct variation. Ever heard of it? Don't worry if you haven't; we're going to break it down super easily. Direct variation is just a fancy way of saying that two things are related in a specific way: as one thing changes, the other changes in proportion. In our case, we're looking at how y changes as x changes. We've got a table showing some values of x and y, and our mission is to find out what y is when x is 5. Sounds like a fun challenge, right? So, let's put on our math hats and get started!

Decoding Direct Variation

Okay, so what exactly is direct variation? In the world of mathematics, direct variation describes a relationship between two variables where one is a constant multiple of the other. Think of it like this: if you double one variable, the other variable doubles too. If you triple one, the other triples, and so on. This constant multiple is super important, and we call it the constant of variation, often represented by the letter k. The relationship can be written in a neat little equation: y = kx. This equation is the key to solving our problem. It tells us that y is directly proportional to x, and k is the magic number that connects them. To really nail this down, let's look at our table again. We have pairs of x and y values. If we can figure out what k is for these pairs, we'll be golden. The beauty of direct variation is that this k stays the same no matter which pair of x and y we pick from the table. It's like the secret sauce that makes the relationship tick. So, our first step is to find this k. How do we do that? Well, we just need to pick a pair of x and y values from the table and plug them into our equation. Let's walk through this step-by-step, so it's crystal clear.

Finding the Constant of Variation (k)

Let's grab the first pair of values from our table: x = 1 and y = 3. We're going to use these values and our direct variation equation, y = kx, to solve for k. It's like detective work, but with numbers! So, we substitute y with 3 and x with 1 in the equation. This gives us: 3 = k * 1. Now, this looks pretty simple, right? To find k, we just need to isolate it. In this case, k is already pretty much on its own. Since anything multiplied by 1 is itself, we can see that k = 3. Awesome! We've found our constant of variation. But, just to be sure, let's try another pair of values from the table. Let's pick x = 3 and y = 9. We do the same thing: substitute these values into our equation: 9 = k * 3. Now, to get k by itself, we need to divide both sides of the equation by 3. This gives us 9 / 3 = k, which simplifies to k = 3. Bingo! We got the same value for k again. This confirms that we're on the right track and that the relationship between x and y is indeed a direct variation with a constant of variation k = 3. This is a crucial step, guys, because without k, we can't find the value of y when x is 5. Now that we have k, the fun really begins!

Calculating 'y' When x is 5

Alright, now that we've cracked the code and found our constant of variation, k = 3, we're ready to tackle the main question: What is the value of y when x is 5? This is where our direct variation equation, y = kx, really shines. We know k, we know x, and we just need to plug them in and solve for y. It's like following a recipe – we have all the ingredients, and now we just need to mix them together in the right way. So, let's substitute k with 3 and x with 5 in our equation. This gives us: y = 3 * 5. Now, this is a straightforward multiplication. Three times five is fifteen, so we have y = 15. There you have it! When x is 5, y is 15. We've successfully navigated the world of direct variation and found our answer. But, let's not stop here. It's always a good idea to double-check our work, just to make sure we haven't made any silly mistakes. We can do this by thinking about the relationship between x and y in our table. We noticed that y is always three times x. So, if x is 5, then y should indeed be 3 times 5, which is 15. Perfect! Our answer checks out. We've not only found the value of y, but we've also confirmed that our understanding of direct variation is solid. This is the kind of problem-solving that makes math fun and rewarding. Now, let's recap what we've done and solidify our understanding even further.

Recapping the Steps

Let's take a moment to recap the steps we took to solve this problem. This will help solidify our understanding of direct variation and make sure we can tackle similar problems in the future. First, we understood what direct variation means. We learned that it's a relationship where one variable is a constant multiple of the other, and we met the constant of variation, k. We also got familiar with the equation y = kx, which is the key to unlocking direct variation problems. Next, we focused on finding the constant of variation k. We picked a pair of values from our table (x and y) and plugged them into the equation. We solved for k and, just to be sure, we tried another pair of values to confirm that k was consistent. This is a great practice to ensure accuracy. Once we had k, we moved on to the main event: calculating y when x is 5. We plugged in the values of k and x into our equation and solved for y. We got our answer: y = 15. Finally, we double-checked our work by thinking about the relationship between x and y in the table. We saw that y is always three times x, so our answer made perfect sense. By following these steps, we not only solved the problem but also deepened our understanding of direct variation. Remember, guys, math isn't just about getting the right answer; it's about understanding the process and building a solid foundation for future challenges. Now, let's wrap things up with a final summary of our solution and some key takeaways.

Conclusion: y is 15 When x is 5

So, to wrap it all up, we've successfully found the value of y when x is 5 in our direct variation problem. The answer, as we discovered, is y = 15. We started by understanding the concept of direct variation and the importance of the constant of variation, k. We then used the given table of x and y values to calculate k, which turned out to be 3. With k in hand, we plugged in the value of x = 5 into our equation, y = kx, and solved for y. We even took the extra step of verifying our answer to ensure accuracy. This whole process has been a fantastic journey into the world of direct variation, and we've learned some valuable skills along the way. We now know how to identify a direct variation relationship, find the constant of variation, and use this information to solve for unknown values. These are skills that will come in handy in many different math problems and real-world situations. Remember, direct variation is all about the proportional relationship between two variables, and the equation y = kx is our trusty tool for navigating this relationship. So, the next time you encounter a problem involving direct variation, remember the steps we've discussed, and you'll be well-equipped to tackle it with confidence. Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!

Repair Input Keyword

Given the direct variation relationship between y and x as shown in the table, what is the value of y when x equals 5?