Direct Variation: Find Y When X = 2

Hey guys! Let's break down this direct variation problem step by step. Direct variation problems are super common in math, and once you get the hang of them, they're actually pretty straightforward. This article will walk you through how to solve this type of question, making sure you understand each step along the way.

Understanding Direct Variation

When we say that y varies directly as x, it means that y is proportional to x. Mathematically, we write this relationship as:

${y = kx}$

where k is the constant of variation. This constant tells us how y changes with respect to x. In simpler terms, it's the factor by which x is multiplied to get y. Finding this constant is the key to solving direct variation problems. In our case, understanding direct variation is the foundational piece that allows us to solve the problem efficiently and accurately. Without grasping this concept, the subsequent steps would lack context, potentially leading to confusion. With direct variation, it is important to ensure that you fully understand the formula behind the variation. Understanding the formula allows you to determine the value of a given variable. Remember, direct variation problems will have a k constant that you must solve for. Once you have solved for this k constant, you can solve for any other value of the equation. To succeed, you need to fully understand the equation and how to manipulate it, using basic algebra skills, in order to succeed.

Finding the Constant of Variation

We're given that y = 48 when x = 6. We can use this information to find the constant of variation, k. Plug these values into our equation:

${48 = k(6)}$

To solve for k, divide both sides by 6:

${k = \frac{48}{6} = 8}$

So, the constant of variation, k, is 8. This means that y is always 8 times x. Finding this k constant is important. We can say that finding the constant of variation is similar to using a recipe. In a recipe, if you understand all of the measurements, you can use the recipe to make as little or as much as possible. The k constant is a direct recipe for all the values of x and y in the equation. When you have a strong handle on how to find the constant of variation, you have mastered direct variation problems. Direct variation problems will have different constants. Keep in mind that even though the approach to solving the problems will be the same, do not expect that all of your answers are the same. The constant of variation is based on how y related to x in the equation, so make sure you understand how to find it, and you will be set for success.

Setting Up the Equation

Now that we know k = 8, we can write the specific equation for this relationship:

${y = 8x}$

This equation tells us exactly how y and x are related in this problem. Setting up the equation is one of the most important things you can do to ensure you are on your way to success. Some tips in setting up the equation include, but are not limited to, re-reading the problem, checking your work, and ensuring you have the correct terms in your equation. It is important to not rush, but also not second guess yourself. If you work through it, and believe you have found the correct equation, work through your problem. If you get stuck later in the problem, this is a good time to check your equation. This is because you may have set up something improperly that you might have missed originally. Setting up the equation is a critical step to ensuring that you can solve the problem correctly. I like to think of this as ensuring your foundation is built correctly before you continue to build your house. If the foundation is not set up correctly, you are going to have a bad time.

Finding y when x = 2

We want to find the value of y when x = 2. We simply plug x = 2 into our equation:

${y = 8(2) = 16}$

So, when x is 2, y is 16. Finding y when x = 2 is a simple process if you follow all the steps. Make sure you keep track of your equation and the values you have found for both the k constant and the x value. It is easy to overlook these values. When solving any math problem, writing down your work is an excellent way to ensure that you are able to track with your solution. It is also an excellent way to check your work when trying to find errors in the problem. Finding y when x = 2 can be done with a simple multiplication. It is important to not get bogged down in this step, as you have already done most of the heavy lifting already. Keep up the great work, and you will be able to solve problems like this with ease!

Identifying the Correct Expression

Now, let's look at the given options to see which one matches our method:

A. ${y=\frac{48}{6}(2)}$ B. ${y=\frac{6}{48}(2)}$ C. ${y=\frac{(48)(6)}{2}}$ D. ${y=\frac{2}{(48)(6)}}$

Option A, ${y=\frac{48}{6}(2)}$, is the correct one. Here’s why:

• ${\frac{48}{6}}$ is the constant of variation, k, which we found to be 8.
• We then multiply this constant by the new value of x, which is 2.

So, ${y=\frac{48}{6}(2)}$ is the correct expression. Make sure that you understand all the steps we have worked through in this explanation so that you can identify the correct expression on your own when completing these problems on your own. When taking a test or quiz, it is important to take the time to ensure you have selected the correct answer. This step can be one of the most important, as it is the point in the problem that you select the correct answer. I suggest that you take the time to mark all other answers as wrong so that you can focus on the correct answer. This will help you organize your thoughts, and focus on what matters most. Keep up the great work!

Conclusion

So, the correct answer is A. ${y=\frac{48}{6}(2)}$. I hope this explanation helped you guys understand how to solve direct variation problems. Remember, the key is to find the constant of variation first, and then use it to find the unknown value. Keep practicing, and you'll master these in no time! Remember that the conclusion of the problem solving process is just as important as all the other steps. Take your time and ensure that you have reviewed your work and that you understand the problem and the answer you selected. Doing so will enable you to have the confidence and courage to take on new types of math problems. With patience and practice, you will be a mathlete in no time! Keep up the great work!