Solving Direct Variation: A Step-by-Step Guide

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Hey math enthusiasts! Ever stumbled upon a problem involving direct variation? It might seem a bit intimidating at first, but trust me, once you grasp the concept, it's a piece of cake. In this article, we'll dive deep into a specific direct variation problem. We'll break it down step by step, making sure you understand how to tackle these types of questions with confidence. So, grab your pens and notebooks, and let's get started! This problem involves finding the direct variation equation when given a specific point.

First things first, let's clarify what direct variation means. Essentially, two quantities are in direct variation if they increase or decrease together at a constant rate. Think of it like this: as one quantity doubles, the other doubles too; if one halves, the other halves as well. This constant relationship is key. Now, let's get to the problem. The problem provides a starting point for understanding the question, In a direct variation equation, f(x) = 6 when x = 4.

So, the problem provides specific information: f(x) = 6 when x = 4. This gives us a concrete point on the line representing the direct variation. This point is (4, 6). What does this mean, and how can it help us find the equation? The general form of a direct variation equation is f(x) = kx, where k is the constant of variation. Our goal is to find the value of k because once we do, we have our equation. The value of k will determine the specific relationship between x and f(x). Let's look at how we can find k.

Understanding Direct Variation and Finding the Constant of Variation

Alright, guys, let's zoom in on the core idea behind this problem – direct variation. As we mentioned earlier, direct variation describes a relationship where two variables move in sync. If one goes up, the other does too, and vice versa. It's like a seesaw; when one side goes down, the other follows suit. The best way to visualize it is through an equation. The standard form for a direct variation equation is f(x) = kx. In this equation:

  • f(x) represents the dependent variable (the output).
  • x represents the independent variable (the input).
  • k is the constant of variation. This is the magic number that defines the specific relationship between x and f(x). It's the slope of the line in a linear equation.

So, to find the direct variation equation, we need to figure out the value of k. We have a point (4, 6) where x = 4 and f(x) = 6. We're going to use this point to solve for k. Substitute the values into the direct variation formula.

Using the given values, substitute x = 4 and f(x) = 6 into the formula f(x) = kx. This becomes 6 = k * 4. The next step is to isolate k to get its value. To do this, divide both sides of the equation by 4. This will give you k = 6 / 4 which simplifies to k = 1.5. Now that we know k = 1.5, we can rewrite the direct variation formula as f(x) = 1.5x. Now that we understand the concept of direct variation and how to find the constant of variation, we are ready to look at the answers.

Analyzing the Multiple-Choice Options

Alright, let's put on our detective hats and analyze the multiple-choice options, shall we? Now that we've done all the hard work, the rest should be a breeze! Remember, we are looking for the equation that represents the direct variation relationship. We previously solved for the constant of variation k=1.5. The general form is f(x) = kx. This equation tells us that f(x) is directly proportional to x. In other words, as x changes, f(x) changes proportionally. Let's see which of the given options match our findings. Now, let's break down the options:

  • A. f(x) = 6x: This equation would mean that f(x) is six times x. This clearly doesn't match our answer.
  • B. f(x) = 0.67x: This suggests that f(x) is about two-thirds of x. This is not the correct answer.
  • C. f(x) = 1.5x: This is the exact equation we found when solving for k. The constant of variation is 1.5, and this equation matches our findings.
  • D. f(x) = 4x: This would indicate that f(x) is four times x. This doesn't align with the direct variation relationship.

Step-by-Step Solution

Let's recap our approach to solving this problem. Here's a step-by-step breakdown:

  1. Understand Direct Variation: Grasp the concept that two quantities change in proportion to each other. The equation is f(x) = kx.
  2. Identify the Given Point: We know that f(x) = 6 when x = 4, giving us the point (4, 6).
  3. Find the Constant of Variation (k): Substitute the values into the direct variation formula, 6 = k * 4. Solve for k by dividing both sides by 4, resulting in k = 1.5.
  4. Write the Equation: Use the value of k to write the direct variation equation: f(x) = 1.5x.
  5. Choose the Correct Answer: Compare the equation with the multiple-choice options. Option C, f(x) = 1.5x, is the correct answer.

Conclusion: Mastering Direct Variation

And there you have it! You've successfully navigated a direct variation problem. Remember, the key is to understand the concept of direct variation, identify the constant of variation (k), and then write the equation. This is a foundational skill in mathematics, and with practice, you'll become a pro at solving these types of problems. Always remember the steps: Understand, Identify, Solve, and Choose. Keep practicing, and you'll be acing these questions in no time! Understanding direct variation opens the door to understanding other mathematical concepts. Keep up the great work, guys, and keep exploring the fascinating world of mathematics!