Finding Direct Variation Equations: A Step-by-Step Guide

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Hey math enthusiasts! Today, we're diving into the world of direct variation – a super useful concept in algebra. We'll figure out how to spot a direct variation equation and, more specifically, how to find the one that includes the ordered pair (2, 7). Get ready to flex those math muscles! Let's get started, shall we?

What is Direct Variation? Let's Break It Down!

First things first: What exactly is direct variation? In simple terms, it's a relationship between two variables, typically denoted as x and y, where one variable is a constant multiple of the other. This means that as one variable increases, the other increases proportionally, and as one decreases, the other decreases proportionally as well. Think of it like a recipe: if you double the ingredients, you double the output, right? That's direct variation in action!

The general form of a direct variation equation is y = kx, where k is the constant of variation. This k is a non-zero number. It's the magic number that links x and y together. Graphically, direct variation equations always pass through the origin (0, 0). This is a great way to identify them visually!

Now, let's look at the options provided and see which one fits the bill. The key here is to understand the definition of direct variation and how it relates to the given ordered pair. Remember, the equation must be in the form of y = kx, and it must satisfy the condition that when x is 2, y must be 7. Let's start with option A.

Analyzing the Options: Which Equation Works?

Let's get down to the nitty-gritty and analyze each option. We'll substitute the values of x and y from the ordered pair (2, 7) into each equation to see if it holds true. This is like a process of elimination, but with math! It's actually kind of fun when you get the hang of it.

Option A: y = 4x - 1

If we substitute x = 2 into this equation, we get y = 4(2) - 1 = 8 - 1 = 7. This seems promising initially, right? However, this equation does not represent a direct variation. The presence of the '-1' indicates a y-intercept that is not at the origin (0, 0). Direct variation equations must be in the form y = kx, and this one has an added constant. So, option A is out!

Option B: y = 7/x

In this equation, y is inversely proportional to x, not directly proportional. When x = 2, y = 7/2 which is not equal to 7. Therefore, option B is incorrect. This is an example of an inverse variation, not a direct variation. Inverse variations have a different relationship: as x increases, y decreases, and vice versa. Keep this in mind, guys!

Option C: y = (2/7)x

Let's see if this one hits the mark. If x = 2, then y = (2/7) * 2 = 4/7. This does not equal 7. Therefore, option C is also incorrect. Although this is in the form of a linear equation, it does not satisfy the given point. The value of k in this equation is 2/7.

Option D: y = (7/2)x

This is where the magic happens! If we plug in x = 2, we get y = (7/2) * 2 = 7. Awesome! This equation fits the bill perfectly. It's in the form y = kx, where k = 7/2, and it includes the ordered pair (2, 7). This equation represents a direct variation and satisfies the given conditions. High five!

The Final Answer and Why It Matters

So, the correct answer is option D: y = (7/2)x. This equation represents a direct variation because it's in the form y = kx, and it passes through the point (2, 7). When x is 2, y is indeed 7. We've confirmed this through substitution and by understanding the characteristics of direct variation equations.

Understanding direct variation is crucial because it helps us model many real-world scenarios. For example, the cost of buying multiple items when each item has the same price or the relationship between distance, speed, and time when the speed is constant. It's a fundamental concept that builds the foundation for more advanced topics in algebra and calculus.

By working through this problem step-by-step, we've strengthened our ability to recognize and manipulate direct variation equations. We’ve also seen how to test if a given ordered pair satisfies the equation. Keep practicing, and you'll become a pro at these problems in no time! Keep in mind that understanding these basics helps a lot in understanding more advanced concepts as well.

Tips for Success: Mastering Direct Variation

To become a direct variation whiz, here are a few extra tips:

  • Memorize the Form: Always remember that a direct variation equation is in the form y = kx. This is your starting point.
  • Find k: To find the constant of variation (k), divide y by x (k = y/x). This gives you the magic number.
  • Graph It: Always graph the equation to confirm that it passes through the origin (0, 0). This confirms your answer visually.
  • Practice, Practice, Practice: The more problems you solve, the more comfortable you'll become. Practice with different ordered pairs and equations.
  • Understand the Concept: Make sure you understand the relationship between x and y. As one changes, the other changes proportionally.

By following these tips, you'll be well on your way to mastering direct variation problems. Keep up the excellent work, and always remember that math can be fun and rewarding. Good luck, and keep exploring the amazing world of mathematics! You got this!