Direct Variation: Finding Y When X = 2

by Andrew McMorgan 39 views

Hey Plastik Magazine readers! Today, we're diving into the world of direct variation and tackling a classic problem. We're going to figure out how to find the value of y when x is 2, given that y varies directly as x, and y is 48 when x is 6. Sounds like fun, right? Let's break it down step by step.

Understanding Direct Variation

Before we jump into the problem, let's make sure we're all on the same page about what direct variation actually means. In simple terms, when y varies directly as x, it means that y is directly proportional to x. This can be expressed mathematically as:

y = kx

Where:

  • y and x are our variables.
  • k is the constant of variation. This constant represents the ratio between y and x, and it stays the same no matter what the values of y and x are.

Think of it this way: if x doubles, y doubles as well. If x triples, y triples, and so on. This constant relationship is what defines direct variation.

In our case, we know that y varies directly as x, which means we have a relationship like y = kx. Our goal is to find the value of y when x is 2, but first, we need to figure out what the constant of variation, k, is. This is a crucial step in solving direct variation problems, and it sets the stage for finding any value of y for a given x.

Finding the Constant of Variation (k)

The problem tells us that y is 48 when x is 6. This is our golden ticket to finding k! We can plug these values into our equation y = kx:

48 = k(6)

Now, to isolate k, we simply divide both sides of the equation by 6:

k = 48 / 6

k = 8

So, we've found our constant of variation! k is 8. This means that for this particular direct variation relationship, y is always 8 times x. This understanding of how to calculate k is a fundamental skill in dealing with direct variation problems. Now that we know k, we can write the specific equation that relates y and x for this problem:

y = 8x

Calculating y When x is 2

Now that we have our equation, y = 8x, finding y when x is 2 is a piece of cake! We just substitute 2 for x in the equation:

y = 8(2)

y = 16

So, when x is 2, y is 16. We've successfully navigated the direct variation and found our answer! Isn't math satisfying when it all comes together?

This entire process illustrates the power of understanding the underlying principles of direct variation. By first identifying the relationship and then finding the constant of proportionality, we can solve for any value within that relationship.

Identifying the Correct Expression

The original question asked us which expression could be used to find the value of y when x is 2. Let's look at the options again, keeping in mind that we found k to be 8, and our equation is y = 8x.

We know that y = 16 when x = 2. Now let's analyze the given options to see which one leads us to the correct calculation:

A. y = (48/6)(2) B. y = (6/48)(2) C. y = (48)(8)/2 D. y = 2/(48)(6)

Let's evaluate each one:

  • Option A: y = (48/6)(2)
    • This simplifies to y = (8)(2) = 16. This is exactly what we found when we calculated y using our equation y = 8x. So, this expression correctly represents the solution.
  • Option B: y = (6/48)(2)
    • This simplifies to y = (1/8)(2) = 1/4. This is incorrect.
  • Option C: y = (48)(8)/2
    • This simplifies to y = 384/2 = 192. This is also incorrect.
  • Option D: y = 2/(48)(6)
    • This simplifies to y = 2/288 = 1/144. Definitely not the right answer!

Therefore, the correct expression to find the value of y when x is 2 is A. y = (48/6)(2). This option correctly uses the ratio of y to x (which is 48/6, or 8) and multiplies it by the new value of x (which is 2) to find the corresponding value of y.

Key Takeaways

Let's recap the key steps we took to solve this direct variation problem:

  1. Understanding the Concept: We started by defining what direct variation means (y = kx).
  2. Finding the Constant of Variation (k): We used the given information (y = 48 when x = 6) to calculate k.
  3. Creating the Equation: We plugged k back into the equation y = kx to get the specific equation for this problem.
  4. Solving for y: We substituted x = 2 into the equation to find the value of y.
  5. Identifying the Correct Expression: We analyzed the given options and chose the one that correctly calculated y.

These steps provide a solid framework for tackling any direct variation problem. Remember, the key is to understand the relationship between the variables and to find that crucial constant of variation.

Practice Makes Perfect

Direct variation problems are a staple in algebra and mathematics, guys. The more you practice, the more comfortable you'll become with them. Try working through similar problems, changing the given values and seeing how it affects the solution. You can even create your own problems to challenge yourself! The beauty of direct variation is that once you grasp the core concept, you can apply it to a wide range of situations. Keep practicing, and you'll become a direct variation master in no time!

So, there you have it! We've successfully navigated this direct variation problem and learned how to find the value of y when x is 2. Keep exploring the world of math, and remember, practice makes perfect!