Solving Direct Variation Problems Finding Y When X Is 7

Hey guys! Today, we're diving into the fascinating world of direct variation. It's all about understanding how two quantities relate to each other, and we're going to use a table of values to crack the code. We'll walk through the steps together, making sure you grasp the concept so you can tackle similar problems with confidence. So, let's jump right in and figure out what happens to 'y' when 'x' hits 7, given the direct variation relationship shown in our table.

Understanding Direct Variation: The Key to Proportions

So, what exactly is direct variation? In simple terms, two quantities, let's call them x and y, vary directly if one is a constant multiple of the other. Think of it like this: as x increases, y increases at a proportional rate, and vice versa. The magic ingredient here is the constant of variation, often denoted by k. This constant tells us the exact relationship between x and y. Mathematically, we express direct variation as y = kx. This equation is the backbone of our problem-solving approach.

Now, why is this concept so important? Direct variation pops up everywhere in the real world! Imagine the distance you travel at a constant speed – that's direct variation. The amount you earn per hour of work? Direct variation again! Understanding this relationship allows us to make predictions and solve practical problems. In our case, we have a table showing how x and y change together, and we need to find the missing piece of the puzzle. By identifying the constant of variation from the given data, we can predict the value of y for any given x, including our target value of 7. This ability to extrapolate and interpolate values is a powerful tool in mathematics and beyond.

To make it even clearer, let's visualize it. Imagine a graph where x is on the horizontal axis and y is on the vertical axis. A direct variation relationship will always appear as a straight line passing through the origin (the point where x and y are both zero). The slope of this line is precisely our constant of variation, k. A steeper line means a larger k, indicating a stronger direct relationship between x and y. Conversely, a flatter line implies a smaller k, suggesting a weaker relationship. This graphical representation provides an intuitive way to understand how changes in x directly influence changes in y. So, keep this visual in mind as we move forward – it'll help solidify your understanding of direct variation.

Deciphering the Table: Finding the Constant of Variation

Alright, let's get our hands dirty with the table you provided. We have pairs of x and y values, and our mission is to find the constant of variation, k. Remember, the equation y = kx is our guiding star. To find k, we can simply rearrange this equation to k = y/x. This means we can take any pair of x and y values from the table and divide y by x to calculate k. The beauty of direct variation is that this value of k should be the same for all pairs of x and y in the table. This consistency is the hallmark of a direct variation relationship, and it's what allows us to confidently predict values.

Let's put this into action. Looking at the table, we see the first pair is x = 1 and y = 14. Plugging these values into our formula, we get k = 14/1 = 14. So, our constant of variation appears to be 14. But, we're not done yet! To be absolutely sure, we need to verify this value with other pairs in the table. Let's take the second pair: x = 3 and y = 42. Calculating k again, we get k = 42/3 = 14. Awesome! The constant of variation remains the same. This reinforces our belief that we're indeed dealing with a direct variation relationship. Let's do one more check, just to be super confident. Using the pair x = 6 and y = 84, we find k = 84/6 = 14. Bingo! We've confirmed that the constant of variation is consistently 14 across all the given data points.

Now that we've confidently determined k = 14, we've essentially unlocked the secret code of this direct variation relationship. We can now express the relationship between x and y as the equation y = 14x. This equation is our key to finding the value of y for any given x. It's like having a magic formula that connects x and y. So, with this powerful equation in our arsenal, let's move on to the final step: finding y when x is 7.

Solving for Y: Plugging in X = 7

Okay, guys, we're in the home stretch! We've successfully identified the direct variation relationship and found our constant of variation, k = 14. We even have the equation y = 14x that perfectly describes the connection between x and y. Now comes the fun part: using this equation to answer the original question – what is the value of y when x is 7? This is where all our hard work pays off, as we simply plug in the value of x into our equation and solve for y.

So, let's do it! We have y = 14x, and we're given x = 7. Substituting 7 for x in the equation, we get y = 14 * 7. Now it's just a matter of simple multiplication. 14 multiplied by 7 is 98. Therefore, when x is 7, y is equal to 98. Ta-da! We've found our answer. This demonstrates the power of understanding direct variation – once we know the constant of variation, we can predict the value of one variable given the other.

To recap, we started with a table of values and a question about the relationship between x and y. We recognized that the table represented a direct variation, meaning x and y are proportional. We then calculated the constant of variation, k, by dividing y by x for several pairs of values in the table, confirming that k was consistently 14. This allowed us to write the equation y = 14x, which captures the direct variation relationship. Finally, we plugged in x = 7 into this equation to find the corresponding value of y, which turned out to be 98. This entire process highlights the step-by-step approach to solving direct variation problems, and it's a method you can apply to a wide range of similar situations.

The Answer and Why It Matters

So, drumroll please… the value of y when x is 7 is 98! Looking at the options provided, the correct answer is D. 98. We successfully navigated the world of direct variation, found the constant of proportionality, and used it to solve for our missing value. Give yourselves a pat on the back for a job well done!

But beyond just getting the right answer, let's think about why this matters. Understanding direct variation isn't just about solving textbook problems; it's about developing a powerful problem-solving mindset. The skills we used here – identifying relationships, finding patterns, and using equations to make predictions – are valuable in countless real-world scenarios. Whether you're calculating recipe ingredients, determining travel time based on speed, or even understanding financial growth, the principles of direct variation come into play.

Furthermore, mastering direct variation lays the foundation for more advanced mathematical concepts. It's a stepping stone to understanding other types of variations, such as inverse variation, and it's closely related to linear functions and their graphs. By grasping the fundamentals of direct variation, you're building a solid mathematical base that will serve you well in future studies and applications. So, take pride in your accomplishment today, and remember that every problem you solve is a step towards becoming a more confident and capable problem-solver.

Remember guys, math is not just about numbers and equations; it's about understanding the world around us. And direct variation is one of those key concepts that helps us make sense of the relationships we see every day. Keep practicing, keep exploring, and keep that mathematical curiosity burning bright!