Direct Variation: Find The Equation That Includes (2, 7)

by Pedro Alvarez 57 views

Hey there, math enthusiasts! Let's dive into the world of direct variation and figure out which equation proudly contains the ordered pair (2, 7). We'll break down what direct variation means, how to spot it, and then test each equation to find our winner. So, buckle up and get ready to explore the relationship between x and y!

Understanding Direct Variation

When we talk about direct variation, we're describing a special kind of relationship between two variables, usually x and y. In simple terms, y varies directly with x if y is a constant multiple of x. This means as x increases, y increases proportionally, and as x decreases, y decreases proportionally. Think of it like this: the more you work, the more you get paid (hopefully!).

The general form of a direct variation equation is y = kx, where k is the constant of variation. This constant represents the factor by which x is multiplied to get y. It's a crucial part of understanding the relationship, as it tells us the rate at which y changes with respect to x. So, identifying direct variation boils down to spotting this y = kx structure.

Now, to solidify your understanding, let's delve deeper into the characteristics of direct variation. First and foremost, the graph of a direct variation equation is always a straight line that passes through the origin (0, 0). This is because when x is 0, y is also 0, as y = k * 0 = 0. This characteristic gives us a visual cue to identify direct variation relationships. Imagine a line perfectly balanced, pivoting at the center of your graph – that's direct variation in action!

Secondly, the constant of variation, k, plays a pivotal role. It represents the slope of the line. Remember slope? It's the measure of steepness, indicating how much y changes for every unit change in x. A larger k means a steeper line, signifying a more rapid change in y compared to x. Conversely, a smaller k means a gentler slope, indicating a more gradual change.

Finally, let's look at some real-world examples. Think about the relationship between the number of hours you work and your earnings (if you have a fixed hourly rate). The more hours you put in, the more money you make – a perfect example of direct variation! Or consider the relationship between the distance you travel and the time it takes, assuming you're moving at a constant speed. The farther you go, the more time you'll need. These everyday scenarios help illustrate the practical applications of direct variation.

Identifying the Correct Equation

Okay, guys, we've got a solid grasp on direct variation now. We know it looks like y = kx, and we know what that k means. Now, let's get to the heart of the problem. We need to find the equation that not only represents direct variation but also contains the ordered pair (2, 7). Remember, an ordered pair (x, y) is just a specific point on the graph of the equation. So, if the equation contains the ordered pair (2, 7), it means that when we plug in x = 2, we should get y = 7.

Let's examine the given equations one by one:

  1. y = 4x - 1:
    • This equation looks linear, but it's not direct variation! Why? Because of that “- 1” hanging out at the end. Direct variation equations have the form y = kx, with nothing added or subtracted. This equation shifts the line down by one unit, so it won't pass through the origin. Let's plug in x = 2 and see what happens: y = 4(2) - 1 = 8 - 1 = 7. Interestingly, it does give us y = 7! But remember, it's not direct variation, so it's not our answer.
  2. y = 7/x:
    • This is a totally different beast! This equation represents inverse variation, not direct variation. In inverse variation, as x increases, y decreases, and vice versa. The relationship is multiplicative, not additive. So, this one is definitely not in the form y = kx. Let's try plugging in x = 2 anyway: y = 7/2 = 3.5. Nope, doesn't give us y = 7. Scratch this one off the list.
  3. y = (2/7)x:
    • Aha! This looks promising! It has the form y = kx, where k is 2/7. This is a direct variation equation! But does it contain the point (2, 7)? Let's check: y = (2/7)(2) = 4/7. Nope! When x is 2, y is 4/7, not 7. So, this one is out.
  4. y = (7/2)x:
    • This one also has the y = kx form, with k equal to 7/2. Another direct variation contender! Let's plug in x = 2 and see if it works: y = (7/2)(2) = 7. Bingo! When x is 2, y is 7. This equation contains the ordered pair (2, 7) and represents direct variation.

The Winner: y = (7/2)x

Drumroll, please! The equation that represents a direct variation and contains the ordered pair (2, 7) is y = (7/2)x. We found our match! We systematically analyzed each equation, checked for the y = kx form, and then verified whether the ordered pair (2, 7) satisfied the equation. This methodical approach is key to solving these kinds of problems.

So, there you have it! We've successfully navigated the world of direct variation and pinpointed the equation that fits our criteria. Remember, understanding the fundamental concepts and applying them step-by-step is the secret to math success. Keep practicing, and you'll become a direct variation pro in no time!

Why is y = (7/2)x the Answer?

Let's recap why y = (7/2)x is the correct answer and really hammer home the concepts we've discussed. This is crucial for understanding the why behind the answer, not just the answer itself. Knowing the why will help you tackle similar problems with confidence.

First, let's revisit the definition of direct variation. Direct variation, as we know, is a relationship where one variable is a constant multiple of another. The equation that embodies this relationship is y = kx. Notice the simplicity of this equation: y is directly proportional to x, and k is the constant of proportionality.

Now, let's look at our winning equation: y = (7/2)x. Does it fit the y = kx mold? Absolutely! We can clearly see that k, the constant of variation, is 7/2. This immediately tells us that y varies directly with x, and for every unit increase in x, y increases by 7/2 units. This constant of variation is the key to understanding the relationship between x and y.

But that's not all. The problem specifically asks for an equation that contains the ordered pair (2, 7). This means that if we substitute x = 2 into the equation, we must get y = 7. Let's test it out: y = (7/2)(2) = 7. It works! This confirms that the point (2, 7) lies on the line represented by the equation y = (7/2)x.

So, the equation y = (7/2)x satisfies both conditions: it represents a direct variation relationship, and it contains the ordered pair (2, 7). This is why it's the correct answer.

Think of it like a puzzle. We had two pieces: the direct variation requirement and the (2, 7) requirement. Only one equation fit both pieces perfectly. By understanding the definition of direct variation and how ordered pairs relate to equations, we were able to assemble the puzzle and find the solution.

Furthermore, let's consider the implications of the constant of variation, 7/2. This value tells us the slope of the line represented by the equation. A slope of 7/2 means that for every 2 units we move to the right on the graph (increase in x), we move 7 units up (increase in y). This steep slope visually represents the direct proportionality between x and y. The larger the constant of variation, the steeper the line, and the more dramatic the change in y for a given change in x.

In contrast, the other equations failed to meet one or both of the conditions. The equation y = 4x - 1, while passing through the point (2, 7), wasn't a direct variation because of the “- 1” term. The equation y = 7/x represented inverse variation, a completely different relationship. And the equation y = (2/7)x, while being a direct variation, didn't pass through the point (2, 7).

Therefore, y = (7/2)x stands out as the sole solution that perfectly embodies direct variation and includes the specified ordered pair. This comprehensive understanding reinforces the importance of grasping the fundamental principles and applying them systematically to solve problems.

Final Thoughts and Practice Tips

Alright, mathletes, we've conquered this direct variation problem! Hopefully, you've gained a deeper understanding of what direct variation is, how to identify it, and how to find the right equation that fits specific conditions. Remember, math is like a muscle – the more you exercise it, the stronger it gets. So, let's talk about some final thoughts and practice tips to keep your direct variation skills sharp.

First, solidify your understanding of the core concept: Direct variation is all about that y = kx relationship. Memorize it, understand it, and love it! When you see this form, you know you're dealing with direct variation. Think about how y changes directly with x, influenced by the constant k. This understanding is your foundation.

Second, practice, practice, practice! The more problems you solve, the more comfortable you'll become with identifying direct variation equations and working with ordered pairs. Look for problems that ask you to:

  • Identify direct variation equations from a list.
  • Find the constant of variation given a direct variation equation and a point.
  • Write the equation of a direct variation given a point or the constant of variation.
  • Solve real-world problems involving direct variation (like distance and time, earnings and hours worked, etc.).

Third, visualize the graphs: Remember that direct variation equations always graph as straight lines passing through the origin. Visualizing these lines can help you understand the relationship between x and y and the significance of the slope (the constant of variation). Try graphing some direct variation equations and observe how the slope affects the steepness of the line.

Fourth, don't be afraid to test and check: When you're working through a problem, always double-check your work. Plug in values, solve equations, and make sure your answers make sense in the context of the problem. If you're unsure, try a different approach or ask for help. There's no shame in seeking clarification!

Fifth, connect direct variation to other concepts: Math is interconnected. Direct variation is related to linear equations, slope, and proportionality. Understanding these connections will deepen your overall mathematical knowledge and make it easier to solve a wider range of problems. For example, think about how direct variation is a special case of a linear equation where the y-intercept is zero.

Sixth, break down complex problems: If you encounter a challenging problem, don't get overwhelmed. Break it down into smaller, more manageable steps. Identify the key information, write down the relevant equations, and work through the steps one at a time. This methodical approach will make the problem seem less daunting and increase your chances of success.

And finally, remember to celebrate your successes! Math can be challenging, but it's also incredibly rewarding. When you solve a problem, take a moment to appreciate your accomplishment. This positive reinforcement will motivate you to keep learning and growing.

So, go forth and conquer direct variation! With a solid understanding of the concepts, consistent practice, and a positive attitude, you'll be well on your way to mastering this important mathematical skill.