Which Relation Is A Direct Variation That Contains The Ordered Pair { (2, 7)$}$?A. { Y = 4x - 1$}$ B. { Y = \frac{7}{x}$}$ C. { Y = \frac{2}{7}x$}$ D. { Y = \frac{7}{2}x$}$
Introduction
In mathematics, a direct variation is a type of relationship between two variables where one variable is a constant multiple of the other. This means that as one variable increases or decreases, the other variable also increases or decreases at a constant rate. In this article, we will explore the concept of direct variation and use it to determine which of the given relations contains the ordered pair (2, 7).
What is Direct Variation?
Direct variation is a relationship between two variables, x and y, where y is a constant multiple of x. This can be represented by the equation y = kx, where k is the constant of variation. The value of k determines the rate at which y changes in response to changes in x.
Example of Direct Variation
Let's consider an example of direct variation. Suppose we have a relationship where the cost of a product is directly proportional to the number of units sold. If the cost of one unit is $5, then the cost of 2 units would be $10, the cost of 3 units would be $15, and so on. This is an example of direct variation, where the cost is a constant multiple of the number of units sold.
Ordered Pairs and Direct Variation
An ordered pair is a pair of values that represent the coordinates of a point on a graph. In the context of direct variation, an ordered pair can be used to determine the constant of variation. For example, if we have the ordered pair (2, 7), we can use it to find the constant of variation.
Using the Ordered Pair to Find the Constant of Variation
Let's use the ordered pair (2, 7) to find the constant of variation. We know that y = kx, so we can substitute the values of x and y into the equation to get:
7 = k(2)
To solve for k, we can divide both sides of the equation by 2:
k = 7/2
Evaluating the Options
Now that we have found the constant of variation, we can use it to evaluate the options. We need to find the relation that contains the ordered pair (2, 7).
Option A: y = 4x - 1
Let's substitute the values of x and y into the equation to see if it matches the ordered pair:
7 = 4(2) - 1
Expanding the equation, we get:
7 = 8 - 1
Simplifying the equation, we get:
7 = 7
This equation is true, so option A is a possible solution.
Option B: y = 7/x
Let's substitute the values of x and y into the equation to see if it matches the ordered pair:
7 = 7/2
This equation is not true, so option B is not a possible solution.
Option C: y = (2/7)x
Let's substitute the values of x and y into the equation to see if it matches the ordered pair:
7 = (2/7)(2)
Expanding the equation, we get:
7 = 4/7
This equation is not true, so option C is not a possible solution.
Option D: y = (7/2)x
Let's substitute the values of x and y into the equation to see if it matches the ordered pair:
7 = (7/2)(2)
Expanding the equation, we get:
7 = 7
This equation is true, so option D is a possible solution.
Conclusion
In conclusion, we have found that both option A and option D contain the ordered pair (2, 7). However, we need to determine which one is the correct relation.
Which Relation is the Correct One?
To determine which relation is the correct one, we need to examine the equations more closely. Option A is y = 4x - 1, while option D is y = (7/2)x. We can see that option D is a direct variation, where y is a constant multiple of x. This means that as x increases or decreases, y also increases or decreases at a constant rate.
On the other hand, option A is not a direct variation. It is a linear equation, but it is not a direct variation. This means that as x increases or decreases, y does not increase or decrease at a constant rate.
Therefore, we can conclude that option D is the correct relation. The ordered pair (2, 7) is a direct variation that contains the relation y = (7/2)x.
Final Answer
Introduction
In our previous article, we explored the concept of direct variation and used it to determine which of the given relations contains the ordered pair (2, 7). In this article, we will answer some frequently asked questions about direct variation.
Q: What is direct variation?
A: Direct variation is a type of relationship between two variables where one variable is a constant multiple of the other. This means that as one variable increases or decreases, the other variable also increases or decreases at a constant rate.
Q: How do I determine if a relation is a direct variation?
A: To determine if a relation is a direct variation, you need to check if it can be written in the form y = kx, where k is a constant. If the relation can be written in this form, then it is a direct variation.
Q: What is the constant of variation?
A: The constant of variation is the value of k in the equation y = kx. It determines the rate at which y changes in response to changes in x.
Q: How do I find the constant of variation?
A: To find the constant of variation, you need to use an ordered pair that satisfies the relation. You can substitute the values of x and y into the equation to solve for k.
Q: What is the difference between direct variation and linear equation?
A: A direct variation is a type of relationship between two variables where one variable is a constant multiple of the other. A linear equation, on the other hand, is an equation that can be written in the form y = mx + b, where m is the slope and b is the y-intercept. While a direct variation is a specific type of linear equation, not all linear equations are direct variations.
Q: Can a direct variation have a negative constant of variation?
A: Yes, a direct variation can have a negative constant of variation. For example, the relation y = -2x is a direct variation with a negative constant of variation.
Q: Can a direct variation have a fractional constant of variation?
A: Yes, a direct variation can have a fractional constant of variation. For example, the relation y = (1/2)x is a direct variation with a fractional constant of variation.
Q: How do I graph a direct variation?
A: To graph a direct variation, you need to use the equation y = kx. You can plot a few points on the graph and then draw a line through them. The line will represent the direct variation.
Q: Can a direct variation have a zero constant of variation?
A: No, a direct variation cannot have a zero constant of variation. If the constant of variation is zero, then the relation is not a direct variation.
Conclusion
In conclusion, direct variation is a type of relationship between two variables where one variable is a constant multiple of the other. It is a specific type of linear equation and can be written in the form y = kx, where k is the constant of variation. We hope that this article has helped to answer some of the frequently asked questions about direct variation.
Final Answer
The final answer is that direct variation is a type of relationship between two variables where one variable is a constant multiple of the other.