Which relation is a direct variation that contains the ordered pair $(2,7)$?
$y=4 x-1$
$y=\frac{7}{x}$
$y=\frac{2}{7} x$
$y=\frac{7}{2} x$
Answer (2)
Direct variation has the form y = k x .
Check each option with the point ( 2 , 7 ) .
y = 2 7 x satisfies the condition since 7 = 2 7 ( 2 ) .
The correct relation is y = 2 7 x .
Explanation
Understanding Direct Variation A direct variation is a relationship between two variables where one is a constant multiple of the other. This can be expressed in the form y = k x , where k is the constant of variation. We need to find the equation in this form that is satisfied by the ordered pair ( 2 , 7 ) . This means when x = 2 , y must equal 7 .
Testing the Options Let's test each option:
y = 4 x − 1 : If x = 2 , then y = 4 ( 2 ) − 1 = 8 − 1 = 7 . However, this is not in the form y = k x because of the − 1 .
y = x 7 : If x = 2 , then y = 2 7 = 3.5 . This does not equal 7, so the ordered pair ( 2 , 7 ) does not satisfy this equation. Also, this is not a direct variation.
y = 7 2 x : If x = 2 , then y = 7 2 ( 2 ) = 7 4 . This does not equal 7, so the ordered pair ( 2 , 7 ) does not satisfy this equation.
y = 2 7 x : If x = 2 , then y = 2 7 ( 2 ) = 7 . This equation is in the form y = k x , and the ordered pair ( 2 , 7 ) satisfies this equation.
Conclusion Therefore, the direct variation that contains the ordered pair ( 2 , 7 ) is y = 2 7 x .
Examples
Direct variation is used in many real-world scenarios, such as calculating the distance traveled at a constant speed. For example, if a car travels at a constant speed of 60 miles per hour, the distance traveled is directly proportional to the time spent traveling. This relationship can be expressed as d = 60 t , where d is the distance and t is the time. Another example is currency conversion, where the amount in one currency is directly proportional to the amount in another currency, given a fixed exchange rate. Understanding direct variation helps in making predictions and calculations in these types of situations.
The relation that represents a direct variation containing the ordered pair ( 2 , 7 ) is y = 2 7 x . This equation fits the form y = k x and accurately gives the correct output for the input value of x .
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