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 relation to see if it is a direct variation.
Substitute the ordered pair ( 2 , 7 ) into the direct variation equations.
The relation that satisfies both conditions is y = 2 7 x , so the answer is y = 2 7 x .
Explanation
Understanding Direct Variation A direct variation is a relationship between two variables, x and y , that can be written in the form y = k x , where k is a constant of proportionality. This means that y is directly proportional to x . We need to find which of the given relations is a direct variation and contains the ordered pair ( 2 , 7 ) . This means that when x = 2 , y must be equal to 7 .
Checking Each Relation Let's examine each relation:
y = 4 x − 1 : This is a linear equation, but it has a y -intercept of − 1 . Direct variations must pass through the origin ( 0 , 0 ) , so this is not a direct variation.
y = x 7 : This is an inverse variation, not a direct variation.
y = 7 2 x : This is a direct variation, but let's check if the point ( 2 , 7 ) satisfies the equation. If x = 2 , then y = 7 2 ( 2 ) = 7 4 . Since 7 4 = 7 , this is not the correct relation.
y = 2 7 x : This is a direct variation. If x = 2 , then y = 2 7 ( 2 ) = 7 . Since the point ( 2 , 7 ) satisfies this equation, this is the correct relation.
Final Answer 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. For example, the distance you travel at a constant speed is directly proportional to the time you travel. If you travel at a speed of 60 miles per hour, the equation is d = 60 t , where d is the distance and t is the time. Another example is the relationship between the number of items you buy and the total cost, assuming each item has the same price. If each item costs 5 , t h e t o t a l cos t i s C = 5n , w h ere C i s t h e t o t a l cos t an d n$ is the number of items.
The correct relation that is a direct variation and contains the ordered pair ( 2 , 7 ) is y = 2 7 x . This confirms that y scales directly with x in this equation. Therefore, the answer is y = 2 7 x .
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