What is the constant of variation, [tex]k[/tex], of the line [tex]y=k x[/tex] through (3,18) and (5,30)?
A. [tex]\frac{3}{5}[/tex]
B. [tex]\frac{5}{3}[/tex]
C. 3
D. 6
Answer (2)
Substitute the point ( 3 , 18 ) into the equation y = k x to get 18 = 3 k .
Solve for k : k = 3 18 = 6 .
Verify with the point ( 5 , 30 ) : 30 = 5 k , so k = 5 30 = 6 .
The constant of variation is 6 .
Explanation
Understanding the Problem We are given that the line y = k x passes through the points ( 3 , 18 ) and ( 5 , 30 ) . We need to find the constant of variation k .
Using the First Point We can use either point to find the value of k . Let's use the point ( 3 , 18 ) . Substituting x = 3 and y = 18 into the equation y = k x , we get 18 = k "."3 .
Solving for k Now, we solve for k by dividing both sides of the equation 18 = 3 k by 3: 3 18 = 3 3 k 6 = k So, k = 6 .
Using the Second Point Let's verify this value of k using the second point ( 5 , 30 ) . Substituting x = 5 and y = 30 into the equation y = k x , we get 30 = k "."5 .
Solving for k (again) Now, we solve for k by dividing both sides of the equation 30 = 5 k by 5: 5 30 = 5 5 k 6 = k So, k = 6 .
Conclusion Since we obtained the same value of k using both points, we can be confident that the constant of variation is indeed 6.
Examples
The concept of direct variation is used in many real-world scenarios. For example, the distance traveled by a car moving at a constant speed varies directly with the time traveled. If a car travels at a constant speed of 60 miles per hour, the distance d traveled in t hours is given by d = 60 t . Here, 60 is the constant of variation. Similarly, the amount you earn at a job that pays an hourly wage varies directly with the number of hours you work. If you earn 15 p er h o u r , yo u r t o t a l e a r nin g s E f or w or kin g h h o u rs i s g i v e nb y E = 15h$, where 15 is the constant of variation.
The constant of variation k for the line passing through the points ( 3 , 18 ) and ( 5 , 30 ) is 6 . This value is derived by substituting the coordinates into the equation y = k x and confirming it with both points. Therefore, the correct answer is D .6 .
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