For each equation, determine whether it represents a direct variation, an inverse variation, or neither.
Find the constant of variation when one exists and write it in simplest form.
[tex]8 y=5 x[/tex]
Direct variation
Constant of variation: [tex]k=[/tex] $\square$
Inverse variation
Constant of variation: [tex]k=[/tex] $\square$
Neither
[tex]5 x y=15[/tex]
Direct variation
Constant of variation: [tex]k=[/tex] $\square$
Inverse variation
Constant of variation: [tex]k=[/tex] $\square$
Neither
Answer (1)
The equation 8 y = 5 x represents a direct variation.
Rewrite the equation as y = 8 5 x , so the constant of variation is k = 8 5 .
The equation 5 x y = 15 represents an inverse variation.
Rewrite the equation as x y = 3 , so the constant of variation is k = 3 .
k = 8 5 , k = 3
Explanation
Problem Analysis We are given two equations and asked to determine if they represent a direct variation, an inverse variation, or neither. We also need to find the constant of variation if it exists.
Analyzing the First Equation The first equation is 8 y = 5 x . To check for direct variation, we need to rewrite the equation in the form y = k x , where k is the constant of variation. Dividing both sides of the equation by 8, we get y = 8 5 x . This is in the form y = k x , so it represents a direct variation with k = 8 5 .
Analyzing the Second Equation The second equation is 5 x y = 15 . To check for inverse variation, we need to rewrite the equation in the form x y = k , where k is the constant of variation. Dividing both sides of the equation by 5, we get x y = 3 . This is in the form x y = k , so it represents an inverse variation with k = 3 .
Conclusion Therefore, the first equation represents a direct variation with a constant of variation k = 8 5 , and the second equation represents an inverse variation with a constant of variation k = 3 .
Examples
Direct and inverse variations are useful in many real-world scenarios. For example, the distance traveled at a constant speed is a direct variation with the speed. The time it takes to travel a fixed distance is an inverse variation with the speed. Understanding these relationships helps in making predictions and solving problems in physics, engineering, and economics.
