Constant of Variation for Direct Variation:
A direct variation between two variables x and y is given by the equation,
y = kx.
Here, k is a fixed number and it is known as the constant of variation. By the above equation, we get the formula
k = y/x.
In order to find the constant of variation we simply substitute the values for x and y in the above formula.
Constant of Variation for Inverse Variation:
An inverse variation between two variables x and y is given by the equation,
y=k/x.
Here, k is the constant of variation. By the above equation, we get the formula
k=xy.
Example 1: Is y=2x a direct variation equation?
Since we have a relationship of the form y = kx we see that it is indeed a direct variation. By comparing the equations, we see that the constant of variation k is equal to 2.
Example 2: What is the constant of variation, k, of the direct variation, y = kx, through (5, 8)?
We use the formula k=y/x for the case of direct variation. We have that k=8/5 and so the constant of variation is k=1.6.
Example 3: For the inverse variation equation xy = k, what is the value of x when y = 4 and k = 7?
Substituting the values of y and k in the equation we get that,
xy = k ⟹ 4x =7 ⟹ x = 7/4 = 1.75.
So the value of x is 1.75.
Example 4: What is the constant of variation, k, of the direct variation, y = kx, through (–3, 2)?
We substitute the values of x and y in the equation below,
k = y/x = 2/-3 = -0.67.
Therefore the constant of variation is k=-0.67.