Inverse Variation
The statement "y varies inversely as x means that when x increases, ydecreases by the same factor. In other words, the expression xy is constant:
where
k is the constant of variation.
We can also express the relationship between
x and
y as:
y = 

where
k is the constant of variation.
Since k is constant, we can find k given any point by multiplying the xcoordinate by the ycoordinate. For example, if y varies inversely as x, and x = 5 when y = 2, then the constant of variation is k = xy = 5(2) = 10. Thus, the equation describing this inverse variation is xy = 10 or y = .
Example 1: If y varies inversely as x, and y = 6 when x = , write an equation describing this inverse variation.
k = (6) = 8
xy = 8 or y =
Example 2: If y varies inversely as x, and the constant of variation is k = , what is y when x = 10?
xy =
10y =
y = × = × =
k is constant. Thus, given any two points (x_{1}, y_{1}) and (x_{2}, y_{2}) which satisfy the inverse variation, x_{1}y_{1} = k and x_{2}y_{2} = k. Consequently, x_{1}y_{1} = x_{2}y_{2} for any two points that satisfy the inverse variation.
Example 3: If y varies inversely as x, and y = 10 when x = 6, then what is y when x = 15?
x_{1}y_{1} = x_{2}y_{2}
6(10) = 15y
60 = 15y
y = 4
Thus, when x = 6, y = 4.