The statement "*y* varies inversely as *x* means that when *x* increases, *y**decreases by the same factor.* In other words, the expression *xy* is constant:

xy = k |

where

We can also express the relationship between

y = |

where

Since *k* is constant, we can find *k* given any point by multiplying the x-coordinate by the y-coordinate. 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* =

10*y* =

*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) = 15*y*

60 = 15*y*

*y* = 4

Thus, when *x* = 6, *y* = 4.