The statement "*y* varies directly as *x*," means that when *x* increases, *y* increases *by the same factor.* In other words, *y* and *x* always have the same ratio:

= k |

where

We can also express the relationship between

y = kx |

where

Since *k* is constant (the same for every point), we can find *k* when given any point by dividing the y-coordinate by the x-coordinate. For example, if *y* varies directly as *x*, and *y* = 6 when *x* = 2, the constant of variation is *k* = = 3. Thus, the equation describing this direct variation is *y* = 3*x*.

*Example 1*: If *y* varies directly as *x*, and *x* = 12 when *y* = 9, what is the equation that describes this direct variation?

*k* = =

*y* = *x*

*Example 2*: If *y* varies directly as *x*, and the constant of variation is *k* = , what is *y* when *x* = 9?

*y* = *x* = (9) = 15

As previously stated, *k* is constant for every point; i.e., the ratio between the *y*-coordinate of a point and the *x*-coordinate of a point is constant. Thus, given any two points (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) that satisfy the equation, = *k* and = *k*. Consequently, = for any two points that satisfy the equation.

*Example 3*: If *y* varies directly as *x*, and *y* = 15 when *x* = 10, then what is *y* when *x* = 6?

=

=

6() = *y*

*y* = 9