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

= k
where is the constant of variation.
We can also express the relationship between and as:
y = kx
where is the constant of variation.

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

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

k =  =
y =  x

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

y =  x = (9) = 15

As previously stated, is constant for every point; i.e., the ratio between the -coordinate of a point and the -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 varies directly as , and y = 15 when x = 10 , then what is when x = 6 ?

=
=
6() = y
y = 9