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 k is the constant of variation.
We can also express the relationship between x and y as:
y = kx where k is the constant of variation.
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 = 3x .
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