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:
|y = kx|
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 (x1, y1) and (x2, y2) 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