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:

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

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 (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

An equation of the form y = kx can be thought of as an equation of the form y = mx + b where m = k and b = 0. Thus, a direct variation equation is an equation in slope- intercept form which passes through (0, 0) and has a slope equal to the constant of variation.

Therefore, to graph a direct variation equation, start at (0, 0) and then proceed as you would in graphing a slope. Or, if you know one point, draw a straight line between (0, 0) and that point, and extend the line on both sides.

Example 4: y varies directly as x. If the constant of variation is , graph the line which represents the variation, and write an equation that describes the variation.

To calculate the constant of variation, given a graph of direct variation, simply calculate the slope.