Direct Variation

When two variables change proportionally to each other, they are said to be in direct variation. This can also be called directly proportional.

For example, a toothbrush costs $2. Purchasing five toothbrushes would cost $10; purchasing 10 toothbrushes would cost $20. No matter how many toothbrushes purchased, the ratio will always remain: $2 per toothbrush. Thus we can say that cost varies directly as the value of toothbrushes.

Direct variation is easily illustrated using a linear graph. Knowing that the relationship between two variables is constant, we can show their relationship as :

$\frac yx=k$

where k is a constant known as the constant of proportionality.

This can be rearranged to slope-intercept format:

$y=kx$

In this case, the y-intercept is equal to 0.

Revisiting the example with toothbrushes and dollars, we can define the x axis as number of toothbrushes and the y axis as number of dollars. Doing so, the variables would abide by the relationship:

$\frac yx=2$

Any augmentation of one variable would lead to an equal augmentation of the other. For example, doubling y would result in the doubling of x.