# Direct Variation

## Key Questions

• Take note of the behavior of one variable and compare with the corresponding behavior of the other.

If you increase the value of one variable, what happens to the value of the other variable? If it increases, you have a direct variation.
If it decreases, you have a inverse variation

• Direct Variation

If $y$ is directly proportional to $x$, then we can write

$y = k x$,

where $k$ is the constant of proportionality.

If you solve for $k$, we have

$k = \frac{y}{x}$,

which is the ratio of $y$ to $x$.

Hence, the constant of proportionality is the ratio between two quantities that are directly proportional.

Inverse Variation

If $y$ is inversely proportional to $x$, then we can write

$y = \frac{k}{x}$,

where $k$ is the constant of proportionality.

If you solve for $k$, we have

$k = x y$,

which is the product of $x$ and $y$.

Hence, the constant of proportionality is the product of quantities that are inversely proportional.

I hope that this was helpful.

• Direct Variation

If $y$ varies directly as $x$, then we can write

$y = k \cdot x$,

where $k$ is a constant of variation.

I hope that this was helpful.

• When you have a direct variation, we say that as your variable changes, the resulting value changes in the same and proportional manner.

A direct variation between $y$ and $x$ is typically denoted by

$y = k x$

where $k \in \mathbb{R}$

This means that as $x$ goes larger, $y$ also tends to get larger.
The opposite is also true. As $x$ goes smaller, $y$ tend to get smaller.