Inverse Variation

By the end of this lesson, you will be able to identify and write inverse variation equations, find the constant of variation, distinguish direct from inverse variation, and apply joint and combined variation to real-world problems.

The Big Idea

Inverse variation describes a relationship where one quantity increases as another decreases proportionally — their product is constant. This is the mathematical model for pressure-volume gas laws, electric resistance, gravitational force at a distance, and many other physical phenomena.

Direct vs. Inverse Variation

Direct variation: $y = kx$ — as $x$ increases, $y$ increases proportionally; $\dfrac{y}{x} = k$ constant.

Inverse variation: $y = \dfrac{k}{x}$ — as $x$ increases, $y$ decreases; $xy = k$ constant.

Constant of variation $k$: found from any known pair $(x, y)$.

Identifying Inverse Variation

If $xy = k$ for all data points, the relationship is inverse variation.

$x$	$y$	$xy$
2	12	24
4	6	24
6	4	24
8	3	24

$xy = 24$ for all pairs → inverse variation with $k = 24$: $y = \dfrac{24}{x}$

Example 1 — Writing the Equation

$y$ varies inversely as $x$. When $x = 5$, $y = 8$. Find the equation and find $y$ when $x = 20$.

$k = xy = 5 \times 8 = 40$

Equation: $y = \dfrac{40}{x}$

When $x = 20$: $y = \dfrac{40}{20} = 2$

Example 2 — Finding $x$

$y$ varies inversely as $x$. When $x = 3$, $y = 12$. Find $x$ when $y = 4$.

$k = 3 \times 12 = 36$

$4 = \dfrac{36}{x} \implies x = 9$

Joint Variation

$z$ varies jointly as $x$ and $y$: $z = kxy$

Both $x$ and $y$ are in the numerator.

Example: The volume of a prism $V = \frac{1}{3}Bh$ — $V$ varies jointly as the base area $B$ and height $h$ (with $k = \frac{1}{3}$).

Combined Variation

Combines direct, inverse, and joint variation in one equation.

$z$ varies directly as $x$ and inversely as $y$:

$$z = \frac{kx}{y}$$

Example 3 — Combined Variation