Variation — Definition, Formula & Examples

Variation is a relationship between two or more variables where a change in one variable produces a predictable change in another. The most common types are direct variation (both increase together) and inverse variation (one increases while the other decreases).

A variation is a functional relationship between variables expressible in the form $y = kx^n$ (direct), $y = \frac{k}{x^n}$ (inverse), or combinations thereof (joint/combined), where $k$ is a nonzero constant called the constant of variation and $n$ is a positive rational number.

How It Works

To solve a variation problem, first identify the type of variation from the wording. "Varies directly" means

y = kx

; "varies inversely" means

y = \frac{k}{x}

; "varies jointly" means

y = kxz

(or more variables). Next, substitute a known pair of values to solve for

k

. Finally, use your equation with the new value to find the unknown.

Worked Example

Problem: Suppose y varies inversely as x, and y = 12 when x = 3. Find y when x = 9.

Write the model: Because y varies inversely as x, the equation is:

y = \frac{k}{x}

Find k: Substitute the known values y = 12 and x = 3:

12 = \frac{k}{3} \implies k = 36

Solve for y: Use the equation with x = 9:

y = \frac{36}{9} = 4

Answer: When x = 9, y = 4.

Why It Matters

Variation models appear throughout physics and engineering — gravitational force varies inversely with the square of distance, and Ohm's law (V = IR) is a direct variation. Recognizing the variation type lets you set up the equation quickly in Algebra 2 and precalculus courses.

Common Mistakes

Mistake: Confusing direct and inverse variation when translating word problems.

Correction: "y varies directly as x" means y = kx (both increase together). "y varies inversely as x" means y = k/x (one increases, the other decreases). Focus on the keyword: 'directly' → multiply, 'inversely' → divide.