Variation describes how one quantity changes with another. It turns word statements into useful algebraic models.

Direct variation

\(y\) varies directly as \(x\): \[ y\propto x \quad\Rightarrow\quad y=kx \] where \(k\) is the constant of variation.

Example: If \(y=18\) when \(x=6\), then \(k=3\), so \(y=3x\).

Inverse variation

\(y\) varies inversely as \(x\): \[ y\propto\frac1x \quad\Rightarrow\quad y=\frac{k}{x}. \]

Example: If \(y=5\) when \(x=4\), then \(k=20\), so \(y=\frac{20}{x}\).

Variation with powers

You may see statements like:

• \(y\propto x^2 \Rightarrow y=kx^2\)
• \(y\propto \sqrt{x} \Rightarrow y=k\sqrt{x}\)
• \(y\propto \frac{1}{x^2} \Rightarrow y=\frac{k}{x^2}\)

Joint variation

If \(y\) varies jointly as \(x\) and \(z\): \[ y\propto xz \Rightarrow y=kxz. \]

Example: \(y=24\) when \(x=2,z=3\Rightarrow k=4\), so \(y=4xz\).

Partial variation

If \(y\) varies partly as \(x\), model has constant and variable part: \[ y=mx+c,\quad c\ne0. \]

This appears in pricing: fixed fee + per-unit charge.

Solving workflow

1. Translate words into proportional equation with \(k\).
2. Use given data to find \(k\).
3. Substitute required values to answer question.
4. Check units and reasonableness.

Common word-problem forms

• “\(A\) is directly proportional to \(B\)”
• “\(A\) varies inversely as \(B\)”
• “\(A\) varies jointly as \(B\) and \(C\)”
• “\(A\) varies directly as square/cube of \(B\)”

Exam strategy

• Always introduce \(k\) first before substituting numbers.
• Keep symbolic form until \(k\) is found.
• For inverse models, domain often excludes zero.
• State final answer with units when applicable.