Direct, Inverse & Joint Variation

Variation problems describe how quantities move together, with a constant k.

Direct: y = kx. One goes up, the other goes up proportionally.
Inverse: y = k/x. One goes up, the other goes down. Their product stays fixed.
Joint: y = kxz. Directly with two or more variables at once.
Combined: mixes them, such as y = kx/z.

The routine is always the same: write the equation, use the given values to find k, then use the completed equation.

Speed and time for a fixed distance are inversely related: double the speed, halve the time.

Variation statements translate to equations with a constant of proportionality determined by one data point.

Direct variation y = kx is linear through the origin, and its graph is a line. Inverse variation y = k/x, equivalently xy = k, is a hyperbola with both axes as asymptotes, so it is a rational function rather than a linear one.

Distinguishing them from data is a matter of which combination is constant: a fixed quotient y/x indicates direct variation, a fixed product xy indicates inverse.

Powers may be involved: "y varies inversely with the square of x" is y = k/x², the inverse-square law of gravitation and light intensity. Joint and combined variation extend this to several variables, as in the ideal gas law V = kT/P, which is direct in T and inverse in P simultaneously.

Once k is determined, the model is fully specified and any other value can be computed, which is why these problems always supply exactly one complete data point.