Direct, Inverse, Joint, and Combined Variation

A step-by-step guide to direct, inverse, joint, and combined variation

In a mathematical equation that establishes a relationship for some type of parameter, there are usually two types of quantities. One is a constant that does not change with the changes of other parameters in the equation and the other is a variable that changes for different situations. Changing variable parameters is called variation. Variation can have different types according to the pattern of change or relationships of variables:

Direct variation:

Direct variation between two variables exists when one quantity is directly dependent on the other, i.e. if one quantity increases relative to the other and vice versa. It is a relationship between two variables where one variable is a constant multiple of the other. Since these two variables are directly related to each other, it is also called directly proportional.

When two quantities are directly proportional to each other, they are represented using the symbol “\(∝\)”. Suppose there are two quantities \(x\) and \(y\) that are in direct variation, then they are expressed as follows:

\(y∝x\)

When the proportionality sign is removed, the direct variation formula is given as follows:

\(\color{blue}{y=kx}\)

Where \(k\) is the constant of proportionality.

Inverse variation:

If one quantity is increasing and the other quantity is decreasing at the same time, the two quantities are said to be in inverse variation. It is expressed as \(y∝\frac{1}{x}\). The inverse variation formula is given by:

\(\color{blue}{y=\frac{k}{x}}\)

Joint variation:

Joint variation describes a situation in which one variable is dependent on two (or more) other variables and, when the others are held constant, varies directly as each of them. We say that \(z\) varies jointly as \(x\) and \(y\) if:

\(\color{blue}{z=kxy}\)

For some constant \(k\).

Combined variation:

Combined variation describes a situation in which one variable is dependent on two (or more) other variables, varying directly with some of them and inversely with others (when all other variables are held constant). For example, if \(z\) varies directly as \(x\) and inversely as \(y\), we have the following combined variation equation:

\(\color{blue}{z=k(\frac{x}{y})}\)

Direct, Inverse, Joint, and Combined Variation – Example 1:

Let x and y be in direct variation, \(x = 6\) and \(y = 28\). Find the equation of the direct variation.

Solution: As \(x\) and \(y\) are in a direct variation thus \(y = kx\) or \(k =\frac{y}{x}\).

\(k=\frac{28}{6}=\frac{14}{3}\)

The direct variation equation is \(y=\frac{14}{3}x\)