How to Solve Inverse Variation?

Step-by-step guide to inverse variation

There are two types of proportionalities. These are direct variation and inverse variation. Two non-zero quantities have an inverse variation if their product yields a constant term (constant of proportionality). In other words, if one quantity is directly proportional to the reciprocal of the other quantity, the two quantities follow inverse variations. This means that an increase in one value leads to a decrease in the other value while a decrease in one value leads to an increase in the other value.

Inverse variation formula

The symbol “\(∝\)” is used to indicate proportionality. If two quantities x and y follow an inverse variation, they are represented as:

\(\color{blue}{x\:∝\:\frac{1}{y}\:or\:y\:∝\:\frac{1}{x}}\)

To convert this expression into an equation, a constant or proportionality factor must be introduced. Therefore, the inverse variation formula is presented as follows:

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

Here, \(k\) is the constant of proportionality. Also, \(x ≠ 0\) and \(y ≠ 0\).

Product rule for inverse variation

Suppose the two solutions of inverse variation are \((x_1, y_1)\) and \((x_2, y_2)\). This can also be expressed as \(x_1y_1=k\) and \(x_2 y_2=k\).

Using these two equations:

\(x_1y_1 = x_2y_2\) and \(\frac{x_1}{x_2}=\frac{y_2}{y_1}\).

This is the product rule for inverse variation.

Inverse variation graph

The graph of an inverse variation is a rectangular hyperbola. If there are two quantities \(x\) and \(y\) are in inverse variation then their product will be equal to a constant \(k\). Since neither \(x\) nor \(y\) can be zero, the graph never crosses the \(x\)-axis or the \(y\)-axis. The graph of an inverse variation with the function \(y= \frac{k}{x}\) is given below:

Inverse Variation – Example 1:

If \(x = 15\) and \(y = 4\) follow an inverse variation, find the constant of proportionality.

Solution: Since \(x\) and \(y\) are in inverse variation therefore, \(xy=k\):

\(k= 15\times 4=60\)