Inverse Variation

1.2

Inverse Variation

Definition inverse variation

In inverse variation, variable \(y\) increases when the variable \(x\) decreases at the same rate, and vice versa. This relation can be written as \(y\) varies inversely as \(x\).

In general,For an inverse variation, \(y\) varies inversely as \(x^n\) can be written as \(x^n\)

\(\begin{aligned}y\propto \frac{1}{x^n}\end{aligned}\hspace{1mm}\text{(variation relation)}\) or

\(\begin{aligned} y=\frac{k}{x^n} \end{aligned} \hspace{1mm} \text{(equation relation)}\)

where \(\begin{aligned} n=1,2,3,\frac{1}{2},\frac{1}{3} \end{aligned}\) and \(k\) is a constant.

Example 2

Given \(y=3\) when \(x=7\).

Express \(y\) in terms of \(x\) if

a) \(y\) varies inversely as \(x\).

b) \(y\) varies inversely as \(x^2\).

Solution:

\(\begin{aligned}a)\hspace{1mm}& y\propto \frac{1}{x}\implies y = \frac{k}{x} \dots (1) \end{aligned}\)

Substitute \(y=3\) and \(x=7\) into \((1)\):

\(\begin{aligned}3&=\frac{k}{7}\implies k=(3)(7)\\\\&=21.\\\\&\therefore y=\dfrac{21}{x}. \end{aligned}\)

\(\begin{aligned}b)\hspace{1mm}& y\propto \frac{1}{x^2}\implies y = \frac{l}{x^2} \dots (2) \end{aligned}\)

Substitute \(y=3\) and \(x=7\) into \((2)\):

\(\begin{aligned}3&=\frac{l}{7^2}\implies k=(3)(49)\\\\&=147.\\\\ &\therefore y=\dfrac{147}{x^2}. \end{aligned}\)