Direct Variation

1.1

Direct Variation

Definition of direct variation

Direct variation explains the relationship between two variables, such that when variable \(y\) increases, then variable \(x\) also increases at the same rate and vice versa.

This relation can be written as \(y\) varies directly as \(x\) .

In general, for a direct variation, \(y\) varies directly as \(x^n\) can be written as

\(\begin{aligned}x\propto x^n\end{aligned}\hspace{1mm}\text{(variation relation)}\) or \(\begin{aligned} x=kx^n \end{aligned} \hspace{1mm} \text{(equation relation)}\)

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

Example 1

Given \(m=12\) when \(n=3\).

Express \(m\) in terms of \(n\) if

a) \(m\) varies directly as \(n\).

b) \(m\) varies directly as \(n^3 \).

Solution:

a) \(n\implies m = kn \dots (1)\).

Substitute \(m=12\) and \(n=3\) into \((1)\)

\(12=k(3)\implies k=\dfrac{12}{3}=4\).

\(\therefore m=4n\).

b) \(m\propto n^3\implies m = ln^3 \dots (2).\)

Substitute \(m=12\) and \(n=3\) into \((2)\):

\(12=l(3)^3\implies l=\dfrac{12}{27}=\dfrac{4}{9}\)

\(\therefore m=\dfrac{4}{9}n^3.\)