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 \(n^3 \)

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 \((2)\):

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

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


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 \(x^2\).

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

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 \((2)\):

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


Joint Variation

Definition joint variation
In general, for a combined variation\(y\) varies directly as \(x^m\) and inversely as \(z^n\) can be written as

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

\(\begin{aligned} x=\frac{kx^m}{z^n} \end{aligned} \hspace{1mm} \text{(equation relation)}\)


such that

\(\begin{aligned} m&=1,2,3,\frac{1}{2},\frac{1}{3},\hspace{1mm} \\\\n&=1,2,3,\frac{1}{2},\frac{1}{3} \end{aligned}\)

and \(k\) is a constant.

Example 3
Given that \(y\) varies directly as then square of \(x\)  varies inversely as square root of \(z\). If \(y=8\) when \(x=4\) and \(z=36\), express \(y\) in terms of \(x\) and \(z\).

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

Substitute \(y=8\)\( x=4\), and

\(z=36\) into \((1)\):

\(\begin{aligned}8&=\frac{k4^2}{\sqrt{36}}\implies k=\frac{(8)(6)}{16}\\\\&=3.\\\\ &\therefore y=\frac{3x^2}{\sqrt{z}}. \end{aligned}\)