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}