 # 5.2 Inverse Variation

5.2 Inverse Variation
If a quantity y  varies inversely as another quantity x, then
(a) y increases when x decreases,
(b) y decreases when x increases

5.2b Expressing an inverse variation in the form of an equation
An inverse variation can be written in the form of an equation, $y=\frac{k}{x}$ where k is a constant which can be determined.

Example 1:
Given y  varies inversely as x and y = 4 when x =10. Write an equation which relates x and y.

Solution:
$\begin{array}{l}y\propto \frac{1}{x}\to y=\frac{k}{x}\\ 4=\frac{k}{10}\to k=40\\ \therefore y=\frac{40}{x}\end{array}$

Example 2:
Given that y = 3 when x = 6, find the equation relates x and y if:

Solution:
$\begin{array}{l}\left(a\right)\text{}y\propto \frac{1}{{x}^{2}}\text{}\to y=\frac{k}{{x}^{2}}\\ 3=\frac{k}{{6}^{2}}\\ k=3×36=108\\ \therefore y=\frac{108}{{x}^{2}}\\ \\ \text{(b)}y\propto \frac{1}{{x}^{3}}\text{}\to y=\frac{k}{{x}^{3}}\\ 3=\frac{k}{{6}^{3}}\\ k=3×216=648\\ \therefore y=\frac{648}{{x}^{3}}\\ \\ \text{(c)}y\propto \frac{1}{\sqrt{x}}\text{}\to y=\frac{k}{\sqrt{x}}\\ 3=\frac{k}{\sqrt{6}}\\ k=3×\sqrt{6}=3\sqrt{6}\\ \therefore y=\frac{3\sqrt{6}}{\sqrt{x}}\end{array}$