1.2 Inverse Variation - SPM Mathematics

1.2 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

1.2b 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 \\ ∴ y = \frac{40}{x} \end{array}

Example 2:

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

(a) y \propto \frac{1}{x^{2}} (b) y \propto \frac{1}{x^{3}} (c) y \propto \frac{1}{\sqrt{x}}

Solution:

\begin{array}{l} (a) y \propto \frac{1}{x^{2}} \to y = \frac{k}{x^{2}} \\ 3 = \frac{k}{6^{2}} \\ k = 3 \times 36 = 108 \\ ∴ y = \frac{108}{x^{2}} \\ (b) y \propto \frac{1}{x^{3}} \to y = \frac{k}{x^{3}} \\ 3 = \frac{k}{6^{3}} \\ k = 3 \times 216 = 648 \\ ∴ y = \frac{648}{x^{3}} \\ (c) y \propto \frac{1}{\sqrt{x}} \to y = \frac{k}{\sqrt{x}} \\ 3 = \frac{k}{\sqrt{6}} \\ k = 3 \times \sqrt{6} = 3 \sqrt{6} \\ ∴ y = \frac{3 \sqrt{6}}{\sqrt{x}} \end{array}