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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 \((2)\):

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

 

The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of Pandai.
Chapter : Variations
Topic : Inverse Variation
Form 5 Mathematics
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