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Direct Variation

 
1.1

 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 \(n^3 \). 
 
Solution:
 

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 \((2)\):

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

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

 

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