# How do you find the variation constant and an equation of variation where y varies inversely as x and y = 6 when x = 14?

May 20, 2016

Variation constant is $84$

#### Explanation:

The maths symbol for proportional to is:$\text{ } \alpha$

Given that $\text{ } y \textcolor{w h i t e}{.} \alpha \textcolor{w h i t e}{.} \frac{1}{x}$

Let $k$ be the constant of variation giving

$y \text{ "=" "kxx1/x" "=" } \frac{k}{x}$
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Given condition $\textcolor{b l u e}{\text{ if "x=14" then } y = 6}$

So by substitution

color(brown)(y=k/x" "->" "color(blue)(6)=k/(color(blue)(14))

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Multiply both sides by $\textcolor{g r e e n}{14}$

$\implies \textcolor{b l u e}{6 \textcolor{g r e e n}{\times 14} \textcolor{b r o w n}{=} \frac{\textcolor{b r o w n}{k}}{14} \textcolor{g r e e n}{\times 14}}$

$\implies 84 = k \times \frac{14}{14}$
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But $\frac{14}{14} = 1$

$84 = k \times 1$

$\textcolor{m a \ge n t a}{k = 84 \to \text{variation constant}}$

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So $\text{ "color(brown)(y color(white)(.) alpha color(white)(.) 1/x)" "->" } \textcolor{b l u e}{y = \frac{84}{x}}$