Question

What is the constant of variation,k, of the direct variation,y=kx,through (5,8)

Answer 1

we know that

A relationship between two variables, x, and y, represent a direct variation if it can be expressed in the form $\frac{y}{x}=k$ or $y=kx$

where

k is the constant of variation

in this problem we have

the point $(5,8)$

so

$x=5\\y=8$

substitute

$\frac{y}{x}=k$

$\frac{8}{5}=k$

therefore

the answer is

$k=\frac{8}{5}$

Answer 2

$\bf \qquad \qquad \textit{direct proportional variation} \\\\ \textit{\underline{y} varies directly with \underline{x}}\qquad \qquad y=kx\impliedby \begin{array}{llll} k=constant\ of\\ \qquad variation \end{array}\\\\ -------------------------------\\\\ (\stackrel{x}{5}~,~\stackrel{y}{8})\textit{ we also know that } \begin{cases} x=5\\ y=8 \end{cases}\implies 8=k5\implies \cfrac{8}{5}=k$