# Suppose y is inversely proportional to x. If y = 6 when x = 4, how do you find the constant of proportionality and write the formula for y as a function of x and use your formula to find x when y = 8?

Jul 31, 2017

If $y$ is inversely proportional to $x$, and if $y = 6$ when $x = 4$

then $\underline{y = \frac{24}{x}}$.

Using this formula we find that if $y = 8$ then $\underline{x = 3}$

#### Explanation:

Given, $y$ is inversely proportional to $x$

or $y \propto {x}^{- 1}$

this is if and only if

$y \propto \frac{1}{x}$

$\iff$

$y = \frac{1}{x} k$

Also, we know that if $y = 6$ when $x = 4$

then

$6 = \frac{1}{4} k$

$\iff$ multiply both sides by $4$

$24 = k$ this is our constant of proportionality

$\implies$

this gives us our formula

$y = \frac{24}{x}$

Then consider when $y = 8$

then

$8 = \frac{24}{x}$

$\iff$ multiply both sides by $x$

$8 x = 24$

$\iff$ divide both sides by $8$

$x = 3$

Jul 31, 2017

$K = 24$

$y = \frac{24}{x}$

$x = 3$ when $y = 8$

#### Explanation:

"y is inversely proportional to x":

$\implies y \propto \frac{1}{x}$
$\therefore y = K \frac{1}{x} = \frac{K}{x}$

"If $y = 6$ when $x = 4$, how do you find the constant of proportionality":

$\implies 6 = \frac{K}{4}$
$\therefore K = 24$

"write the formula for y as a function of x"

$K = 24 \implies y = \frac{24}{x}$

"use your formula to find $x$ when $y = 8$"

$y = 8 \implies 8 = \frac{24}{x}$
$\therefore x = \frac{24}{8} = 3$