Question

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?

Answer 1

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

then `underline(y=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 prop x^(-1)`

this is if and only if

`y prop 1/x`

`<=>`

`y=1/x k`

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

then

`6=1/4 k`

`<=>` multiply both sides by `4`

`24=k` this is our constant of proportionality

`=>`

this gives us our formula

`y=24/x`

Then consider when `y=8`

then

`8=24/x`

`<=>` multiply both sides by `x`

`8x=24`

`<=>` divide both sides by `8`

`x=3`

Answer 2

`K = 24`

`y = 24/x`

`x = 3` when `y = 8`

Explanation:

"y is inversely proportional to x":

`=> y prop 1/x`
`:. y = K 1/x = K/x`

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

`=> 6 = K/4`
`:. K = 24`

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

`K = 24 => y = 24/x`

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

`y = 8 => 8 = 24/x`
`:. x=24/8 = 3`