# Suppose that y varies jointly with x and x inversely with z and y=540 when w=15, x=30, and z=5, how do you write the equation that models the relationship?

Nov 8, 2017

$y = 60 x + \frac{90}{z} + 120 w$

#### Explanation:

You have only referenced $w$ by its value and not included any anything linking it to $y$. So I am going to make an assumption about it.

The wording implies:

$y = {k}_{1} x$

$x = {k}_{2} / z$

$\textcolor{w h i t e}{}$

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$\textcolor{b l u e}{\text{Consider the case}}$
y=k_1x color(white)("d")->color(white)("d")540= k_1(30) => k_1=540/30 = 180

$y = 180 x$

$\textcolor{w h i t e}{}$

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$\textcolor{b l u e}{\text{Consider the case}}$
y=k_2/z color(white)("d")->color(white)("d")540= k_2/(5) => k_2=5xx540= 2700

$y = \frac{2700}{z}$

$\textcolor{w h i t e}{}$

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$\textcolor{b l u e}{\text{Consider the case } w = 15}$

Observe that $\textcolor{w h i t e}{\text{d}} 2 \times 15 = 30$
$\textcolor{w h i t e}{\text{ddddddd")=>color(white)("d}} 2 \times w = x$

Thus $\textcolor{w h i t e}{\text{d")y=k_1xcolor(white)("d")->color(white)("d}} y = {k}_{1} \left(2 w\right)$

as ${k}_{1} = 180$ then we have:

$\textcolor{w h i t e}{\text{dddddddddddd")->color(white)("d}} y = 180 \left(2 w\right)$

$\textcolor{w h i t e}{\text{dddddddddddd")->color(white)("d}} y = 360 w$
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$\textcolor{b l u e}{\text{Constructing "ul("an equation")" that links all three}}$

There are a number of equation structures that can link $w , x , y \mathmr{and} z$

Lets pick on the most strait forward.

$y = 180 x = \frac{270}{z} = 360 w$

So we have: $\textcolor{w h i t e}{\text{d}} 3 y = 180 x + \frac{270}{z} + 360 w$

Notice that 3, 18, 27 and 36 are all exactly divisible by 3. Consequently 180, 270 and 360 are also exactly divisible by 3

Dividing all of both sides by 3

$y = 60 x + \frac{90}{z} + 120 w$