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?

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_1x#

#x=k_2/z#

#color(white)()#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Consider the case")#
#y=k_1x color(white)("d")->color(white)("d")540= k_1(30) => k_1=540/30 = 180#

#y=180x#

#color(white)()#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("Consider the case")#
#y=k_2/z color(white)("d")->color(white)("d")540= k_2/(5) => k_2=5xx540= 2700#

#y=2700/z#

#color(white)()#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("Consider the case "w=15)#

Observe that #color(white)("d")2xx15=30#
#color(white)("ddddddd")=>color(white)("d") 2xxw=x#

Thus #color(white)("d")y=k_1xcolor(white)("d")->color(white)("d")y=k_1(2w)#

as #k_1=180# then we have:

#color(white)("dddddddddddd")->color(white)("d")y=180(2w) #

#color(white)("dddddddddddd")->color(white)("d")y=360w#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Constructing "ul("an equation")" that links all three")#

There are a number of equation structures that can link #w,x,y and z#

Lets pick on the most strait forward.

#y=180x=270/z=360w#

So we have: #color(white)("d")3y=180x+270/z+360w#

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=60x+90/z+120w#