Hey guys,

I'm working on a physics lab that is trying to derive a specific equation by using other equations but I'm having problems...

Heres the question:

Derive ρk= ρg+ (ρl - ρg)fk

where:

ml + mg = mk

vg + vl= vk

mg = ρg*vg

ml= ρl*vl

mk = ρk*vk

vl= fk*vk

vg= (1-fk)vk

so I first tried to simplify the first equation by putting every value in terms of m and v but I hit a wall

$\displaystyle \displaystyle \frac {mg+ml}{vg+vl} = \frac {mg}{vg} +( \frac {ml}{vl}- \frac {mg}{vg}) \frac {vl}{vk}$

so looking at this I can make it look a little more pleasant by turing this into a four variable equation if this could help

mg=x ml=y vg=z vl=n

$\displaystyle \displaystyle \frac {x+y}{z+n} = \frac {x}{z} +( \frac {y}{n}- \frac {x}{z}) \frac {n}{z+n}$

I've tried seperating it in several ways with no luck the closest I've gotten is (assuming I didn't mess up)

$\displaystyle \displaystyle {x+y}= \frac {x+y-xn}{1-z}$

Can I simplify this any farther or is there a step that I messed up fully?