1. ## simplification through substitution

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 + l - 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?

2. Wow that looks nasty!

I think you're trying to do too much here. Do it one by one and it turns out to be pretty easy.

Use the first equation

mk = ml + mg

Since mk = pk*vk, ml = pl * vl, mg = pg*vg

pk*vk = pl*vl + pg*vg

Now substitute vl = fk*vk and vg = (1-fk)vk

pk*vk = pl*fk*vk + pg(1-fk)(vk)

Divide through by vk

pk = pl(fk) + pg(1-fk)

Expand and rearrange to get

pk = pg + (pl - pg)fk

Don't be flustered next time. Take it one step at a time.

3. Thanks a lot!
This makes a lot more sense then what I tried at first,
really appreciate it!