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.