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Math Help - simplification through substitution

  1. #1
    Newbie
    Joined
    Sep 2010
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    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 \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 \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 {x+y}= \frac {x+y-xn}{1-z}
    Can I simplify this any farther or is there a step that I messed up fully?
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  2. #2
    Super Member
    Joined
    Jan 2008
    Posts
    588
    Thanks
    87
    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.
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  3. #3
    Newbie
    Joined
    Sep 2010
    Posts
    2
    Thanks a lot!
    This makes a lot more sense then what I tried at first,
    really appreciate it!
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