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Nonlinear simultaneous eqn

  1. Jul 6, 2011 #1
    Hi all,

    I have more or less convinced myself through trial and error that the following three-dimensional non-linear simultaneous equation cannot be solved. However, it would be great if someone could provide me with a proper mathematical reason as to why this is not solvable, rather than me simply stating I can't do it...

    A = x/y
    B = y/z
    C = x/z
    for {x,y,z} in terms of {A,B,C} only

    Any help is greatly appreciated,

  2. jcsd
  3. Jul 6, 2011 #2
    One problem is you have A = x/y, or x = Ay, and C = x/z, or x = Cz. This means Ay = Cz, or y/z = C/A = B (or AB = C, which is all you can really discover, other than the only solution is (0,0,0)).
  4. Jul 6, 2011 #3
    Multiplying your first two equations and dividing by the third gives AB/C = 1. If that is not true, there is no solution. If AB/C = 1, there are an infinite number of solutions. Choose any value for z you like, then x = Cz, y = Bz is a solution.
  5. Jul 6, 2011 #4
    You can also look at it from a spatial persepctive. The first equation gives a plane passing through the z axis defined by y=Ax. The second gives another plane passing through the x axis defined by y = Bz. The third gives a plane passing through the y axis defined by x = Cz. The intersection of three planes, none of which are parallel, coplanar, etc., is a point. The only solution is the point (0, 0, 0), regardless of what A, B and C are.

    Edited: By only solution, I mean the only solution regardless of what A, B and C are. You can have other solutions that depend on what A, B and C are.
    Last edited: Jul 6, 2011
  6. Jul 6, 2011 #5
    Great, thanks for the help is visualizing that!
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