Nonlinear simultaneous eqn

1. Jul 6, 2011

natski

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...

Solve...
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,

natski

2. Jul 6, 2011

daveb

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)).

3. Jul 6, 2011

pmsrw3

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.

4. Jul 6, 2011

daveb

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
5. Jul 6, 2011

natski

Great, thanks for the help is visualizing that!