Can Nonlinear Simultaneous Equations Be Solved?

[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

 

[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)).

 

[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.

 

[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.

 

[natski]

Great, thanks for the help is visualizing that!