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Simultaneous Real Satisfaction (Posted on 2015-11-28) Difficulty: 3 of 5
Find all possible real numbers M and N that satisfy this system of simultaneous equations:

M + (3M-N)/(M2 + N2) = 3, and:
N = (M+3N)/(M2 + N2)

No Solution Yet Submitted by K Sengupta    
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Solution Solution Comment 1 of 1
The second equation can be solved as M*N = 1/2 +/- sqrt[-N^4+3N^2+1/4].

Solving both equations for M^2+N^2 yields -(3M-N)/(M-3) = (M+3N)/N.  This eventually solves to M = -3N+3/2 +/- sqrt[10N^2+9/4]

After lengthy algebra, three answers emerge: (M,N) = (2,1), (0,0), (1,-1).  The (0,0) must be discarded for division by zero, leaving (2,1) and (1,-1) as the final answers.

  Posted by Brian Smith on 2016-12-18 18:04:56
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