Determine all possible real pairs (m,n) satisfying the following system of equations:

mn² = 15m²+ 17mn + 15n²

m²n = 20m² + 3n²

Rewriting the equations:

  mn(n-17) = 15(m^2 + n^2)        (1)

  m^2*(n-20) = 3n^2               (2)

Eq. (2) implies n != 20 and n != 17.

Multiplying (1) by n-20 gives

  mn(n-17)(n-20) = 15(m^2^[n-20] + n^2*[n-20])

                 or

  mn(n-17)(n-20) = 15(3n^2 + n^2*[n-20])

                 or

  mn(n-20) = 15n^2                (3)    

Squaring eq. (3) gives

  n^2*m^2*(n-20)(n-20) = 15^2*n^4

                 or

  n^2*3n^2*(n-20) = 15^2*n^4

                 or

  n^4*(n-95) = 0

Which implies n = 0 or n = 95.

Eq. (2) implies (m,n) = (0,0), (-19,95), and
(19,95) are solutions. But, eq. (1) implies
(-19,95) is not a solution. Therefore,

         (m,n) = (0,0) and (19,95)
             

Posted by Bractals on 2007-04-27 12:05:06