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Integer Solutions (Posted on 2006-03-30) Difficulty: 3 of 5
Determine all integer solutions (x,y,z) for the system of equations

x²z + y²z + 4xy = 40,

x² + y² + xyz = 20

See The Solution Submitted by Bractals    
Rating: 2.5000 (6 votes)

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kiss- corrected and reinforced | Comment 15 of 16 |

LET

X^2+Y^2=W  AND XY=T

WE GET:      WZ+4T=40             (1)

AND             W+TZ=20            ( 2)  multiply by z and

SUBTRACT   (1)-(2)*Z   ====>  T=20/(2+Z)

 

CHECK FOR  Z =  0,2,3,8 AND 18   to get integer T and then evaluate W.  ONLY Z=3 AND Z=18 QUALIFY.

The rest follows easily.T=4 ===>x=y +2 or -2

                                 T=1           x=y +1 or -1

                                               

    remark:

There is much shorter   way to solve- by reasoning that symmetry implies x=y (writing x instead of y and vice versa leaves the set of equations unchanged...)

So  ( 2x^2)*z+4*x^2=40

         the other equation becomes redundant, - both imply

x^2=20/(z+2  )  and we throw away the non-integer solutions

ady

Edited on April 1, 2006, 4:04 am
  Posted by Ady TZIDON on 2006-04-01 03:49:27

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