Question 1164209
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For linear  Diophantine equations  (one equation with integer coefficients for two integer unknowns)  the trial an error method 

finding a solution is absolutely and totally  LEGAL  (and standard)  method.



For such equations,  any version of trial and error search is  ALLOWABLE,  and nobody of mature mathematicians 

do not really care by  WHICH  concretely searching method the solution is obtained.



You may use an  Excel table/spreadsheet for it and to find the solution in seconds  (as I usually do it . . . )


Nobody cares about it . . . 



Or,  alternatively,  you can find it via  30  lines formula transformations.   It is also good,

but again,  NOBODY CARES about it . . . 



What is  REALLY  IMPORTANT  for such equations,  it is the necessary and sufficient condition for  {{{highlight(existing)}}}  the solution.


This condition  REQUIRES  that the right side constant term must be a MULTIPLE of the Greatest Common Divisor of the coefficients

of the equation at  "x"  and  "y".


If this condition is satisfied,  then the solution  DOES EXIST,  and all the troubles just do stop at this point.