So, if the sign of a, b and c is the same and opposite to that of m, the plane doesn't pass through the first "octant" and there is no solution.

If a, b and c are not all of the same sign, the extension of the plane through the first octant is infinite. If a, b, c and m are rational, the number of solutions is infinite, too.

If the coefficients are real but not rational, the problem becomes more difficult and has no general solution, I believe.

If a, b, c and m are all positive, the portion of the plane in the first octant is limited to a triangle and there is a finite number of solutions. But I have to work more to understand how to calculate this number. If someone has an idea ...

What do you mean ?

Is it the number of solutions for the triplet (x,y,z) as a function of the given parameters a,b,c and m ?

?

And, must be a, b, c and m integers, too ? Or just rational ?

Ok !

Your expression is the equation of a plane in R^3

So, if the sign of a, b and c is the same and opposite to that of m, the plane doesn't pass through the first "octant" and there is no solution.

If a, b and c are not all of the same sign, the extension of the plane through the first octant is infinite. If a, b, c and m are rational, the number of solutions is infinite, too.

If the coefficients are real but not rational, the problem becomes more difficult and has no general solution, I believe.

If a, b, c and m are all positive, the portion of the plane in the first octant is limited to a triangle and there is a finite number of solutions. But I have to work more to understand how to calculate this number. If someone has an idea ...