Find the number of non-negative integer solutions to the equation.
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 ...
And, must be a, b, c and m integers, too ? Or just rational ?
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 ?