Hi everyone, I would like to ask your help in the following problem, I want to find nonnegative solutions for a system of linear Diophantine equations using some conditions on the constant terms. For example, in the following linear system the variables are $x,y_1,y_2,z_1,z_2,w$, and $n,a,b,c_1,c_2$ are nonnegative integers: $$x+w =a $$ $$ y_1+y_2+nw=b$$ $$y_1+w-z_1=c_1$$ $$y_2+w-z_2=c_2$$ and we have the next conditions for the constants: $$0\leq b-an, \;\; 0\leq a-c_1, \;\; 0\leq a-c_2, \;\; \mbox{and} \;\; 0\leq b-c_1-c_2.$$ I tried to use FindInstance[ ] but I had mistakes even if I don't try to write the conditions for the integers $n,a,b,c_1,c_2$: FindInstance[ x + w == a && y1 + y2 + n w == b && y1 + w - z1 == c1 && y2 + w - z2 == c2 && 0 <= x && 0 <= y1 && 0 <= y2 && 0 <= w && 0 <= z1 && 0 <= z2, {x, y1, y2, w, z1, z2}, Integers] FindInstance::exvar: "The system contains a nonconstant expression a independent of variables {x,y1,y2,w,z1,z2}" If I try to add the conditions for the constants, then I obtain the same mistake. Thanks.

Seems to me you are trying to use FindInstance to find a symbolic instance. I don't think it does that.

Thank you for your comment, I didn't know Thread. That was a problem with FindInstance, such constraints for the constants made the mistake. I just try with Thread before FindInstance and the mistake remains, but before Solve and considering only the linear system everything works fine. Indeed, I didn't use SetAttributes and requiring integer solutions, in the conditional expression obtained appears the hypothesis that $a,b,c_1,c_2,n$ are integer numbers.

Hi. I think you are missing some constraints. const = Thread[ {a, b, n, c1, c2, b - a*n, a - c1, a - c2, b - c1 - c2} >= 0 ]

Solve system of linear Diophantine equations with conditions