How to find the exact number of nonnegative integer solutions of the following set of equations :
$$x_1 + x_2 + x_3 + x_4 + x_5 + x_6=6 $$ $$ 2x_1 + x_2 + x_3 = 4$$ $$ x_2 + 2x_4 + x_5 = 4$$ $$ x_3 + x_5 +2x_6 =4$$
How to find the exact number of nonnegative integer solutions of the following set of equations : $$x_1 + x_2 + x_3 + x_4 + x_5 + x_6=6 $$ $$ 2x_1 + x_2 + x_3 = 4$$ $$ x_2 + 2x_4 + x_5 = 4$$ $$ x_3 + x_5 +2x_6 =4$$ 

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This is going to be closed, but I'll answer anyway: Here's how to solve your problem using 'zsolve' from 4ti2 (according to the manual www.4ti2.de/4ti2_manual.pdf) To solve a linear system $Ax = b$ under nonnegativity $x\in\mathbb{N}_0^n$ create a file A.mat with the matrix:
The first two rows are the dimensions. Then create A.rhs to save the right hand side (as a row vector):
Finally, create A.sign to encode nonnegativity (there are other choices too...)
Now run
Voilà, the file A.zinhom contains your 15 points:


