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Since your equations are linear, and your constraints about the variables is that they are integer, i strongly suggest to look at linear programming in Sage:

• http://doc.sagemath.org/html/en/thematic_tutorials/linear_programming.html
• http://doc.sagemath.org/html/en/reference/numerical/sage/numerical/mip.html

Do not hesitate to ask for more details if something is not clear (with your attempts so that we can discuss them).