I have a system of linear equations with some constraints. I would appreciate it if someone could help me solving this system of equations in Python.

Should be fairly easy to solve with sympy: http://docs.sympy.org/dev/modules/solvers/solveset.html#sympy.solvers.solveset.linsolve

Add all your equations to a list. Then pass that list to linsolve(). You'll probably loop through each value of j to generate those equations.