Integer programming, not linear programming. Linear programming (without the integrality constraint) is in P, so you cannot use it to solve general SAT problems

Integer programming, not linear programming. Linear programming (without the integrality constraint) is in P, so you cannot use it to solve general SAT problems, which are NP-complete (unless a major and highly surprising theoretical breakthrough is found). The free SAT solvers are very good, and much better than commercial IP solvers at solving problems that are a natural fit for SAT. (Obviously, you can encode any IP as an SAT formula and vice versa, and the IP solvers are better at solving the problems where you actually have meaningful arithmetic.)