NP-Completeness of deciding the feasibility of Linear Equations over binary-variables with coefficients and constants that are 0, 1, or -1

We convert, within polynomial-time and sequential processing, NP-Complete Problems into a problem of deciding feasibility of a given system S of linear equations with constants and coefficients of binary-variables that are 0, 1, or -1. S is feasible, if and only if, the NP-Complete problem has a feasible solution. We show separate polynomialtime conversions to S, from the SUBSET-SUM and 3-SAT problems, both of which are NP-Complete. The number of equations and variables in S is bounded by a polynomial function of the size of the NP-Complete problem, showing that deciding the feasibility of S is strongly-NP-Complete. We also show how to apply the approach used for the SUBSET-SUM problem to decide the feasibility of Integer Linear Programs, as it involves reducing the coefficientmagnitudes of variables to the logarithm of their initial values, though the number of variables and equations are increased.