0/1 Integer Programming and Karp's Reduction
I believe there's a typo in Karp's definition of the problem. If we replace $Ax=b$ with $Ax \geq b$, the reduction goes through as is. The easiest way I can think of to show the hardness of your problem (the one as written in Karp's paper), is via 1-in-3 SAT or via hypergraph perfect matching.