Betti numbers and linear covers of points

We prove that for a finite set of points $X$ in the projective $n$-space over any field, the Betti number $\beta_{n,n+1}$ of the coordinate ring of $X$ is non-zero if and only if $X$ lies on the union of two planes whose sum of dimension is less than $n$. Our proof is direct and short, and the inductive step rests on a combinatorial statement that works over matroids.