We consider subsets of the natural numbers that contains infinitely many aritmetic progressions (APs) of any given length - such sets will be called AP-sets and we know due to the Green-Tao Theorem that the primes is an AP-set. We prove that the equation

where $M$ is an integer matrix whose null space has dimension at least $2$, has infinitely many solutions in any AP-set such that the coordinates of each solution are elements in the same AP, if and only if $(1,1,\ldots,1)$ is a solution.

We will furthermore prove that AP-sets are exactly the sets that has infinitely many solutions to a homogeneous system of linear equations, whenever the sum of the columns is zero.