The Maker-Breaker Rado game on a random set of integers

Given an integer-valued matrix $A$ of dimension $\ell \times k$ and an integer-valued vector $b$ of dimension $\ell$, the Maker-Breaker $(A,b)$-game on a set of integers $X$ is the game where Maker and Breaker take turns claiming previously unclaimed integers from $X$, and Maker's aim is to obtain a solution to the system $Ax=b$, whereas Breaker's aim is to prevent this. When $X$ is a random subset of $\{1,\dots,n\}$ where each number is included with probability $p$ independently of all others, we determine the threshold probability $p_0$ for when the game is Maker or Breaker's win, for a large class of matrices and vectors. This class includes but is not limited to all pairs $(A,b)$ for which $Ax=b$ corresponds to a single linear equation. The Maker's win statement also extends to a much wider class of matrices which include those which satisfy Rado's partition theorem.