  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.
