Let's define a linear equation as a tuple $\left\langle a, b, x, /, \cdot, =, \sim \right\rangle$, expressing an equation $x \cdot a = b$ (with respect to the unknown $x$) and a solution $x \sim b / a$.
Here, $a \in A, b \in B$, and $x \in X$ are elements of some sets, $\cdot: X \times A \rightarrow B$ and $/: B \times A \rightarrow X$ are some binary operations, $=$ is an equality relation on $B$, and $\sim$ is an equality relation on $X$.
When $A = B = X$, we have an entity called quasigroup. Though, there are cases when this abstract structure is not general enough: mind matrices and vectors. Given an invertible matrix $A$ and a vector $b$, the equation $A x = b$ has a solution $x = A^{-1} b$. This is obviously not a quasigroup (if we don't embed vectors into the space of square matrices in a special way).
Is there a name for the general structure of solvable linear equations?