The Backward Euler scheme: \begin{align} u^{n+1} &= u^n + \Delta t (a u^{n+1} + b v^{n+1})\\ v^{n+1} &= v^n + \Delta t (c u^{n+1} + d v^{n+1}) \end{align} which is a 2\times 2 linear system: \begin{align} (1 - \Delta t a)u^{n+1} + bv^{n+1} &= u^n \\ c u^{n+1} + (1 - \Delta t d) v^{n+1} &= v^n \end{align}
Crank-Nicolson also gives a 2\times 2 linear system.