The Backward Euler method gives a system of algebraic equations

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.