How to solve coupled linear 1st order PDE

It is fairly straight forward to solve linear 1st order PDEs by the method of characteristics. For example, if

${\mathrm{\partial}}_{t}f+a{\mathrm{\partial}}_{x}f=bf$

we have that $\frac{df}{dt}}=bf$ on the characteristic curve of $\frac{dx}{dt}}=a$. From this we deduce that $f(t,x)=g(C){e}^{bt}$ where $x=at+C$.

Now, how does this work when $f$ is multidimensional. Can I solve equations on the following form by characteristics, or by any other means?

${\mathrm{\partial}}_{t}{f}_{i}(t,x)+\sum _{j}{A}_{ij}{\mathrm{\partial}}_{x}{f}_{j}(t,x)=\sum _{j}{B}_{ij}{f}_{j}(t,x)$

where the components of $A$ and $B$ might be dependent on $x$ and $t$.

In particular, I am trying to solve the following,

$\{\begin{array}{l}{\mathrm{\partial}}_{t}f+{\displaystyle \frac{c}{t}}{\mathrm{\partial}}_{x}g=-(a+{\displaystyle \frac{1}{t}})f\\ {\mathrm{\partial}}_{t}g+{\displaystyle \frac{c}{t}}{\mathrm{\partial}}_{x}f=-(b+{\displaystyle \frac{1}{t}})g\end{array}$

where $f$ and $g$ are functions of $x$ and $t$ , where $t>{t}_{0}>0$, $c\ne 0$. Any help is highly appreciated.