I know how to rewrite systems of 2nd order DEs to a system of 1st order DEs and would like to do the same with this system: \begin{align} a x' + b y' + c x & = z(t) \\ d x' + e y' + f y & = 0 \end{align} where a to f are constants.
EDIT:
Proposed solution in a now deleted comment was to isolate like this: \begin{align} x' & = \frac{z(t) - b y' - c x}{a} \\ y' & = \frac{- d x' - f y}{e} \end{align}
and then insert them into the original system like this: \begin{align} a x' + b \frac{- d x' - f y}{e} + c x & = z(t) \\ d \frac{z(t) - b y' - c x}{a} + e y' + f y & = 0 \end{align}
and then carry on from there.
Thanks for your time!
You are almost done. Isolate $x'$ and $y'$ on the left hand side of the equalities. For instance $$ a\,x' + b \frac{- d\,x' - f\,y}{e} + c\,x = z(t)\implies \Bigl(a-\frac{b\,d}{e}\Bigr)x'=-c\,x+\frac{b\,f}{e}\,y+z(t). $$ Observe that this is solving the original system for $x'$ and $y'$.