Convert system of second order PDEs to a system of first order PDEs I am attempting so convert
$$H_t = D_H H_{xx} - vH_x - \alpha H Z$$
$$Z_t = D_Z Z_{xx} + \beta H Z $$
into a system of first order equations. $D_H$, $D_Z$, $\alpha$ and $v$ are constants and $H$ and $Z$ describe the population of humans and zombies which interact as with the equations above.  
Following the post from Harry49 in >>here<< I first tried to rewrite the equations as a linear system of balance laws $\mathbf{q_t + A ~ q_x = S ~ q}$ where $\mathbf{q}$ = $(H, H_x, H_t, Z, Z_x, Z_t)^T$.
This gives me 
$$\begin{bmatrix} H_t \\ H_{xt} \\ H_{tt} \\ Z_t \\ Z_{xt} \\ Z_{tt} \end{bmatrix} + \mathbf{A} \begin{bmatrix} H_x \\ H_{xx} \\ H_{tx} \\ Z_x \\ Z_{xx} \\ Z_{tx} \end{bmatrix}  = \mathbf{S} \begin{bmatrix} H \\ H_{x} \\ H_{t} \\ Z \\ Z_{x} \\ Z_{t} \end{bmatrix}$$
My next step would be to determine the coefficients of the matrices $\mathbf{A}$ and $\mathbf{S}$, which is where I get confused due to the coefficients that couple the equations. I am not even sure if the approach I chose is a legitimate one for my kind of problem and would appreciate any kind of help!
Thank you very much!

The basic idea is to introduce new variables $X = H_x$, $Y = Z_x$, that gives directly:
$$
H_t = D_H X_x - v H_x - \alpha H Z \\
X   = H_x \\
Z_t = D_Z Y_x + \beta H Z \\
Y   = Z_x
$$
