Classification of linear systems of partial differential equations

I'm working with the linear equations of poroelasticity (describing flow and deformation processes in porous media). The particular equations i'm working with (in 1D) are of the form

$$-\left(\lambda + 2\mu\right)\frac{d^2u}{dx} + \alpha\frac{dp}{dx} = f$$ $$\frac{d}{dt} \left(c_0p + \alpha \frac{du}{dx}\right) -\frac{1}{\mu_f} \frac{d}{dx}\left(k(x)\frac{dp}{dx}\right)=h$$

I'm familiar with PDE Theory as it applies to elliptic, parabolic, and hyperbolic equations (of a single dependent variable), but I don't know if this theory extends to include systems of linear PDE's as well. Is there an extension of PDE theory to linear systems of PDE's? In particular, how do I classify a system as opposed to a single equation?

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