We study systems of partial differential-algebraic equations
(PDAEs) of first order.  Classical solutions of the theory of Hyperbolic Partial Differential
 equation such as discontinuities (shock and contact discontinuities),
rarefactions and diffusive traveling waves  are extended for variables
living on a surface $\mathcal{S}$, which is defined as solution of a set of
algebraic equations.  We propose here an alternative formulation to study
numerically and theoretically the PDAEs by changing the algebraic conditions into
partial differential equations with relaxation source terms (PDREs).
The solution of such relaxed systems is proved to tend to the surface $\mathcal{S}$, i.e., to satisfy the algebraic equations for long times.
We formulate a unified numerical scheme for  systems of PDAEs and PDREs. This scheme is naturally parallelizable and has faster convergence.
Evidence of its  effectiveness  is presented by means of   simulations for physical and synthetic problems.