  We use partial differential equations (PDEs) to describe physical systems. In
general, these equations include evolution and constraint equations. One method
used to find solutions to these equations is the Free-evolution approach, which
consists in obtaining the solutions of the entire system by solving only the
evolution equations. Certainly, this is valid only when the chosen initial data
satisfies the constraints and the constraints are preserved in the evolution.
In this paper, we establish the sufficient conditions required for the PDEs of
the system to guarantee the constraint preservation. This is achieved by
considering quasi-linear first-order PDEs, assuming the sufficient condition
and deriving strongly hyperbolic first-order partial differential evolution
equations for the constraints. We show that, in general, these constraint
evolution equations correspond to a family of equations parametrized by a set
of free parameters. We also explain how these parameters fix the propagation
velocities of the constraints.
  As application examples of this framework, we study the constraint
conservation of the Maxwell electrodynamics and the wave equations in arbitrary
space-times. We conclude that the constraint evolution equations are unique in
the Maxwell case and a family in the wave equation case.
