This paper presents preconditioned methods for solving the incompressible and low-speed compressible equations. The methods involve introducing artificial time derivatives to accelerate convergence to the steady state. The compressible equations are considered in conservation form with slow flow, and two arbitrary functions, α and β, are introduced in the general preconditioning. An analysis of this system is presented, and an optimal value for β is determined given a constant α. It is shown that the resultant incompressible equations form a symmetric hyperbolic system and are well-posed. Several generalizations to the compressible equations are presented, which generalize previous results. The paper discusses the incompressible flow equations and their extension to compressible flow with very low Mach numbers. It shows that the system is stiff due to the large ratio of acoustic and convective time scales. The paper generalizes previous approaches to preconditioning these equations. It also discusses the implementation of these methods for both explicit and implicit schemes, and shows that the preconditioning helps to equilibrate wave speeds and accelerate convergence. The paper also discusses the use of curvilinear coordinates and the symmetrization of the system. The results show that the preconditioned system is well-posed and that the optimal β is determined based on the flow velocity. The paper concludes that the preconditioned methods are effective for solving both incompressible and low-speed compressible equations.This paper presents preconditioned methods for solving the incompressible and low-speed compressible equations. The methods involve introducing artificial time derivatives to accelerate convergence to the steady state. The compressible equations are considered in conservation form with slow flow, and two arbitrary functions, α and β, are introduced in the general preconditioning. An analysis of this system is presented, and an optimal value for β is determined given a constant α. It is shown that the resultant incompressible equations form a symmetric hyperbolic system and are well-posed. Several generalizations to the compressible equations are presented, which generalize previous results. The paper discusses the incompressible flow equations and their extension to compressible flow with very low Mach numbers. It shows that the system is stiff due to the large ratio of acoustic and convective time scales. The paper generalizes previous approaches to preconditioning these equations. It also discusses the implementation of these methods for both explicit and implicit schemes, and shows that the preconditioning helps to equilibrate wave speeds and accelerate convergence. The paper also discusses the use of curvilinear coordinates and the symmetrization of the system. The results show that the preconditioned system is well-posed and that the optimal β is determined based on the flow velocity. The paper concludes that the preconditioned methods are effective for solving both incompressible and low-speed compressible equations.