Conventional approaches for solving the Navier-Stokes equations of incompressible fluid dynamics are the primitive-variable approach and the vorticity-velocity approach. In this paper, an alternative approach is presented. In this approach, pressure and one of the velocity components are eliminated from the governing equations. The result is one higher-order partial differential equation with one unknown for two-dimensional problems or two higher-order partial differential equations with two unknowns for three-dimensional problems. A meshless collocation method based on radial basis functions for solving the Navier-Stokes equations using this approach is presented. The proposed method is used to solve a two- and a three-dimensional test problem of which exact solutions are known. It is found that, with appropriate values of the method parameters, solutions of satisfactory accuracy can be obtained