The primitive physical equations for elastic problems and incompressible Newtonian fluid flows are first-order partial differential systems: the stress-displacement and the stress-velocity-pressure formulations, respectively. Traditionally, these first-order systems are converted into second-order partial differential systems with less variables through differentiation and elimination. Computations are then based on these well-known second-order systems. Although substantial progress in numerical methods and in computations has been achieved, these problems may still be difficult and expensive to solve. In this talk, we will first derive the stress-velocity formulation for incompressible Newtonian fluid flows since the pressure can be represented in terms of the normal stresses. We will then introduce a least-squares method applied to the first-order system: the stress-displacement (velocity) formulation. The principal attractions of our least-squares method include freedom in the choice of finite element spaces, fast MULTIGRID solver for the resulting algebraic equations, a practical and sharp a posteriori error indicator for adaptive mesh refinements at no additional cost, and the same approaches of spatial discretization and fast solution solver for both solid and fluid problems. Numerical results for a benchmark test problem of planar elasticity will be presented.