In this paper, a simple trick is worked out which immediately leads on to the derivation of the Fredholm integral equations in isotropic and anisotropic theory of elasticity for the first and the second boundary value problems. This trick is based on the circumstance that the system of equations of anisotropic theory of elasticity has simple complex characteristics. This circumstance is crucial, since it leads on to a simple system of linear algebraic equations. This approach can be extended to arbitrary second order elliptic systems with constant coefficients in the plane, when boundary operators contain only zero order or the first order derivatives. It is specific that this approach does not require a knowledge of the fundamental solution.