Several necessary and sufficient conditions for the existence of uniform or C¹-approximation of functions on compact subsets of ℝ² by solutions of elliptic systems of the form c₁₁u_x1x1 + 2c₁₂u_x1x2 + c₂₂u_x2x2 = 0 with constant complex coefficients c₁₁, c₁₂, and c₂₂ are obtained. The proofs are based on a refinement of Vitushkin's localization method, in which one constructs localized approximating functions by 'gluing together' some special many-valued solutions of the above equations. The resulting conditions of approximation are of a topological and metric nature.