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We address the problem of the best uniform approximation by linear combinations of a finite system of functions. If the system is Chebyshev and the problem is unconstrained, then the classical Remez algorithm provides a fast and precise solution. For non-Chebyshev systems, this problem may offer a great resistance. The same happens to approximations under linear constraints. We propose a solution by modifying the concept of alternance and of the Remez iterative procedure. A criterion of the best approximation is proved and the full set of polynomials of best approximation (which may not be unique in the non-Chebyshev case) is characterized. The method of finding the best polynomial is applicable for arbitrary functional systems under arbitrary linear constraints. The efficiency is demonstrated in examples with systems of complex exponents, Gaussian functions, and lacunar polynomials. As an application, the Markov-Bernstein type inequalities are obtained for those systems. Applications to signal processing, linear ODEs, switching dynamical systems are considered.