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Linear I/O systems are a fundamental tool in systems theory, and have been used to design complex circuits and control systems in a variety of settings. Here we present a principled design method for implementing arbitrary linear I/O systems with biochemical reactions. This method relies on two levels of abstraction: first, an implementation of linear I/O systems using idealised chemical reactions, and second, an approximate implementation of the ideal chemical reactions with enzyme-free, entropy-driven DNA reactions. The ideal linear dynamics are shown to be closely approximated by the chemical reactions model and the DNA implementation. We illustrate the approach with integration, gain and summation as well as with the ubiquitous robust proportional-integral controller.