Linear demand systems and quasi-linear quadratic utility models are widely used in industrial economics. We clarify the link between the two settings and explain their exact origin as it seems to be little known by practitioners. We offer practical recommendations to achieve consistency, tractability and a reasonable degree of generality when using the linear demand framework. We show that all tractable versions of the model used in practice are (almost) identical and have a mean-variance structure. We provide concise, ready-to-use formulae for the symmetric model. Finally, we revisit and extend the asymmetric model of Shubik and Levitan.