Systems in which a control action can only be taken when a certain resource η is available can be found in various applications, i.e. in biological, engineering and economic sciences. In many of these applications, it is also the case that a smaller value of available resources will grant us a smaller influence on the system through the control action, i.e., the control input u will scale with the magnitude of η, thus yielding bi-linear expressions of the form ηu. Simultaneously, the resource is consumed depending on the physical type of resource and the applied input trajectories in an integral fashion. We study such systems with underlying linear dynamics in terms of their reachable set and demonstrate the connection between the shape of the reachable set and the physical type of resource, deriving explicit upper and lower approximations of the reachable set for systems with stable origin. A biological example demonstrates how these ideas can be used to extract essential mechanisms from a complex system in order to generate simple hypotheses.