We deal with the Hyers-Ulam stability problem of linear mappings from a vector space into a Banach one with respect to the following functional equation: $$f(frac{-x+y}{3})+f(frac{x-3z}{3})+f(frac{3x-y+3z}{3})=f(x)$$. We then combine this equation with other ones and establish the Hyers-Ulam stability of several kinds of linear mappings, among which the algebra (*-) homomorphisms, the derivations, the multipliers and others. We thus repair and improve some previous assertions in the literature.