Varieties defined by linear equations have the amalgamation property

A variety is a class of algebraic structures axiomatized by a set of equations. An equation is linear if there is at most one occurrence of an operation symbol on each side. We show that a variety axiomatized by linear equations has the strong amalgamation property. Suppose further that the language has no constant symbol and, for each equation, either one side is operation-free, or exactly the same variables appear on both sides. Then also the joint embedding property holds. Examples include most varieties defining classical Maltsev conditions. In a few special cases, the above properties are preserved when further unary operations appear in the equations.