It is shown that given a Lagrangian for a system with a finite number of degrees of freedom, the
existence of a variational symmetry is equivalent to the existence of coordinates in the extended
configuration space such that one of the coordinates is ignorable. The proof given here, which only
requires multivariable calculus, provides an elementary derivation of the partial differential
equation that determines the variational symmetries of a given Lagrangian, which is obtained in
treatises on the Lie theory of symmetries of differential equations or of variational calculus.