In this work we propose a new, general and computationally cheap way to tackle parametrized PDEs defined on domains with variable shape when relying on the reduced basis method. We easily describe a domain by boundary parametrizations, and obtain domain deformations by solving a solid extension through a linear elasticity problem. The procedure is built over a two-stages reduction: (i) first, we construct a reduced basis approximation for the mesh motion problem; (ii) then, we generate a reduced basis approximation of the state problem, relying on finite element snapshots evaluated over a set of reduced deformed configurations. A Galerkin-POD method is employed to construct both the reduced problems, although this choice is not restrictive. To deal with unavoidable non affinities arising in both cases, we apply a matrix version of the discrete empirical interpolation method, allowing to treat geometrical deformations in a non-intrusive, efficient and purely algebraic way. In order to assess the numerical performances of the proposed technique we consider the solution of a parametrized (direct) Helmholtz scattering problem where the parameters describe both the shape of the obstacle and other relevant physical features.