Abstract: Let M be a filtered manifold, i.e. a manifold together with a filtration of the tangent bundle by subbundles, which is compatible with the Lie bracket of vector fields. In particular a contact manifold is a filtered manifold. For the sections of a vector bundle over a filtered manifold we will introduce a notion of weighted jets and define the weighted symbol of a differential operator. We will show how this concept of weighted order can be used to rewrite a certain class of (semi)linear systems of partial differential equations on filtered manifolds as first order systems in closed form.