Geometric invariant theory for graded additive groups

We consider geometric invariant theory for graded additive groups, groups of the form $G_a^r{⋊}_w{G}_m$ such that the $G_m$-action on $G_a^r$ is a scalar multiplication with weight $w\in N_{+}$. We provide an algorithm of equivariant birational modifications, such that we can apply the geometric invariant theory of Bérczi-Doran-Hawes-Kirwan. In particular, the geometric $G_a^r$-quotient exists. This complements Bérczi-Doran-Hawes-Kirwan, in the special case of one grading weight.