  We consider geometric invariant theory for \emph{graded additive groups},
groups of the form $\mathbb{G}_a^r\rtimes_w\mathbb{G}_m$ such that the
$\mathbb{G}_m$-action on $\mathbb{G}_a^r$ is a scalar multiplication with
weight $w\in\mathbb{N}_+$. We provide an algorithm of equivariant birational
modifications, such that we can apply the geometric invariant theory of
B\'erczi-Doran-Hawes-Kirwan. In particular, the geometric
$\mathbb{G}_a^r$-quotient exists. This complements B\'erczi-Doran-Hawes-Kirwan,
in the special case of one grading weight.
