Let G be a solvable Lie group. Let $\chi$ be a character on G with values in $\mathbb R$. Let $\mathfrak B$ be a Banach space on which G is made to act by bounded operators, with the equality $\Vert gb\Vert=\chi(g)\Vert b\Vert$ satisfied for all $g\in G$ and $b\in\mathfrak B$. A criterion is given on the pair $(G,\chi)$ for the cohomology group $H^1(G,\mathfrak B)$ to vanish. Using this criterion, we classify up to Cr conjugacy all the volume-preserving locally free Cr actions of G on closed manifolds with dimension $\dim(G)+1$, for G belonging to a wide class of solvable Lie groups.