  Given a place $\omega$ of a global function field $K$ over a finite field,
with associated affine function ring $R_\omega$ and completion $K_\omega$, the
aim of this paper is to give an effective joint equidistribution result for
renormalized primitive lattice points $(a,b)\in {R_\omega}^2$ in the plane
${K_\omega}^2$, and for renormalized solutions to the gcd equation $ax+by=1$.
The main tools are techniques of Goronik and Nevo for counting lattice points
in well-rounded families of subsets. This gives a sharper analog in positive
characteristic of a result of Nevo and the first author for the
equidistribution of the primitive lattice points in $\ZZ^2$.
