Let $X=\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$ and $a(t)=\mathrm{diag}(t^2,t^{-1},t^{-1})$. The expanding horospherical group $U^+$ is isomorphic to $\mathbb{R}^2$. A result of Shah tells us that the $a(t)$-translates of a non-degenerate real-analytic curve in a $(U^+)$-orbit get equidistributed in $X$. It remains to study degenerate curves, i.e. planar lines $y=ax+b$. In this talk, we give a Diophantine condition on the parameter $(a,b)$ which serves as a necessary and sufficient condition for equidistribution. \\ \\ Joint work with Kleinbock, Saxcé and Shah. If time permits, I will also talk about generalisations to $\mathrm{SL}_n(\mathbb{R})/\mathrm{SL}_n(\mathbb{Z})$. Joint work with Shah.