Show that $\Delta \ne 0$ and LI of $G_4^3$ and $G_6^2$ are equivalent.

Question

In Complex Analysis by Freitag it is claimed that $\Delta \ne 0$ and LI of $G_4^3$ and $G_6^2$ are equivalent; that is, if $\Delta = (60G_4)^3 - 27(140G_6)^2 \ne 0$ then $G_4^3$ and $G_6^2$ are linearly independent and conversely. But $\Delta = (60G_4)^3 - 27(140G_6)^2 \ne 0$ means that only one (up to a multiplication of $\Delta$ by a nonzero number) linear combination of $G_4^3$ and $G_6^2$ is nonzero, so how that implies that all linear combination of $G_4^3$ and $G_6^2$ is nonzero? Same question goes for the converse.