$1)$ $\sqrt{3x+1}-\sqrt{6-x}+3x^{2}-14x-8=0$
$2)$ $\sqrt{4x-1}+\sqrt{4x^{2}-1}=\frac{(4x+1)^{2}}{27}$
$3)$ $\begin{cases}2(x^{3}-y^{6})=3(y^{4}-xy^{2}) \\ 4\frac{x^{2}}{y^{4}}+5\frac{x}{y}-8y^{2}+7=0 \end{cases}$
$4)$ $\begin{cases}\sqrt{x-2}-\sqrt{y-1}=27-x^{3} \\ (x-2)^{2}+1=y \end{cases}$