$1. \begin{cases}x^{3}+2y^{3}=2xy+1 \\ x^{3}+y^{3}=3xy-1 \end{cases}$$2. \begin{cases}x^{4}-2x=y^{4}-y \\ (x^{2}+y^{2})^{3}=3 \end{cases}$$3. \begin{cases}y-x+1+\sqrt{2}=\sqrt{x+1}+\sqrt{2-x} \\ 2x^{3}-y^{3}+x^{2}y^{2}=2xy-3x^{2}+3y \end{cases}$$4. \begin{cases}x^{3}+7y=(x+y)^{2}+x^{2}y+7x+4 \\ 3x^{2}+y^{2}+8y+4=8x \end{cases}$$5.\begin{cases}14x^{2}-21y^{2}+22x-39y=0 \\ 35x^{2}+28y^{2}+111x-10y=0 \end{cases}$