[tex]x=0.\overline{7}\ \ \ \ \ |multiply\ both\ sides\ by\ 10\\\\10x=7.\overline{7}\\\\10x-x=7.\overline{7}-0.\overline{7}\\\\9x=7\ \ \ \ \ |divide\ both\ sides\ by\ 7\\\\x=\frac{7}{9}\\\\Answer:\boxed{0.\overline{7}=\frac{7}{9}}[/tex]
[tex]other:\\\\We\ know:0.\overline{1}=\frac{1}{9}\ then\ 0.\overline{7}=\frac{7}{9}[/tex]