Find all the solutions of the polynomial equation $x^5+y^3 \equiv 1 \pmod9$. Find all the solutions of the polynomial equation $$x^5+y^3 \equiv 1\pmod9.$$ I have learned to find the solutions of the polynomials like $x^p \equiv n\pmod m$, but never been like two variables. Please help me to solve this. Thanks.

Guide:
Let $y=3m+r, r=0,1,2$
$y^3=(3m+r)^3=(3m)^3+3(3m)^2r+3(3m)r^2+r^3\equiv r^3 \pmod{9}$
Hence we just have to consider $3$ cases, 
Case $1$: $y \equiv 0 \pmod{3}$ and the problem reduces to $x^5 \equiv 1\pmod{9}$.
Case $2$: $y \equiv 1 \pmod{3}$ and the problem reduces to $x^5 \equiv 0\pmod{9}$.
Case $3$: $y \equiv -1 \pmod{3}$ and the problem reduces to $x^5 \equiv 2\pmod{9}$.
The problems are now of the form of $x^p \equiv n \pmod{m}$
