<s>[INST] Cubic system of equations modulo $n$. What would be the best method to solve the systems of equations? If solving it, please show how to solve for $x$ and $y$ because that's where it's confusing. Thanks.
$5y^3+4x^2-y^2+10x-y$ $=$ $1$ $\pmod {31}$
$2y^3+13x^3+7y^2-x^2+5x-2$ $=$ $1$ $\pmod {31}$
 [/INST] Systematically eliminating $y$ by starting with the cubic term and then moving on to lower powers of $y$ leads to the equation:
$$16 x^9+9 x^8+10 x^7+14 x^6+28 x^5+28 x^4+19 x^3+13 x^2+2 x+26=0 \bmod 31$$
and 
$$y = \frac{15 x^3+11 x^2+29 x+9}{x^3+6 x^2+12 x+15}\bmod 31$$
where the denominator is to be interpreted as the multiplicative inverse of the polynomial $\bmod 31$ there, which can be rewritten as a polynomial (because it doesn't have linear factors), but we don't need to do that. Then we can solve for $x$ using trial and error, we find that $x=11\bmod 31$, substituting in the equation for $y$ yields $y = 16\bmod 31$.
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