<s>[INST] How can these two equations be solved by elimination? In this question, the following two equations were solved using elimination. With a google crash course I sort of get how elimination works, but it seems like these are much too complex to add the left sides together in a way that cancels out x or y.
$$
\frac{3x(3x^2+9)}{2y}
-
\frac{(3x^2+9)}{8y^3}^3
-
y
=
6 \pmod {23}
$$
$$
\frac{(3x^2+9)^2}{(2y)^2}
-
2x
=
12 \pmod {23}
$$
Ultimately I'm trying to figure out any way (the easiest preferably) to reduce these such that x=18, y=10 or x=19, y = 3 in order to convert it to a function in a scripting language.
If elimination is the way to go, what were the steps involved in arriving at the following equation?
$$x^4-48 x^3-18 x^2+13968 x=-86481 \pmod {23}$$
 [/INST] If you want to implement the resultant yourself, you can use the Sylvester matrix (see wikipedia).
In M2:
R=ZZ/23[x,y]
f=3*x*(3*x^2+9)*(4*y^2)-(3*x^2+9)^3-(y+6)*8*y^3;g=(3*x^2+9)^2-(2*x+12)*(2*y)^2;
sylvesterMatrix(f,g,y)

yields 
$${\tiny \begin{pmatrix}{-8}&
  {-2}&
  -10 x^{3}-7 x&
  0&
  -4 x^{6}+10 x^{4}+7 x^{2}+7&
  0\\
  0&
  {-8}&
  {-2}&
  -10 x^{3}-7 x&
  0&
  -4 x^{6}+10 x^{4}+7 x^{2}+7\\
  -8 x-2&
  0&
  9 x^{4}+8 x^{2}-11&
  0&
  0&
  0\\
  0&
  -8 x-2&
  0&
  9 x^{4}+8 x^{2}-11&
  0&
  0\\
  0&
  0&
  -8 x-2&
  0&
  9 x^{4}+8 x^{2}-11&
  0\\
  0&
  0&
  0&
  -8 x-2&
  0&
  9 x^{4}+8 x^{2}-11\\
  \end{pmatrix} }$$
determinant sylvesterMatrix(f,g,y) gives the same output as resultant(f,g,y).
As for factoring the output
$$-11 x^{16}-x^{15}-3 x^{13}-2 x^{12}-3 x^{11}+7 x^{10}-10 x^{9}+x^{8}+8 x^{7}-6 x^{4}+4 x^{3}+10 x^{2}+10 x+8$$
to $$(x+5) (x+4) ({x^{2}+3})^{6} (x^{2}-11 x-8) ({-11})$$
you probably don't need the whole functionality, just check on $x=-11,\ldots,11$ whether the value is divisible by $23$.
</s>