Let's say I have a set of non-linear equations with exactly the same form but different coefficients: $$ 5xy - 2x + 4y - 2z + 7zy = 3 $$ $$ 3xy - 6x + 7y - 2z + 7zy = 2 $$ $$ 5xy - 1x + 8y - 1z + 3zy = 8 $$
Is there a routine method to solve these sorts of equations, either numerically or in closed form? (Note that I just picked these coefficients at random - my actual example is slightly more complicated and I am aware the system might not have any solutions whatsoever).
Thanks!
Even if random, your problem is interesting.
All CAS (Computer Algebra System) will do the work using most of the time Gröbner bases.
The problem is that the results could be far from nice. Using you example with Wolfram Alpha (have a look here). Just numbers.
Now, make some effort.
From $(1)$, you have $$z=\frac{-5 x y+2 x-4 y+3}{7 y-2}$$ Plug it in $(2)$ and simplify to get $$y=\frac{1-4 x}{2 x-3}$$
Plug in $(3)$ and obtain (after removing the denominator) $$184 x^3+492 x^2-648 x+169=0$$
Now, use the method for cubic equations. Three real roots since $\Delta=16840872000$. Using the trigonometric method, they are $$x_k=\frac{1}{46} \left(14 \sqrt{85} \cos \left(\frac{1}{3} \left(2 \pi k-\cos ^{-1}\left(-\frac{21311 }{11662}\sqrt{\frac{5}{17}}\right)\right)\right)-41\right)\quad \text{with} \quad k=0,1,2$$ Frm these, you obtain the exact solutions for the corresponding $y_k$ and $z_k$.
Computing with as many figures as you want $$x_0=+0.6187464041305912173164358949669614741334674637271\cdots$$ $$x_1=+0.4017961260109958985396689999338839910437998943606\cdots$$ $$x_2=-3.6944555736198479854213222862051932912642238798268\cdots$$
What do you prefer ?