solution space for non linear algebraic system of equations

Question

Given a non linear algebraic system of equation of $N$ variables and $m$ equations, such that $N>m$.

Example, ($N=3,m=2$)

$$ x_1+x_2x_3-x_1x_3 = 5$$ $$ x_2x_3 = x_1+9$$

What is the best way to determine the solution space?

Is it possible to determine the ranges for each variable $(x_1,x_2,x_3...x_N)$

Answer 1

For this particular system, you can eliminate $x_2x_3$ by plugging in the first equation,

$$2x_1+9-x_1x_3=5$$ or

$$x_1(2-x_3)=-4.$$

You can choose one of the unknowns (say $x_1$) as an independent parameter, and draw the remaining ones:

$$x_3=\frac4{x_1}+2,\\x_2=\frac{x_1+9}{x_3}.$$

For the first identity to make sense, one must have $x_1\ne0$, and for the second $x_3\ne0$ i.e. $x_1\ne-2$.