# Solving an equation in several variable of the form $a_1x_1x_2+a_2x_2=c$

Consider the following equation in two variables $$a_1x_1x_2+a_2x_2=c$$

where $x_i$ are variables and the other constants can be any real numbers. In three variables, this is $$a_1x_1x_2x_3+a_2x_2x_3+a_3x_3=c$$

In $n$ variables the equation becomes

$$\sum_{i=1}^{n}a_i \prod_{j=i}^{n}x_j=c$$

Solve the equation in $n$ variables for the $x_i$.

The equation doesn't seem to complicate, but the only way to solve it I see, is to use numerical methods. The gradient and Hessian are not difficult to compute, so Newton-Raphson can be used.

Is there a way to analytically find the solution of the equation?

If numerical method are the only way to go, as the equation doesn't seem too complicate, is it possible to guaranted all roots have been found?
In my case $n=30$ and the $x_i$ are in $(0,2)$.

-
"Is there a way to analytically find the solution of the equation?" - You can try Gröbner basis methods... – J. M. Nov 6 '11 at 19:14
When you say the $x_i$ are variables, what to they range over? $\mathbb{N}$ (with or without $0$)? Otherwise you need more equations. – Ross Millikan Nov 6 '11 at 19:16
It can be solved by induction: starting to solve the first equation in two variables (by dividing by $x_2$ if $x_2\neq 0$ we get $a_1 x_1+a_2= c/x_2$) and so on. – user17090 Nov 6 '11 at 19:17
There are typically infinitely many solutions, so it is not clear what is meant by solve. One can write down a parametric solution, but that is not really helpful. – André Nicolas Nov 6 '11 at 19:20
The equation makes essentially no demands on the $x_i$. For simplicity take $n=3$. Pick $x_1$ and $x_2$ almost arbitrarily, and let $x_3=c/(a_1x_1x_2+a_2x_2+a_3$. Then we have a solution unless the denominator is $0$. – André Nicolas Nov 6 '11 at 20:00