Solve 5 unknowns with 5 equations

Question

I have the following set of equations:

$A = x_1x_2x_3x_4x_5$

$B = x_1x_3x_5 + x_1x_4x_5 + x_2x_4x_5 + x_2x_3x_4$

$C = x_3 + x_4 + x_5$

$D = x_1x_2x_3x_4$

$E = x_1x_3 + x_1x_4+x_2x_4$

Where A, B, C, D, and E are known constants. So it is 5 equations and 5 unknowns. Is there some way to solve this quickly? Or is it not possible because the equations aren't linearly independent?

Any suggestions is appreciated.

Cheers

Answer 1

Assuming none of your parameters are zero $$x_5=\frac AD\\ B=\frac {AE}D+x_2x_3x_4=\frac {AE}D+\frac D{x_1} \\ x_1=\frac DB+\frac {AE}{BD}=\frac{D^2+AE}{BD}\\x_3+x_4=C-\frac AD\\E=\frac{D^2+AE}{BD}(C-\frac AD)+x_2x_4$$The second and fifth then give $x_3$, the fourth gives $x_4$ and we are done.

Answer 2

The equation is nonlinear (in fact, glancing at $A$ shows that it is quintic), but one can still make various simplifications to make it more tractable.

At a glance, we may as well write the first equation as $$A = D x_5$$ and the second as $$B = E x_5 + x_2 x_3 x_4,$$ which in turn allows us to write the fourth equation as $$D = x_1 (B - E x_5).$$ Just these changes alone reduce the system to a cubic one. Note too that we can eliminate, say, $x_5$ by solving the (linear) third equation.