[Math] Solve 5 unknowns with 5 equations

algebra-precalculuslinear algebrasystems of equations

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

Best Answer

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.

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