0
$\begingroup$

Does the following system of six simultaneous equations in eight variables $x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4$ have solutions in $\mathbb{R}$? in $\mathbb{C}$? $$x_1y_2-x_2y_1=1$$ $$x_1y_3-x_3y_1=0$$ $$x_1y_4-x_4y_1=0$$ $$x_2y_3-x_3y_2=0$$ $$x_2y_4-x_4y_2=0$$ $$x_3y_4-x_4y_3=1$$

Maple cannot help; I don't know if Mathematica will.

  • 0
    The system is underdetermined on purpose?2011-11-28
  • 0
    That is correct. It is underdetermined.2011-11-28
  • 1
    I'm killing the nonlinear tag because, well, the vast majority of things in life are nonlinear.2011-11-30

1 Answers 1

2

There is no solution. (Check my work.)

First note that $x_1\ne 0$ as if it was, either $y_1=0$ or $x_3$ and $x_4=0$, and both of those would cause violations. Now note that $x_3\ne 0$, as that would force either $x_1$ or $y_3$ to be 0, again vioplating the constraints. Similarly, (but more convolutedly $x_4\ne 0$. So we get $$y_1 = \frac{x_1 y_3}{x_3}$$ $$y_2 = \frac{x_2 y_3}{x_3}$$ as $x_3\ne 0$.

Plugging $y_1$ and $y_2$ into the first equation we get $0=1$, so the answer is no in either field.