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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.

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    I'm killi$n$g the nonlinear tag because, well, the vast majority of things in life are nonlinear.2011-11-30

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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.