Complicated System of Equations

Algebra Level 5

Consider the distinct complex solutions \((x_1,y_1),...,(x_n,y_n)\) to the following system of equations: \[\begin{cases} x^3+y^2+y=1\\ x^2+y^3=2. \end{cases} \] Find the absolute value of the product of all the possible \(x\) values, i.e. \(\left| x_1\cdot x_2\cdot...\cdot x_n \right| .\)

Details and assumptions

The absolute value of a complex number \(z = a+ bi\) is given by \( |z| = \sqrt{a^2+b^2} \).

Due to the restriction of distinct solutions, repeated roots (if they exist) do not contribute to the product. For example, if your roots are \( (2, 3), (2, 3)\) then the answer is 2. If your roots are \((2, 3), (2, -3)\) then the answer is 4.


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