Complex System of Equations

Algebra Level 5

For how many ordered pairs (a,b)(a,b) of complex numbers does the system of equations {x2yx=axy2+y=b \begin{cases} x^2y-x=a\\ xy^2+y=b \end{cases} have exactly one complex number solution?

Details and assumptions

A complex number solution is an ordered pair of complex numbers (x,y)(x,y) that satisfies both equations.

Clarification: a,b,xa, b, x and yy are all complex numbers. As an explicit example, (x,y)=(i,i) (x,y) = (i, i) is a solution to (a,b)=(2i,0)(a,b) = (-2i, 0 ) .


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