Solving system of equations: x^3−3y^2x=−1 and 3yx^2−y^3=1

Rihanna Bentley

Rihanna Bentley

Answered question

2022-11-18

Solving system of equations: x 3 3 y 2 x = 1 and 3 y x 2 y 3 = 1

Answer & Explanation

AtticaPlotowvi

AtticaPlotowvi

Beginner2022-11-19Added 18 answers

Eliminating y, we have
y 2 = x 3 + 1 3 x   , y ( 3 x 2 y 2 ) = 1
and so
x 3 + 1 3 x ( 3 x 2 x 3 + 1 3 x ) 2 = 1   .
Multiplying both sides by 27 x 3 , the equation becomes a cubic in x 3 . The cubic happens to have one rational root, so all roots can easily be found.
kituoti126

kituoti126

Beginner2022-11-20Added 8 answers

hint:
x 3 + 3 x 2 y 3 x y 2 y 3 = 0
( x y ) ( x 2 + x y + y 2 ) + 3 x y ( x y ) = 0 ( x y ) ( x 2 + x y + y 2 + 3 x y ) = 0

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