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I need to solve the following system of $(x,y)$: \begin{cases} 3y^3+3x\sqrt{1-x}=5\sqrt{1-x}-2y\\ x^2-y^2\sqrt{1-x}=\sqrt{2y+5}-\sqrt{1-x} \end{cases}

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Substitute $z=\sqrt{1-x}$, $u=\sqrt{2y+5}$. This will give you a polynomial in $z$. – Michael Hoppe Oct 3 '13 at 12:27
How about $u$? I have already done but there is no more results. Can you explain more clearly? Thanks. – Dinh Anh Thi Oct 3 '13 at 13:21

you can try the solve it numerically first with newton algorithm:

$$ f(x,y) = (3y^3-2x\sqrt{1-x}-2y, x^2+(1-y^2)\sqrt{1-x}-\sqrt{2y+5}) $$

$$ (x_{n+1}, y_{n+1}) = (x_n,y_n) +Df(x_n,y_n)^{-1}.f(x_n,y_n) $$

with $$(x_0,y_0) =(0,0) $$

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Oh, my! This looks (looks ...) as discouraging hopeless as the original equations...or maybe more! – DonAntonio Oct 3 '13 at 12:12
It's really quite complicated. Actually, there exists a method which can solve this system more simply but I can't find now. – Dinh Anh Thi Oct 3 '13 at 12:29

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