How to solve this system of equation? 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}
Let's resurrect this old question. Two solutions to
\begin{cases}
3y^3+3xz=5z-2y\\
x^2-y^2z=\sqrt{2y+5}-z\end{cases}
are $(x,\,y)= (-3,\,2)$ for $z=\sqrt{1-x}$ and,
$$x = \text{Real root}(7 - x - 3 x^2 + 7 x^3 - 8 x^4 + 2 x^5 - x^6 + x^7=0)$$
$$y = \text{Real root}(2 + y + 2 y^2 + 4 y^3 - 3 y^5 + y^7=0)$$
for $z=-\sqrt{1-x}$.

Unless your system has special symmetries, I'm afraid the only way to find solutions that algebraic numbers of high degree is to use resultants.
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 $$(x_0,y_0) =(0,0) $$