<s>[INST] Solve the system $ \left\lbrace \begin{array}{ccc} \sqrt{1-x} + \sqrt{1-y} &=& \frac{y-x}{2} \\ x+y &=& 2xy \end{array}\right. $ I have to solve the following system:
$$
\left\lbrace \begin{array}{ccc}
\sqrt{1-x} + \sqrt{1-y} &=& \frac{y-x}{2} \\
x+y  &=& 2xy
\end{array}\right.
$$
I've showed that it is equivalent to
$$
\left\lbrace \begin{array}{ccc}
 \sqrt{1-x} - \sqrt{1-y} &=& 2 \\
 x+y  &=& 2xy
\end{array}\right. \text{ or } x=y= 1
$$
And i can't conclude nothing frome this system:
$$
\left\lbrace \begin{array}{ccc}
 \sqrt{1-x} - \sqrt{1-y} &=& 2 \\
 x+y  &=& 2xy
\end{array}\right.
$$
 [/INST] We can still solve the system from its original form. From the first equation you get: $y \le 1$, and $y - x \ge 0$ or $y \ge x$. This leads to: $2x \le x+y = 2xy \le 2x\implies 2x \le 2xy \le 2x\implies 2x = 2xy \implies 2x(1-y) = 0\implies x = 0$ or $y = 1$ or both. If $x = 0$ then $y = 0$ from the second equation, but the first equation is not satisfied. So $y = 1$, and then $x = 1$. So the only solution is $x = y = 1$.
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