Originally Posted by

**I-Think** Buenas noches. I have a question here that I have answered, so I hope that the forum could rate my solution, detect errors and provide alternatives. Thanks.

Question: If $\displaystyle x$ and $\displaystyle y$ are distinct real numbers such that$\displaystyle x^3=1-y$ and $\displaystyle y^3=1-x,$ find all possible values of $\displaystyle xy$ .

Solution: $\displaystyle x^3=1-y$

$\displaystyle y^3=1-x,$

$\displaystyle x^3-y^3=x-y$

(Dividing by $\displaystyle (x-y)$ ): $\displaystyle x^2+xy+y^2=1$

$\displaystyle (x+y)^2-xy=1 \Rightarrow(x+y)^2-1^2=xy

$

$\displaystyle (x+y+1)(x+y-1)=xy

$

**Isomorphism: Whatever follows is wrong. **

Case 1: $\displaystyle x+y+1=x$, $\displaystyle x+y-1=y$,

$\displaystyle y=-1,x=1$

Case 2: $\displaystyle x+y+1=y$, $\displaystyle x+y-1=x$,

$\displaystyle x=-1,y=1$

Only possible value of $\displaystyle xy$ is $\displaystyle -1$

First post with latex.