Given x, y are real number that are not equal to 1. Find all x, y that satisfy:

Question

Given x, y are real numbers that are not equal to 1. Find all x, y that satisfy:

$x^8+y^7=x(1+y^7)$ and $y^8+x^7=y(1+x^7)$

Answer 1

The resultant of the two polynomials $f=x^8+y^7-x(1+y^7)$ and $g=y^8+x^7-y(1+x^7)$ is given by a polynomial in $x$, which has four real roots, namely $x=0,1,-1$ and $x=0.502017055178$, which is the real root of $$ x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x - 1=0. $$ The roots are the common solutions for $f=g=0$.

Hence the real solutions are
$$ (x,y)=(0,0), (1,1), (-1,-1) $$ and $$ (x,y)=(0.502017055178, 1), (1,0.502017055178). $$