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 February 20th, 2018, 12:44 PM #1 Senior Member   Joined: Dec 2015 From: Earth Posts: 238 Thanks: 27 System equation Solve the system $\displaystyle x^2 + x -1=y$ $\displaystyle y^2 +y-1=z$ $\displaystyle z^2 +z-1=x$
 February 20th, 2018, 12:53 PM #2 Math Team   Joined: Jul 2011 From: Texas Posts: 2,767 Thanks: 1422 sum of the three equations ... $x^2+y^2+z^2 + x + y + z - 3 = y+z+x$ $x^2+y^2+z^2 = 3$ sphere of radius $r=\sqrt{3}$ centered at the origin ... https://www.wolframalpha.com/input/?...E2%2Bz%5E2%3D3 Thanks from topsquark
February 20th, 2018, 01:15 PM   #3
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Quote:
 Originally Posted by skeeter sum of the three equations ... $x^2+y^2+z^2 + x + y + z - 3 = y+z+x$ $x^2+y^2+z^2 = 3$ sphere of radius $r=\sqrt{3}$ centered at the origin ... https://www.wolframalpha.com/input/?...E2%2Bz%5E2%3D3
This is a start. However, the answer has to be specific points on the surface of the sphere. Symmetry immediately leads to (1,1,1) and(-1,-1,-1) as solutions. I don't believe there are any others.

 February 20th, 2018, 02:21 PM #4 Global Moderator   Joined: Dec 2006 Posts: 19,505 Thanks: 1741 If the solutions don't have to be real, there are more solutions.
 February 20th, 2018, 02:56 PM #5 Senior Member     Joined: Sep 2015 From: USA Posts: 2,090 Thanks: 1086 it ends up being the roots of a degree 6 polynomial
 February 20th, 2018, 07:26 PM #6 Global Moderator   Joined: Dec 2006 Posts: 19,505 Thanks: 1741 If $x$ isn't real, it's a zero of $x^6 + 4x^5 + 5x^4 + 2x^3 + 1$.

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