+36916 Here is a Desmos graphical solution, which might help to find the algebraic solution....which I think might be kind of messy,,,,,,
+128460 x = y^2 + y + 1 (1)
5y = 2 - x - x^2 (2)
Square both sides of (1)
x^2 = (y^2 + y + 1) (y^2 + y + 1)
x^2 = y^4 + y^3 + y^2 + y^3 + y^2 + y + y^2 + y + 1
x^2 = y^4 + 2y^3 + 3y^2 + 2y+ 1 (3)
Sub (1) and (3) into (2) for x^2 and x
5y = 2 - (y^2 + y + 1) - ( y^4 + 2y^3 + 3y^2 +2y + 1)
5y = 2 - y^2 - y - 1 - y^4 - 2y^3 - 3y^2 - 2y - 1
5y = -y^4 - 2y^3 - 4y^2 -3 y
y^4 + 2y^3 + 4y^2 + 8y = 0 factor
y^3 ( y + 2) + 4y ( y + 2) = 0
(y^3 + 4y) ( y + 2) = 0
y ( y^2 + 4) ( y + 2) = 0
The second factor does not provide real solutions
The other two solutions are
y = 0 and y + 2 = 0
y = -2
When y = 0 x = (0)^2 + 0 + 1 = 1
When y = -2 x = (-2)^2 - 2 + 1 = 3
So....the solutions are
(1, 0) and ( 3, -2) .....as EP found graphically !!!!
