A Nice Polynomial System #algebra

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[paulortega]

it is interesting to look at the general case
x^2 = ax + by
y^2 = bx + ay
solutions for x are
x = 0
x = a + b
x = ( (a - b) +/- sqrt((a - b)*(a + 3b)) )/ 2
even further for "nice" answers let (a + 3b) = m^2 and (a - b) = n^2
a = (m^2 +3*n^2)/4 and b = (m^2 - n^2)/4
then the second set of solutions are x = n(n ± m)/2
in this case m=5 and n=3 so x = 12, -3

[rob]

x^2 - y^2 = 9(x - y)
(x + y)(x - y) = 9(x - y)
x = y or x + y = 9
x^2 = 13x + 4(9 - x) = 9x + 36
x^2 - 9x - 36 = 0
(x - 12)(x + 3) = 0
x = 12, y = -3 or x = -3, y = 12
or x^2 = 17x
x = y = 0 or x = y = 17
{(0, 0) (17, 17) (12, -3) (-3, 12)}

[RyanLewis-Johnson-wqxs]

x^2=13x+4y y^2=4x+13y y=0 y=12 y=-3 y=17 x=12 x=-3 x=0 x=17