Given that x² = y + z, y² = z + x, and z² = x + y, we can rewrite the equations as:
x² - y - z = 0
y² - z - x = 0
z² - x - y = 0
We can rearrange the equations to get:
x² - y - z = 0
y² - z - x = 0
z² - x - y = 0
Adding all three equations together, we get:
x² + y² + z² - 2(x + y + z) = 0
This can be further simplified as:
(x + y + z)² - 3(x + y + z) = 0
Therefore, (x + y + z)(x + y + z - 3) = 0
This implies that either x + y + z = 0 or x + y + z = 3
If x + y + z = 0, then 1/x + 1/y + 1/z = 0
If x + y + z = 3, then
1/x + 1/y + 1/z = x + y + z / xyz
= 3 / xyz
= 1
Therefore, 1/x + 1/y + 1/z = 1
