Explanation:

x² + y² + 1 = 2x
⇒ x² + y² + 1 – 2x = 0
⇒ x² – 2x + 1 + y² = 0
⇒ (x – 1)² + y² = 0
In the above eq. the L.H.S. can only become zero when the base of terms; (x – 1) and y become zero because for any other value the sum of their squares will always be a positive integer.
Taking (x – 1) = 0
and y = 0
Therefore, x = 1 and y = 0
∴     x⁵ + y⁷ = 1 + 0 = 1.