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X³ + 3 x y² = 14 --- (1)

y³ + 3 x² y = 13 ---- (2)

Add (1) and (2) => (x + y)³ = 14+ 13 = 3³

=> x + y = 3 --- (3)

Subtract (2) from (1) : x³ - y³ + 3 x y² - 3 x² y = 14 - 13 = 1

=> (x - y) (x² + x y + y²) + 3 x y (y - x) = 1

=> (x - y) (x² + x y + y² - 3x y) = 1

=> (x - y) (x - y)² = 1

=> x - y = 1 ---(4)

from (3) and (4) we get: x = 2 and y = 1.

so x² + y² = 5.

y³ + 3 x² y = 13 ---- (2)

Add (1) and (2) => (x + y)³ = 14+ 13 = 3³

=> x + y = 3 --- (3)

Subtract (2) from (1) : x³ - y³ + 3 x y² - 3 x² y = 14 - 13 = 1

=> (x - y) (x² + x y + y²) + 3 x y (y - x) = 1

=> (x - y) (x² + x y + y² - 3x y) = 1

=> (x - y) (x - y)² = 1

=> x - y = 1 ---(4)

from (3) and (4) we get: x = 2 and y = 1.

so x² + y² = 5.