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Solve the system of equations: 1/(y - 1) - 1/(y + 1) = 1/x and y² - x - 5 = 0.

1. x = 4, y = +-3
2. x = 4, y = 3 only
3. x = -4, y = +-3
4. x = 2, y = +-3

Correct answer: x = 4, y = +-3

Solution

1/(y-1) - 1/(y+1) = [(y+1) - (y-1)]/(y² - 1) = 2/(y² - 1) = 1/x, so x = (y² - 1)/2. From the second equation x = y² - 5. Equate: (y² - 1)/2 = y² - 5 => y² - 1 = 2y² - 10 => y² = 9 => y = +-3. Then x = y² - 5 = 4. Both signs of y are valid (y = +-1 would be excluded, but +-3 are fine).

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