Question

Choose the solution(s) of the following system of equations:
x2 + y2 = 6
x2 – y = 6

no solution

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infinitely many solutions

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Answer 1

Subtract the second equation from the first:
(x^2 + y^2) − (x^2 − y) = 6 − 6 ⇒ y^2 + y = 0 ⇒ y(y+1)=0.

So y = 0 or y = −1.

If y = 0: x^2 − 0 = 6 ⇒ x^2 = 6 ⇒ x = ±√6.
If y = −1: x^2 − (−1) = 6 ⇒ x^2 = 5 ⇒ x = ±√5.

Solutions: (√6, 0), (−√6, 0), (√5, −1), (−√5, −1).