SOLUTION: what are all ordered pairs of numbers (x, y) which satisfy x^2 � xy + y^2 = 7 and x- xy + y = -1 ?

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Question 564213: what are all ordered pairs of numbers (x, y) which satisfy x^2 � xy + y^2 = 7 and x- xy + y = -1 ?

Answer by richard1234(7193) (Show Source):
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Add 3xy to both sides of the 1st equation.

The second equation is equivalent to

. Substituting this into the first equation,

This is a quadratic in x+y, and the solutions are x+y = 5 and x+y = -2. If x+y = 5, substituting into the second equation we obtain

And if x+y = -2, substituting into the second equation yields

Therefore we have two cases: x+y = 5 & xy = 6, and x+y = -2 & xy = -1. The first case yields {3,2}; the second case we can let x = -2-y, in which

By the quadratic formula, , the solution obtained is