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Daniel Ludwin (ao460@freenet.hamilton.on.ca) wrote:
: 
: What are the four solutions to the two equations
:     x^2-xy+y^2=7   and   x-xy+y=-1   ?
: 
: I know that two of them are (2,3) and (3,2),
: but I can't get the other two.

The first thing I do is sketch the two graphs to see roughly where the other
two solutions are.  However I won't get into the matrix theory I used since
it takes a while to get the the exact solution.

Second idea, since the second equation is linear in 'y', it is easily
solved for 'y' (in terms of 'x').  Substitute this expression for both 'y's
in the first equation and you get a fourth degree equation in 'x' only.
Since you already know two solutions, factor them out to get a quadratic.
The Quadratic Formula gives the other two solutions.

Better idea (doesn't require that you already know two solutions!)
This would be easier without the 'xy' terms, so move them to the right
hand sides of the equations.  Now you can eliminate the 'x^2' and 'y^2'
terms by squaring the second equation and subtracting the first equation
from it.  The result is an easy quadratic equation in 'xy'.
So you get two possible values for 'xy'.  Substitute each value into
the original two equations and solve.  (Solve the linear equation for y,
substitute into the quadratic equation and use the quadratic formula to
get x.)

Just a hint to check your arithmetic.  The other two solutions are in
the 2nd and 4th quadrants and both involve the square root of 2.

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