SOLUTION: Solve y=1/x and x^2+y^2=2 simultaneously. Begin by subtracting the equation of the hyperbola from the equation of the circle.

I can solve it this way (substitution):









Substitute 1/x for y in the 2nd eqn: 





Simplifies to:





Factors to:





Solutions: 

  --> x = -1   and/or   x = 1



For x=-1  we get y=-1   and 

For x=1  we get y=1     and 



Both solutions (1,1) and (-1,-1) satisify the equations.



 

Our starting equations are


    x^2 + y^2 = 2      (1)

    y =              (2)


From equation (2),  

    xy = 1             (3)



So, I will multiply equation (3) by 2 (both sides) and then subtract it from equation (1).  I will get then


    x^2 - 2xy + y^2 = 0,

or

    (x-y)^2 = 0.


It means  x = y,  and then from equation (1) I have


    2x^2 = 2,  x^2 = 1, x = +/- 1.


Thus the two solutions are  

    x = y = 1   and/or   x = y = -1.