SOLUTION: x^2/y +y^2/x =9 ........eq1 1/x +1/y= 3/4............eq2 solve the simultaneous equations

 = 9       (1)

 = .       (2)


In (1), multiply both sides by x*y. In 2), do the same. You will get an equivalent system (equivalent under the condition xy =/= 0):

 = 9xy,       (3)

x + y = .     (4)


Now,  =  =  = .


Next, replace here all instances of (x+y) by , according to (4), and you will get (3) in the form

 = 9xy.


Cancel 3xy in both sides. You will get

 = 12.


Multiply both sides by 16. You will get

 = 0,

 = 0.    (5)

Introduce new variable u = xy. Then (5) becomes

 = 0.

Solve this quadratic equation using the quadratic formula. The roots are 

 = 8  and/or   = .


Thus the system (3),(4) is reduced to two independent and much simpler systems:


1)  First system is 

    x + y = 6,
    xy    = 8


2) The second system is 

    x + y = -2,
    xy    = 


The major reduction is done.

The first system has two solutions  (x,y) = (2,4)  and  (x,y) = (4,2).


The second system has two solutions  (x,y) = (,)  and  (x,y) = (,).