SOLUTION: 3x + 2y = 19 xy=15

Your starting equations are

    3x + 2y = 19,    (1)

    xy = 15.         (2)


From equation (1),  express  2y = 19-3x.

Write equation (2) in the form

    2xy = 30,  or  x*(2y) = 30,   and replace there 2y by 19-3x.


You will get

    x*(19-3x) = 30.


Reduce to the standard form quadratic equation

    19x - 3x^2 = 30,

    3x^2 - 19x + 30 = 0.    (3)


Calculate the discriminant  d = b^2 - 4ac = (-19)^2 - 4*3*30 = 1

and apply the quadratic formula

     =  = .


The solutions for x are  

     =  =  = ;

     =  =  = 3.


If  x= ,  then  2y = 19-3x =  = 19 - 10 = 9,  so  y = .


If  x= 3,  then  2y = 19-3x =  = 19 - 9 = 10,  so  y =  = 5.



ANSWER.  The system of the given equations has two solutions:

         (x,y) = (,)  and  (x,y) = (3,5).