Solving system of multivariable 2nd-degree polynomials x 2 + 3 x y − 9 = 0 ( 1 ) 2 y 2 − 4 x y + 5 = 0 ( 2 ) where ( x , y ) ∈ C 2 .More generally, how would you solve any set of equations of the form: a x 2 + b x y + c = 0 d y 2 + e x y + f = 0 where a , b , c , d , e , f ∈ Q and ( x , y ) ∈ C 2 .

Question

Solving system of multivariable 2nd-degree polynomials                                      x                  2                +        3        x        y        −        9        =        0                (        1        )                            2                          y                  2                −        4        x        y        +        5        =        0                (        2        )            where   (  x  ,  y  )  ∈        C      2  .More generally, how would you solve any set of equations of the form:                                          a            x                            2                +        b        x        y        +        c        =        0                            d                          y                  2                +        e        x        y        +        f        =        0            where   a  ,  b  ,  c  ,  d  ,  e  ,  f  ∈  Q and   (  x  ,  y  )  ∈        C      2  .

Abigail Palmer · Accepted Answer

Multiply first equation by   5.Multiply second equation by   9.Add both Equations.Divide this equation by       y    2  .Let   t  =      x    y  .You get a quadratic in   tThe steps are[Generalization]:Reduce the two equations to one            a      x          2    +  b  x  y  +            c      y        2    =  0Then divide by       y    2        a                    x        2                    y        2              +  b      x    y    +  c  =  0Replace       x    y    =  t            a          t          2    +  b  t  +  c  =  0