Find all real solutions x, y, z of the following system of equations:

x³ + y = 12x + 20
2y³ + z = 24y + 36
3z³ + x = 36z + 52

> x:='x';
> y:='y';
> z:='z';

                                x := x


                                y := y


                                z := z

> f := (x,y) -> x^3+y-(12*x+20);
> g := (x,z) -> 3*z^3+x-(36*z+52);
> h := (y,z) -> 2*y^3+z-(24*y+36);

                                  3
                  f := (x, y) -> x  + y - 12 x - 20


                                   3
                 g := (x, z) -> 3 z  + x - 36 z - 52


                                   3
                 h := (y, z) -> 2 y  + z - 24 y - 36

> z:=solve(h(y,z)=0,z);

                                 3
                        z := -2 y  + 24 y + 36

> x:=solve(g(x,z)=0,x);

           9        7         6          5          4          3
  x := 24 y  - 864 y  - 1296 y  + 10368 y  + 31104 y  - 18216 y

                   2
         - 186624 y  - 279072 y - 138620

> p := y -> f(x,y);

                          p := y -> f(x, y)

> p(y);

       9        7         6          5          4          3
  (24 y  - 864 y  - 1296 y  + 10368 y  + 31104 y  - 18216 y

                   2                     3                    9
         - 186624 y  - 279072 y - 138620)  + 3348865 y - 288 y

                  7          6           5           4           3
         + 10368 y  + 15552 y  - 124416 y  - 373248 y  + 218592 y

                    2
         + 2239488 y  + 1663420

> assume(y,real);
> y:=fsolve(p(y)=0,y);

                               y := 4.

> (x,y,z);

                              4., 4., 4.

>

Posted by JLo on 2006-07-30 13:29:10