System

x^3  + y^3  =  162 - (x^2 + y^2)

(x + y) (x^2 - xy + y^2)  = 162 - (x^2 + y^2

6( x^2 + y^2)  - 6xy = 162 - (x^2 + y^2)

7 (x^2 + y^2 + 2xy)  -20xy = 162

7 ( x + y)^2  - 20xy = 162

7 (6)^2  - 20xy  =162

90  = 20xy

9/2 = xy

(9/2)/x = y

x + (9/2)/x  = 6

x^2 + (9/2)  = 6x

x^2 - 6x   =  -9/2

x^2 - 6x + 9  = -9/2 + 9

(x - 3)^2  = 9/2

x -3  = sqrt(9/2)         or      x - 3  = -sqrt (9/2)

x = 3 + 3/sqrt (2)                x =  3 - 3/sqrt (2)

(x, y)  = ( 3 + 3/sqrt2 , 3- 3/sqrt 2)  and  ( 3 - 3/sqrt2 , 3 + 3/sqrt 2)