This system can be solve by substitution
Eq 1: #xy= 9#
#=># Solve for #x#
#(xy)/y= 9/y#
#=> color(red)(x= 9/y)# Substitute into Eq.2
Eq 2: #x^2 +y^2= 82#
#color(red)((9/y)^2) +y^2 = 82 #
#=># #(9/y)(9/y)+y^2 = 82#
#=>81/y^2 +y^2 = 82#
#=>color(blue)(y^2)(81/y^2+y^2=82)#
#=>81+y^4 =82y^2#
#=>y^4 -82y^2+81=0#
Factor #(y^2-81)(y^2-1)= 0#
Set each factor equal to #0#
a) #( y^2-81)= 0 #
#y^2 = 81#
#y= -9 , 9#
b) #y^2 -1 = 0 #
#y^2 = 1#
#y= -1, 1#
When #y= -1#
#x= 9/(-1)= -9# ; #(-9, -1)#
When #y= 1#
#x= 9/(10= 9# ; #(9, 1)#
When #y= -9#
#x= 9/(-9) = -1# ; #(-1, -9)#
When #y= 9#
#x= 9/(9) = 1# ; #(1, 9)#