Question #b8484

1 Answer
Sep 20, 2016

Answer:

#((x=-1,y=-1),(x=5-sqrt(6),y=1/3(3-sqrt(6))),(x=5+sqrt(6),y=1/3(3+sqrt(6))))#

Explanation:

Firstly, you should be more careful when copying and publishing problems. The correct post should be:

#{(x^3+1 = 81(y^2+y)) , (x^2+x= 9(y^3 +1)):}#

because so, adding term to term

#{(x^3+1 = 3 xx 27(y^2+y)) , (3(x^2+x)= 27(y^3 +1)):}#

we obtain

#(x+1)^3=27(y+1)^3#

and

#x+1=3(y+1)#

Now solving the system

#{(x+1=3(y+1)),(x^2 + x = 9 (y^3 + 1)):}#

we obtain

#((x=-1,y=-1),(x=5-sqrt(6),y=1/3(3-sqrt(6))),(x=5+sqrt(6),y=1/3(3+sqrt(6))))#

Note:

To solve

#{(x+1=3(y+1)),(x^2 + x = 9 (y^3 + 1)):}# we proceed as follows:

In the second equation

#x^2+x=x(x+1) = x(3(y+1))=9 (y^3 + 1))# then

#x = 9/3(y^3+1)/(y+1) = 3(1-y+y^2)# and finally

#x = 3y+3-1=3-3y+3y^2# or

#3 y^2 - 6 y + 1 =0#