Question

Question #b8484

Answer 1

`((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`