Solve System of 2 Variables: $x^5+y^5=33,\,x+y=3$

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Discussion Overview

The discussion revolves around solving the system of equations $x^5+y^5=33$ and $x+y=3$. The scope includes both real and complex solutions to the equations.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant presents a method to derive a polynomial equation from the given equations, leading to two systems based on the value of $xy$.
  • Another participant notes that the original question also seeks complex solutions, indicating a broader scope than initially considered.
  • The first participant concludes that one of the systems has no real solutions while providing the complex solutions for the other system.

Areas of Agreement / Disagreement

Participants agree on the method of deriving the polynomial equation but differ on the interpretation of the solutions, particularly regarding the inclusion of complex solutions.

Contextual Notes

The discussion does not resolve the implications of the complex solutions fully, nor does it clarify the conditions under which the real solutions were deemed insufficient.

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Solve the system $x^5+y^5=33,\,x+y=3$.
 
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[sp]$(x+y)^5=x^5+y^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4=33+5xy(x^3+y^3)+10x^2y^2(x+y)=

=33+5xy((x+y)^3-3xy(x+y))+30x^2y^2= 33+5xy(27-9xy)+30x^2y^2=243$
or
$15x^2y^2-135xy+210=0$
or
$x^2y^2-9xy+14=0$
And xy=7 or xy=2 impling the following 2 systems of equations :

x+y=3. (A)
xy=2

x+y=3 (B)
xy=7
And (A) gives (x=1,y=2),(x=2,y=1) (B) has no real solutions[/sp]
 
Last edited:
Thanks for participating, solakis! Ah, the question is meant to ask for complex solutions too! (Nod)
 
solakis said:
[sp]$(x+y)^5=x^5+y^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4=33+5xy(x^3+y^3)+10x^2y^2(x+y)=

=33+5xy((x+y)^3-3xy(x+y))+30x^2y^2= 33+5xy(27-9xy)+30x^2y^2=243$
or
$15x^2y^2-135xy+210=0$
or
$x^2y^2-9xy+14=0$
And xy=7 or xy=2 impling the following 2 systems of equations :

x+y=3. (A)
xy=2

x+y=3 (B)
xy=7
And (A) gives (x=1,y=2),(x=2,y=1) (B) has no real solutions[/sp]
[sp]The complex solutions are:
[x=(3+i$\sqrt 19$)/2, y=(3-i$\sqrt 19$)/2]...[x=(3-i$\sqrt 19$)/2 , y=( 3+i$\sqrt 19$)/2][/sp]
 

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