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Carter
10/04/2017 at 08:47
Votes Answers Follow

Find all integer solutions of the system of equations x + y + z =3 and x3 + y3 + z3 =3 .

intergerequations
  • ...
    John 10/04/2017 at 15:29

    We have:

    \(\left(x+y+z\right)^3-\left(x^3+y^3+z^3\right)=3\left(x+y\right)\left(y+z\right)\left(z+x\right)\)

    \(\Rightarrow8=\left(3-z\right)\left(3-x\right)\left(3-y\right)\)   (1)

    Since \(\left(3-x\right)+\left(3-y\right)+\left(3-z\right)=9-\left(x+y+z\right)=9-3=6\)  (2)

    From (1) and (2) we infer \(\left(x,y,z\right)\) would be (1,1,1), (-5,4,4), (4,-5,4), (4,4,-5).

    Carter selected this answer.
  • ...
    FA KAKALOTS 08/02/2018 at 22:03

    We have:

    (x+y+z)3−(x3+y3+z3)=3(x+y)(y+z)(z+x)

    ⇒8=(3−z)(3−x)(3−y)

       (1)

    Since (3−x)+(3−y)+(3−z)=9−(x+y+z)=9−3=6

      (2)

    From (1) and (2) we infer (x,y,z)

     would be (1,1,1), (-5,4,4), (4,-5,4), (4,4,-5).


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