Question

Rezolvati in multimea numerelor intregi sistemul de ecuatii: x+y+z=3; x^3+y^3+z^3=3​

Answer 1

Hei! :)

$\left \{ {{x+y+z=3} \atop {x^{3} +x^{3} +z^{3} =3}} \right. \\=> x+y+z=3\\=> (x+y+z)^{3} ={x^{3} +x^{3} +z^{3}+3(x+y)(x+z)(z+y)$$=>(x+y)(y+z)(z+x)=8\\(3-x)(3-y)(3-z)=8 \\Dar\ (3-x)+(3-y)+(3-z)-3(x+y+z)=6\\=> |3-x|=|3-y|=|3-z|=2$

=> x, y, z ∈ {1, 5}

Dar x+y+z= 3

=> x, y, z ∈ {1}