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Sistem persamaan tiga variabel

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Tentukan semua tripel bilangan real $(x,y,z)$ yang memenuhi sistem persamaan

\begin{align*} x+y+z=0 \\ x^3+y^3+z^3=90 \\ x^5+y^5+z^5=2850 \end{align*}

Tes 3 Nomor 2 Pelatnas Tahap 3 IMO 2016

Edited by -_-

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Coba jawab:

 

Spoiler

Perhatikan bahwa

 

\[x^3+y^3+z^3 - 3xyz = (x+y+z)(x^2+y^2+z^2 - xy - yz - zx)\]

 

Karena $x+y+z = 0$ maka:

 

\[3xyz = x^3+y^3+z^3\]

\[3xyz = 90\]

\[xyz = 30\]

 

Misal ada suatu polinomial derajat 3 dimana $x,y,z$ adalah akar-akarnya:

 

\[t^3+at^2+bt+c = 0\]

Maka $a= -(x+y+z) = 0$, $b=xy+yz+zx$, $c=-(xyz)=-30$, Sehingga persamaan menjadi:

 

\[t^3+bt-30 = 0\]

 

Kalikan kedua ruas dengan $\frac{1}{t}$:

 

\[t^2+b-30(\frac{1}{t}) = 0\]

 

Karena $x,y,z$ adalah akar-akar dari persamaan diatas, nilai $x,y,z$ bisa di substitusikan.

 

\[x^2+b-30(\frac{1}{x}) = 0\]

\[y^2+b-30(\frac{1}{y}) = 0\]

\[z^2+b-30(\frac{1}{z}) = 0\]

 

Jumlahkan ketiga persamaan diatas menjadi:

 

\[(x^2+y^2+z^2) +3b - 30(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}) = 0\]

\[x^2+y^2+z^2 +3b - 30(\frac{xy+yz+zx}{xyz}) = 0\]

 

Karena $xyz = 30$ maka:

 

\[x^2+y^2+z^2 +3b - 30(\frac{xy+yz+zx}{30}) = 0\]

\[x^2+y^2+z^2 +3b - (xy+yz+zx) = 0\]

 

Mengingat $b = xy+yz+zx$:

 

\[x^2+y^2+z^2 +3b-b = 0\]

\[x^2+y^2+z^2 +2b = 0\]

\[x^2+y^2+z^2 = -2b\]

 

Kembali ke persamaan awal:

 

\[t^3+bt-30 = 0\]

 

Kalikan kedua ruas dengan $t^2$:

 

\[t^5+bt^3-30t^2 = 0\]

 

Lalu substitusikan nilai $x,y,z$:

 

\[x^5+bx^3-30x^2 = 0\]

\[y^5+by^3-30y^2 = 0\]

\[z^5+bz^3-30z^2 = 0\]

 

Jumlahkan ketiga persamaan diatas menjadi:

 

\[(x^5+y^5+z^5) + b(x^3+y^3+z^3) - 30(x^2+y^2+z^2) = 0\]

 

karena $x^5+y^5+z^5 = 2850$ dan $x^3+y^3+z^3=90$:

 

\[2850 + 90b - 30(x^2+y^2+z^2) = 0\]

\[2850 + 90b - 30(-2b) = 0\]

\[150b = -2850\]

 

 

Menyebabkan $b=-19$, Sehingga persamaan menjadi:

 

\[t^3-19t-30 = 0\]

\[(t-5)(t+2)(t+3) = 0\]

 

Maka solusi $(x,y,z)$ bilangan real adalah:

 

\[(5,-2,-3),(5,-3,-2),(-2,-3,5),(-3,-2,5),(-2,5,-3),(-3,5,-2)\]

 

Bila ada kesalahan mohon dikoreksi.

Edited by Jun
  • Upvote 4

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