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Problem - $w$ dan $\dfrac{1}{w}$ | Catatan Nurkholis

Problem – $w$ dan $\dfrac{1}{w}$

Problem:
Tentukan semua solusi real $x$, $y$, $z$, $w$ pada sistem:
\begin{eqnarray*}
x+y+z&=&w\\
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}&=&\frac{1}{w}
\end{eqnarray*}

Solusi:
Samakan penyebut pada persamaan ke dua diperoleh:
$$\dfrac{yz+xz+xy}{xyz}=\dfrac{1}{w}$$
dengan mengalikan silang kemudian substitusi persamaan pertama pada $w$, didapat:
$$(x+y+z)(yz+xz+xy)=xyz$$
yang ekuivalen dengan
$$x^2y+x^2z+y^2x+y^2z+z^2x+z^2y+2xyz=0$$
dan dapat disederhanakan menjadi:
$$(x+y)(x+z)(y+z)=0$$
dari sini dapat kita ambil solusinya (sebut saja $y+z=0$, $y=-z$)
dan diperoleh $x=w$.

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