<s>[INST] Solve the system $x^2(y+z)=1$ ,$y^2(z+x)=8$ and $z^2(x+y)=13$ Solve the system of equations in real numbers 
\begin{cases} x^2(y+z)=1 \\  y^2(z+x)=8 \\z^2(x+y)=13 \end{cases}
My try:
Equations can be written as:
\begin{cases}\frac{1}{x}=xyz\left(\frac{1}{y}+\frac{1}{z}\right)\\
\frac{8}{y}=xyz\left(\frac{1}{x}+\frac{1}{z}\right)\\
\frac{13}{z}=xyz\left(\frac{1}{y}+\frac{1}{x}\right)\end{cases}
Let $p=xyz.$ Then we have:
\begin{cases}\frac{1}{x}-\frac{p}{y}-\frac{p}{z}=0\\
\frac{p}{x}-\frac{8}{y}+\frac{p}{z}=0\\
\frac{p}{x}+\frac{p}{y}-\frac{13}{z}=0\end{cases}
Then we get
$$\frac{p+1}{x}=\frac{p+8}{y}=\frac{p+13}{z}$$
Any clue here?
 [/INST] Alternatively, denote $y=ax, z=abx$. Then:
$$\begin{cases} x^2(y+z)=1 \\  y^2(z+x)=8 \\z^2(x+y)=13 \end{cases} \Rightarrow \begin{cases} ax^3(1+b)=1 \\  a^2x^3(1+ab)=8 \\a^2b^2x^3(1+a)=13 \end{cases}.$$
Divide $(2)$ by $(1)$ and $(3)$ by $(2)$:
$$\begin{cases} a(ab+1)=8(1+b) \Rightarrow b=\frac{8-a}{a^2-8} \\  8b^2(1+a)=13(1+ab) \Rightarrow 8\cdot \frac{(8-a)^2}{(a^2-8)^2}\cdot (1+a)=13(1+a\cdot \frac{8-a}{a^2-8}) \Rightarrow \end{cases}\\
8(8-a)^2(1+a)=13\cdot 8\cdot (a-1)(a^2-8) \Rightarrow \\
(3a-10)(2a^2+7a-2)=0.$$
You can find $a$, then $b$, then $x$, $y$ and $z$.
</s>