Elimination of variables from a system of equation. Suppose $a,b,c,d,e,f,g,h,i$ are real numbers. Consider the system of equation
\begin{equation}
\begin{pmatrix}
1 \\ 1 \\ 1
\end{pmatrix}
= \begin{pmatrix}
a + bx  + cy \\ \frac{d}{x} + e + fz \\ \frac{g}{y} + \frac{h}{z} + i
\end{pmatrix}.
\end{equation}
We need to eliminate the variables $x,y$ and $z$ from the above (assuming that solution for the variables $x,y$ and $z$ exists) and get a single  equation in $a,b,c,d,e,f,g,h,i$. I do not know it is possible or not. I hope this is possible but I am unable to get the required. Any suggestion or further help is much appreciated. Thanks in advance.
Multiplying by $x,y,z$ we obtain three polynomial equations in $x,y,z$. Taking resultants we can solve it. The formulas, however, are rather complicated. To show an easy example, take the case of $b=0$. Then we obtain
$$
y=\frac{1-a}{c},\quad z=\frac{h( 1- a )}{ai - a - cg - i + 1}
$$
and
$$
x=\frac{d( - ai + a + cg + i - 1)}{aei - ae - afh - ai + a - ceg + cg -  ei + e + fh + i - 1}.
$$
This example also shows that the coefficients $a,b,\cdots ,i$ need not satisfy a nontrivial polynomial equation in general. In fact, all coefficients are arbitrary in the above formula except for the denominators being nonzero.
Multiplying by $x,y,z$ we obtain three polynomial equations in $x,y,z$. Taking resultants we can solve it. The formulas, however, are rather complicated. To show an easy example, take the case of $b=0$. Then we obtain
$$
y=\frac{1-a}{c},\quad z=\frac{h( 1- a )}{ai - a - cg - i + 1}
$$
and
$$
x=\frac{d( - ai + a + cg + i - 1)}{aei - ae - afh - ai + a - ceg + cg -  ei + e + fh + i - 1}.
$$
This example also shows that the coefficients $a,b,\cdots ,i$ need not satisfy a nontrivial polynomial equation in general. In fact, all coefficients are arbitrary in the above formula except for the denominators being nonzero.