Elimination of variables from a system of equation.

Question

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.

Answer 1

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.

Answer 2

If the solution for the variables $x,y$ and $z$ exist and you're given this: $$ \begin{pmatrix} 1\\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} a + bx + cy \\ \frac{d}{x} + e + fz \\ \frac{g}{y} + \frac{h}{z} + i \end{pmatrix} $$ and then you're asked to find one single equation for $a,b,c,d,e,f,g,h$ and $i$, then you can add up the components and get your desired one single equation like so: $$3 = a + bx + cy + \frac{d}{x} + e + fz + \frac{g}{y} + \frac{h}{z} + i$$

You're welcome.

EDIT

Okay, now I get it, you want to elimnate the variables $x,y,z$. Never mind, if $x,y,z$ are known or not.

This is a different question!!!

Fine, a good way is to do it like Dr. Sonnhard Graubner already described. Solve the first equation for $y$ and you get

$$y = y(x) = 1/c - a/c - (b/c)x$$

Then solve the second equation for $z$ and you get

$$ z = z(x) = 1/f - e/f - d/(fx) $$

Now plug $y=y(x)$ and $z=z(x)$ in the third equation and you get

$$ 1 = g/(1/c - a/c - (b/c)x) + h/(1/f - e/f - d/(fx)) + i $$

or also (just differently written)

$$ 1 = gc(1 - a - bx)^{-1} + hf(1- e - dx^{-1})^{-1} + i $$

If we solve this last equation for $x$ ("the only unknown"), then we're done, because you can plug that $x$ in $y(x)$ and in $z(x)$ and get all $x,y$ and $z$ in terms of the other variables. It can be solved, but it's not handy. Maybe there's a crazy trick, that will help to solve it without much struggle? I will think about it. If I find something, then I will let you know.