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I have a system of equations in a program I'm writing. It's of the form: $$ \left\{ \begin{array}{c} \frac{a_1}{x}+b_1 y+c_1 z=d_1 \\ \frac{a_2}{x}+b_2 y+c_2 z=d_2 \\ \frac{a_3}{x}+b_3 y+c_3 z=d_3 \end{array} \right. $$ $a_n$, $b_n$, $c_n$ and $d_n$ are known coefficients.

I'm going to use Gauss-Jordan elimination to solve it. But I'm not sure how exactly I should form the matrix with $x$ being in the denominator. I guess I could just put in $a_n$ like the other coefficients and take the inverse of the solution for the first variable to get the value of $x$. But it feels like I'm missing something and that it might be an extra step I could avoid somehow. What would be the "correct" way to do this?

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  • $\begingroup$ You can always define $w = 1/x$ and use $w, y, z$ as the variables. After you find the value of $w$, you can calculate $x$ from it. $\endgroup$
    – NickD
    Jun 6, 2020 at 15:55

2 Answers 2

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Simple, take $\frac{1}{x}=u$, the you have three linear simultaneous equations in $u,y,z$, which can be solved by Cramer's method:$$\frac{u}{|A_1|}=\frac{y}{|A_2|}=\frac{z}{|A_3|}=\frac{1}{|A|}, x=\frac{1}{u}, u \ne 0$$. See Cramer's method in:

https://en.wikipedia.org/wiki/Cramer%27s_rule

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Nevermind. I understand what your doing..

So you have some matrix A and an x vector given by

$\begin{bmatrix} 1/x \\ y \\ z\end{bmatrix} $

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