Find $\frac{1}{x-y+z}$ from given fraction equations.

Question

$$\begin{align} \frac{3}{x}\,-\,\frac{4}{y}\,+\,\frac{2}{z}\quad&=\quad3\\ \frac{2}{x}\,-\,\frac{8}{y}\,-\,\frac{1}{z}\quad&=\,-\,8\\ \frac{4}{x}\,-\,\frac{6}{y}\,-\,\frac{3}{z}\quad&=\quad1\\ \frac{1}{x-y+z}\quad&=\quad? \end{align}$$ I tried to find solution by equaling $\frac{1}{x}=a, \frac{1}{y}=b\,$ and $\frac{1}{z}=c\,.$ However, $b=\frac{149}{86}\,$ and $c=\frac{14}{43}$ which led me to ugly solution or no intended correct solution at all. How to find solution to the above-shown fraction equation?
Thanks beforehand.

Answer 1

Go on, go on… $$x=\frac{43}{133}\qquad y=\frac{86}{149}\qquad z=\frac{43}{14}$$ $$\frac1{x-y+z}=\frac{5662}{15953}=0.3549\dots$$ Beauty is not a requirement of mathematical problems – especially in applied problems, and even in pure problems.