<s>[INST] Dealing more efficiently with fractional forms in system of equations As an example, suppose we have to solve the following system of two equations and two unknowns:
$$
\begin{cases}
-\frac{10}{x}-\frac{8}{y} &= \frac{8}{3}  \\ 
-\frac{6}{x}+\frac{6}{y} &= -\frac{1}{3} 
\end{cases}
$$

My approach and solution
I opted to solve it by combination, referring to the first equation as (1) and the second equation as (2), I started by eliminating the y's:
\begin{align}
\frac{6}{8}(1)+(2): -\frac{60}{8x}-\frac{6}{x} &= \frac{6}{3}-\frac{1}{3} \\
\frac{-60-48}{8x} &= \frac{5}{3} \tag{*}\\
40x &=3(-108) \tag{**} \\
x &=-\frac{81}{10} \tag{1'}
\end{align}
Substituting (1') into (2):
\begin{align}
\frac{100}{81}-\frac{8}{y}&=\frac{8}{3} \\
y &= -\frac{162}{29}
\end{align}
So I find the tuple of $(-\frac{81}{10};-\frac{162}{29})$ as solution.

Questions

*

*For such systems, should we not be concerned about the conditions of existence of the system? Namely, in this case both $x\neq 0$ and $y\neq 0,$ in the same way that we would do when solving an equation. If yes, how do we formally write the domain of existence for a system?

*Is my transition from step $(*)$ to $(**)$ allowed? My understanding is that, yes for all $x\neq 0.$

*I am really eager to learn whether there are simple ideas that simplify the system (and other similar systems) before we start solving it. As shown in my approach, it was slightly awkward dealing with the fractions throughout and the "large" numbers, which inherently may render the approach more prone to mistakes. Any alternative, quicker approach (I only know by combination and substitution) would be much appreciated.

 [/INST] $$
\begin{cases}
-\frac{10}{x}-\frac{8}{y} &= \frac{8}{3},\quad{|}\cdot{3}\\ 
-\frac{6}{x}+\frac{6}{y} &= -\frac{1}{3}.\quad{|}\cdot{(-5)}
\end{cases}
\iff
\begin{cases}
-\frac{30}{x}-\frac{24}{y} &= \frac{24}{3},\\ 
\frac{30}{x}-\frac{30}{y} &= \frac{5}{3}.
\end{cases}
\iff
$$
$$
\iff
\begin{cases}
-\frac{54}{y} &= \frac{29}{3},\\ 
\frac{30}{x}-\frac{30}{y} &= \frac{5}{3}.
\end{cases}
\iff
\begin{cases}
{y} = \frac{-162}{29},\\ 
\frac{30}{x}-\frac{30}{y} &= \frac{5}{3}.
\end{cases}
\iff
\begin{cases}
{y}=\frac{-162}{29},\\ 
{x}=\frac{-81}{10}.
\end{cases}
$$
Also the parameters $x\ne{0}$, $y\ne{0}$.
Good luck!
</s>