This method is in fact a change of variable, where we set a new variable $k = y/x$ and convert the problem to an equation for $k$.  Remember that the solution $x$ and $y$ are just numbers, and the variable $k$ simply represents their ratio (as long as $y$ is not $0$).  For multiple solutions for $(x,y)$, we have multiple solutions for $k$.  This is not about $y$ and $x$ (as variables) having a linear relation or something like that.  It is simply a method to convert the problem to solving for a new unknown variable.

If there is a constant term on the right hand side, the change of variable $y=kx$ is still valid, but the resulting new equation is usually not easy to simplify.  Here we don't have a constant term, and we can divide by $x$ and get two simpler equations.  This idea could also work (to simplify the equations) in the general case you mention.