Why is $y=\frac{x+y}{x}$ the same as $y=\frac{x}{x-1}$?
It seems to be the same on graphing calculators, but I don't get why. To generalize my statement, I would like to know how to simplify equations where "$y$" is on both sides, but can't be removed through conventional methods.
On
To answer the question, they are not the same.
The first one is an implicit expression in $y$. It makes no sense for $x=0$. The second one is explicit in $y$, and it is defined for $x=0$. So the graphs are indeed different, since they differ at least in one point.
Yet for many other $x$, their graphs coincide.
A simpler and similar example is $f_1(x) =\frac{x}{x}$ and $f_2 (x)=1$. Almost the same, except for one little detail at $0$.
Your second question is a bit broad. When $y$ is on both sides, there are several methods, depending on how things are involved: variable re-parametrisation, equation solving, higher dimension embedding...
Ah, but $y$ can be removed through conventional methods in this case. We have $$y=\frac{x+y}{x}$$ $$xy = x + y$$ $$xy - y = x$$ $$(x-1)y = x$$ $$y = \frac{x}{x-1}$$