How to rearrange $x = \cfrac s{s-1}$ to find $s$ in terms of $x$?

Question

So, I'm working through an example on Lagrange's method and the immigration-birth process in probability textbook, but I've been unable to follow the example - the reason being the solution required rearranging $x=\cfrac{s}{s-1}$ so that $s$ is the subject. At first sight it seems really basic and irrelevant to what I'm actually learning. However, it doesn't describe how it rearranged it and I can't for the life of me work out how it was found. It simply says...

Writing $x=\cfrac{s}{s-1}$ gives $s=\cfrac{x}{x+1}$

Can anyone tell me how the equation is manipulated to find $s$ in terms of $x$? It's probably really simple, but I can't see how it is done.

Answer 1

You have $$ x = \frac{s}{s-1} = \frac{s - 1 + 1}{s-1} = 1 + \frac{1}{s-1}. $$

Can you take it from here?

Answer 2

Note that

$$x = \frac{s}{s-1} \iff (s-1)x = s\iff sx -x=s\\\iff sx -s=x\iff s(x-1)=x\iff s = \frac{x}{x-1}$$

Answer 3

$x=\cfrac s{s-1}$

$x(s-1)=s$

$sx-x=s$

$sx-s=x$

$s(x-1)=x$

Therefore, $s=\cfrac{x}{x-1}$, NOT $s=\cfrac{x}{x+1}$ as you mentioned.

The inverse of $x=\cfrac s{s-1}$ is $s=\cfrac{x}{x-1}$, and solving for $s$ in terms of $x$ gives you this.