Rearranging Proportional Controller
A few comments first:


It turns out that you have a correct equivalence, as there is no error in your final equation if you assume the first equation is also correct. How you achieved that, without having any of the basic algebraic skills needed and by only "playing around", is a mystery to me. My guess is that somehow you were already able to find the right answer but not through just play.
You haven't shown any of the steps you tried taking. Perhaps that is because of the work you imagine in creating images for each step you tried. But you can use Latex, just fine. If that was your only problem and if you plan on posting here again, I'd recommend learning some Latex and the use of a paired set of \\$ and/or $\$\$$ to inform the web site here that you are writing Latex math.
The fact that you aren't able to proceed from your first equation, then solving for \$G_1\left(s\right)\$, tells me a great deal about your need for a great deal of algebra practice. This kind of work is perhaps a struggle for children just learning algebra. But not someone engaging in transfer functions.


Since you couldn't solve it, I'll assume you need me to take incremental, remedial steps. Follow here:

$$\begin{align*}
G\left(s\right)&amp;=\frac{k\cdot G_1\left(s\right)}{1+k\cdot G_1\left(s\right)}\\ \\
G\left(s\right)\cdot \left[1+k\cdot G_1\left(s\right)\right]&amp;=k\cdot G_1\left(s\right) \\ \\
G\left(s\right)+G\left(s\right)\cdot k\cdot G_1\left(s\right)&amp;=k\cdot G_1\left(s\right) \\ \\
G\left(s\right)&amp;=k\cdot G_1\left(s\right) - G\left(s\right)\cdot k\cdot G_1\left(s\right) \\ \\
G\left(s\right)&amp;=k\cdot G_1\left(s\right)\cdot\left[1 - G\left(s\right)\right] \\ \\
\frac{G\left(s\right)}{1 - G\left(s\right)}&amp;=k\cdot G_1\left(s\right) \\ \\
G_1\left(s\right) &amp;=\frac{1}{k}\cdot\frac{G\left(s\right)}{1 - G\left(s\right)} \\ \\
G_1\left(s\right) &amp;=\frac{1}{k}\cdot\left[\frac{G\left(s\right)}{1 - G\left(s\right)}+1-1\right] \\ \\
G_1\left(s\right) &amp;=\frac{1}{k}\cdot\left[\frac{G\left(s\right)}{1 - G\left(s\right)}+\frac{1 - G\left(s\right)}{1 - G\left(s\right)}-1\right] \\ \\
G_1\left(s\right) &amp;=\frac{1}{k}\cdot\left[\frac{G\left(s\right)+1 - G\left(s\right)}{1 - G\left(s\right)}-1\right] \\ \\
G_1\left(s\right) &amp;=\frac{1}{k}\cdot\left[\frac{1}{1 - G\left(s\right)}-1\right] \\ \\
G_1\left(s\right) &amp;=\frac{\frac{1}{1 - G\left(s\right)}-1}{k}
\end{align*}$$

That's really all there is to the process. It's just algebraic manipulation to solve for a term. A very basic algebraic skill you need to acquire.