I have the following challenge:
Rewrite this fraction:
$$ \frac{a}{x-Sy} $$
where $a$, $x$ and $y$ are constants and $S$ is a variable.
To this fraction:
$$\frac{K}{S}$$
where $K$ is a constant.
If its possible, then what are the steps to rewrite the fraction?
What you're asking for is not possible. If $S=0$, the first expression is $\frac ax$ but the second expression is undefined. Conversely, if $S=\frac xy$, then the first expression is undefined but (assuming $x \neq 0$) the second expression exists.
This is impossible when $a \neq 0$. If this were possible, you'd have $K=\frac{aS}{x-Sy}$ which is supposed to be a constant. With $S=0$ you get $K=0$. Hence $\frac{K}{S}=0$ regardless of what $S$ is, so $\frac{a}{x-Sy} = 0$ regardless of what $S$ is. Hence $a=0$.
When $a=0$ this is possible by taking $K=0$.
As long as $S$ isn’t zero, $$\textrm{anything}=\frac{\textrm{anything}\times S}{S}$$ So necessarily $$K=\frac{aS}{x-Sy}=\frac{a}{\frac{x}{S}-y}.$$ This can’t be constant unless $a$ is zero (in which case $K$ is also zero) or $x$ is zero (in which case $K$ is $-\frac{a}{y}$).
