$xy = 100 $
$y = 100/x$
$x$ is not equal to $0$ and can be represented as $1\cdot x$
However I still do not understand on why $100/x$ does not represent a direct variation. Is it because of the use of division within the right side of the equation is not valid?
On
A direct relation is one of the form $y=mx$, but the relation here is not of that form – $x$ is being divided, not multiplied. Rather, it is an inverse relation.
On
The relation is between the two variables $x$ and $y$. A direct variation means that when $x$ increases $y$ will increase too by an amount specified by the equation, for example:
$y=x$. If $x = 1$ then $y = 1$ and if $x$ increases to be $x=2$ then $y=2$ this is a direct variation and it may be represented like the $f(x) = x$.
Let's change the formula quite a bit $f(x) = x + 1$ this is the same as $y = x + 1$. If $x = 1$ then $y = 1 + 1 = 2$ and if $x = 2$ then $y = 2 + 1 = 3$. This is a direct variation.
Let's change the formula again $f(x) = x - 1$. If $x = 1$ then $y = 1 - 1 = 0$ and if $x = 2$ then $y = 2 - 1 = 1$. This is also a direct variation.
Again $f(x) = x \times 2$. If $x = 1$ then $y = 1 \times 2 = 2$ and if $x = 2$ then $y = 2 \times 2 = 4$. This is also a direct variation.
But what about $f(x) = \frac{1}{x} : x \ne 0$. If $x = 1$ then $y = \frac{1}{1} = 1$ and if $x = 2$ then $y = \frac{1}{2} = 0.5$. This is an inverse variation, as $x$ increases $y$ decreases or their product is a constant.
In the end a direct variation means that as $x$ increases $y$ increases, and an inverse variation means that as $x$ increases $y$ decreases.
Seeing graphs helps in understand so this is the graph of your equation $f(x) = \frac{100}{x}$:
Visually speaking, a directly proportional relationship in the form $y=mx$ is a straight line. Thus the slope, which is $\frac{y}{x}$, must be constant for all $x$.
With $x=1, y=100$ and $x=2, y=50$, the slopes $\frac{y}{x}$ are $100$ and $25$ respectively. Since the slope is not constant, $xy=100$ is not directly proportional.