Justifying why 0/0 is indeterminate and 1/0 is undefined

$\dfrac 00=x$
$0x=0$
$x$ can be any value, therefore $\dfrac 00$ can be any value, and is indeterminate.

$\dfrac 10=x$
$0x=1$
There is no such $x$ that satisfies the above, therefore $\dfrac 10$ is undefined.

Is this a reasonable or naive thought process?
It seems too simple to be true.

Answer

In the context of limits, $0/0$ is an indeterminate form (limit could be anything) while $1/0$ is not (limit either doesn't exist or is $\pm\infty$). This is a pretty reasonable way to think about why it is that $0/0$ is indeterminate and $1/0$ is not.

However, as algebraic expressions, neither is defined. Division requires multiplying by a multiplicative inverse, and $0$ doesn't have one.