Solve and discuss (for the values of $a$ and $b$) the following equation:

$$
\frac{4 a x+1}{b}-5=\frac{3 x}{b}
$$
If $b=0$, the equation makes no sense. If $b \neq 0$, then

$$
4 a x-1-5 b=3 x
$$

from which

$$
x=\frac{5 b-1}{4 a-3}
$$

If $4 a-3=0$, then two cases are possible:

1. $5 b-1=0$, meaning any value of $x$ satisfies the equation, so we have an identity. (Indeed, if $a=\frac{3}{4}$, $b=\frac{1}{5}$, then both sides equal $15 x$.
2. $5 b-1 \neq 0$, then the expression for $x$ is undefined, and the original equation is unsolvable.

If $b \neq 0$, and $a \neq \frac{3}{4}$, then there is always one, and only one solution.