Definition of a linear equation in two variables General form of linear equation using two variables I always read can be written:
$$
 Ax + By + C = 0
$$
where $A$ and $B$ cannot both be $0$.

So does it means like
$$
 1x + 0y + c = 0
$$
is a linear equation of two variables i.e basically what I am asking is that both $A$ and $B$ cannot be zero but can either $A$ or $B$ be $0$ like:
$$
  0x +4y  -5 =0
\quad\text{or}\quad
  3x +0y -15 =0
$$
Are the above equation a Linear equation of 2 variables?

Please clarify my doubt .
Let's discuss the equation $0x+4y-5=0$.  Geometrically this equation specifies a horizontal line whose $y$ intercept is $(0,\frac 54)$.



Technically speaking this equation only has $1$ variable -- and interestingly it's not $y$.  $y$ is completely determined so it is a constant.  The variable is actually $x$.  Even though $x$ doesn't appear anywhere in the simplified equation $4y-5=0$ it is still the only variable in this equation.  But the number of "variables" in this sense isn't really that important.  What's important in determining what this equation really means is the "space" that this equation exists in.  In this case that "space" is the plane.

Perhaps it would be instructive to parametrize this line. Parametrization means finding a formula for the points on the line (rather than an equation relating the $x$ and $y$ values) in terms of other variables called parameters.

Because $y$ is completely determined in this case we don't need to introduce a parameter for it.  But $x$ can take any value. So set $x=t$ for real number $t$ (this is called the parameter). Then we see that every point on the line has the form $(x,y)=(t,\frac 54)$. If you plug in any value of $t$ you like you'll get a point that's on your line.  So even though no $x$'s appear in the simplified equation $4y−5=0$, we can still say that this is an equation relating both $x$ and $y$.
$x=o$  is the equation of the $y$-axis; and every pt. on it is (0,t), where t is a real number...

This is indeed a linear equation in two variables.