How can we derive the standard form of the linear equation: Ax+By+C=0? What do "A",...

Let's say we start with the point-slope form
${\displaystyle y-y_0=m(x-x_0)}$,
where (x,y) represents any point on the line, and ${\displaystyle (x_0,y_0)}$ represents a given point on the line, and m is the given slope of the line. Now,
${\displaystyle y-y_0=m(x-x_0)\iff y-y_0=mx-m{x}_0}$
${\displaystyle \iff y-y_0-mx-m{x}_0=0}$
${\displaystyle \iff (-m)x+y+(-y_0-m{x}_0)=0}$
Now, we can simply let ${\displaystyle A=-m,B=1,C=(-m{x}_0-y_0)}$, which means that
$y+(-m)x+(-y_0-m{x}_0)=0⟺Ax+By+C=0$