That's a nice story. It's mostly true, but the standard form of a line and the form of a line between two points each avoid a direct expression of the slope.
The standard form for a line is

Either both not both a and b may be zero. When b is zero there's no slope, but it's a perfectly good line.
The line through (a,b) and (c,d) is

Again, we have no slope when a=c but a perfectly good equation and line.