In the traditional teaching of Analytical Geometry, the governing equation for a straight line has the following five forms, along with limitations for the first four:
 Point-Slope form: where is a point on the line, and is the slope. The limitation for this form is that it can not represent line perpendicular to the x-axis since it has no slope.
 Slope-Intercept form: where is the slope, is the intersect the line made on y-axis. Its limitation is that it can not represent line perpendicular to the x-axis.
 Two-Point form: where are two points on the line. However, this form can represent neither line perpendicular nor parallel to x-axis due to the fact when or , the form breaks down from dividing by zero.
 Point-Intercept form: where are the intersects the line made on x-axis and y-axis respectively, and . Again, this form can represent neither line perpenticular nor parallel to the x-axis. It does not work for any line that passes the point of origin either.
 General form: , this form can represent all lines.
Here I am presenting a proof to show  is indeed capable of representing all straight lines.
Let us start with an observation:
In a rectangular coordinate system, given two distinct points , and any point on the line connecting and , the area of triangle with vertices and must be zero!
Recall a theorem proved in my blog “Had Heron Known Analytic Geometry“, it means for such and ,
Therefore, we can define the line connecting two distinct points as a set of such that the area of the triangle with vertices and is zero, mathematically written as
is an algebraic representation of the line connecting two distinct points and .
When , (1) becomes
and when , we have
a line perpendicular to the horizontal axis.
When , (1) becomes
a line parallel to the horizontal axis.
Evaluate (1) with yields:
Collecting terms in (1), and letting
(1) can be expressed as
In fact, we can prove the following theorem:
To prove , we need to show
We have already shown (2) by setting the values of and earlier.
We will prove (3) now:
and , we have
Written in matrix form,
then by Cramer’s rule,
is a column vector of zeros,
which contradicts the fact that
are not all zero.
The consequence of is that every point on a line connecting two distinct points satisfies equation for some .
where are not all zero is the governing equation of any straight line.