Theorem 1.3: There is a uniquely determined line going through a given point with a given slope. If the given point is (x1, y1), and the line is vertical, then the equation is
If the line is not vertical, then the equation can be written as
which can be simplified to
where
Proof: If the point is (x1, y1) and the line is a vertical line then its equation is
which all of the points will satisfy, since by definition, all points on a vertical line have the same x coordinate.
If the line is not vertical, then by Theorem 1.2, it satisfies and equation of the form y = mx + b. If (x, y) is any other point on the line, then by Theorem 1.1, the slope between it and (x1, y1) will be m. That is to say.
Clear the denominator
To get this into the form y = mx + b, solve for y. Remove parentheses
Transpose
which is of the form y = mx + b if
