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

x = x1

If the line is not vertical, then the equation can be written as

y - y1 = m(x - x1)

which can be simplified to

y = mx + b

where

b = y1 - mx1

Proof: If the point is   (x1, y1)   and the line is a vertical line then its equation is

x = x1

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

y - y1 = m(x - x1)

To get this into the form   y = mx + b,   solve for   y.   Remove parentheses

y - y1 = mx - mx1

Transpose

y = mx + y1- mx1

which is of the form   y = mx + b   if

b = y1- mx1

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