Theorem 1.4: Two points uniquely determine a line. If the points are   (x1, y1)   and   (x2, y2)   then the equation is

x = x1

if the two points are vertical, and if not then it is of the form

y = mx + b

where

and

Proof: If the two points are vertical, then they both have the same   x-coordinate by definition, so both points are solutions to the equation

x = x1

If they are not vertical, then we find an equation for the form   y = mx + b,   which both points satisfy by solving the following system of two equations in two unknowns

y1 = mx1 + b
y
2 = mx2 + b

for   m   and   b.   If we subtract the top equation from the bottom we get

y2 - y1 = mx2 - mx1

To solve for   m,   factor out the   m   on the right,

y2 - y1 = m(x2 - x1)

Divide.

Now to solve for b.   If we solve the first equation for   b   we get,

b = y1 - mx1

which is the same thing we got in Theorem 1.3. Since we know that

we can substitute and get

Find common denominators.

Multiply the tops and add.

which simplifies to

top

next theorem (1.5)