**Theorem
1.4**: Two points
uniquely determine a line.
If the points are
(*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2})
then the equation is

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

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

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

y

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

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

Divide.

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

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