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