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
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,
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
next theorem (1.5)