Theorem 1.2: A vertical line has infinite slope. A line is not vertical if and only if its equation can be written in the form

Proof: Let (x₁, y₁) and (x₂, y₂) be two points on a vertical line. Since they are two points on the same vertical line, the by the defintion of a vertical line they have the same x coordinates. If they are different points then they must have different y coordinates. So the slope between them will have a zero denominator and a nonzero numerator, and will then have a value of infinity.

If the line satisfies an equation of the form y = mx + b, we must show that the line is not vertical. First, since it will be possible to substitute any real number in for x, we could take any two different real numbers, substitute them in for x and get two points on the line with different x coordinates. So not all points on the line will be vertical. Moreover, for any value of x, there will be a uniquely determined value of y which is obtained by substituting the given value of x into the equation, so above any x on the x-axis there is exactly one point on the line. We must conclude that the line is not vertical.

We must finally show that if the line is not vertical, then it satisfies an equation of the form y = mx + b. By definition, a line is the set of points who's coordinates satisfy a linear equation in one or two unknowns. Such an equation can be written as

where r, s, and t are real numbers where r and s are not both 0. If s = 0, then r must not be zero, because, otherwise, our equation would have no unknowns, and we could solve for x. All of the points on the line would then have the same x-coordinate: x = t/r, and the line would be vertical. So if the line is not vertical, then s is not zero. We could then solve for y. Transpose

and, since s is not 0, divide by s and get an equation of the form

where m = -r/s and b = t/s

next theorem (1.3)