Theorem 2.1: (The parametric representation of a line) Given two points   (x₁, y₁)   and   (x₂, y₂),   the point   (x, y)   is on the line determined by   (x₁, y₁)   and   (x₂, y₂)   if and only if there is a real number   t   such that

and

Proof: Assume that there is a real number   t   such that

and

Then

Remove parentheses

and

If the points are vertical, then    x₂ - x₁ = 0   so

so   x = x₁ = x₂,  and the point is on the same vertical line.

Otherwise, let   m   be the slope of the line between   (x₁, y₁)   and   (x₂, y₂).   Then

and

and   (x, y)   satisfies the point-slope form of the equation of the line going through   (x₁, y₁)   with slope   m, by Theorem 1.3.

Conversely, assume that   (x, y)   is any point on the line. We will first dispose of the case where the line is vertical. In this case,

Let

We can assume the denominator is nonzero because we have two distinct points.

Then

so

and

We must also dispose of the case of a horizontal line where   y₂ = y₁.   Let

We can again assume the denominator is nonzero because we have two distinct points.

Then

so

and

Otherwise, the slope of the line is given by

by Theorem 1.1. Multiply both sides by   x - x₁   and divide by    y₂ - y₁

We can assume that the denominators on both sides are nonzero because we have disposed of those cases.

Define

Then

and

So

and

or

and

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