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### Equation of a Line

```
Date: 02/14/2002 at 00:32:12
From: Victor Squillacioti
Subject: Proofs

Prove the point-slope equation property, which is y-y1 = m(x-x1).

It gave me a hint, but it wasn't very useful to me. It was:

Show that any point (x2,y2) on the line described must satisfy the
equation, and that any point satisfying the equation must be on the
line described.

Please give me any useful information to help me. Thank you in
advance.

Victor Squillacioti
```

```
Date: 02/14/2002 at 16:52:15
From: Doctor Peterson
Subject: Re: Proofs

Hi, Victor.

To show that this is the equation of the line through (x1,y1) with
slope m, you have to show two things:

(1) every point on the line satisfies the equation;
(2) every point that satisfies the equation is on the line.

That is what the hint said.

Now, what does it mean for a point to be on the line? Since we know
the slope of the line, which means that the slope between any two
points on the line has the value m, we can say that the slope between
points (x1,y1), which is given as on the line, and the point (x2,y2)
which we are assuming is on the line, will be m. Write an equation
that says this, and use that to show that the equation will be true
when x = x2 and y = y2. You will have shown that (1) is true.

Now to go the other way, suppose that you have a point (x2,y2) that
satisfies the equation. Write out what the equation tells you about x2
and y2. You want to show that the slope between (x1,y1) and (x2,y2) is
m, which will prove that (x2,y2) is on the line. Write out the
definition of this slope, and see if you can show it is equal to m.

Here is some information on the equations of a line that touches on
this idea:

Formulas for the Equation of a Line
http://mathforum.org/dr.math/problems/michelle9.28.98.html

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Linear Equations

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