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### Equations of a Line given Two Points

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Date: 8/31/96 at 15:42:24
From: Anonymous
Subject: Equations of Lines

Find the equation of the line through the two given points:

(4,-1), (3,-6)

The answer given in the textbook is: y = 5x-21  but we do not know two
things:

a) Is it right?

b) How does one get to this answer?

Zillions of thanks.

Mal. from U.K
```

```
Date: 9/1/96 at 4:6:50
From: Doctor Mike
Subject: Re: Equations of Lines

Greetings Mal.,

For (b) you start by finding the slope of the line.  This is a
fraction where the numerator is the difference of the y-coordinates,
and the denominator is the difference of the x-coordinates.

Make sure you take the differences in the same order, like

-1 -(-6)       -1 + 6        5
-----------  =  --------  =  ---  = 5.
4 - 3         4 - 3        1

The slope is usually written as m, so here m = 5.

The form of the equation of a line you are asking about is called
the slope-intercept form.  This looks like  y = mx + b  where b is
the y-intercept, which is the y-value where the line intersects
the y-axis.  That is, (0,b) is on the line.

What we know so far is that the equation looks like y = 5x + b.
So, what is b ?  You can find that out by considering either of the
points to be on the line.

Let's try the first one (4,-1).  Since the combination of x = 4
and y = -1 must be on the line with equation y = 5x+b, it must be
true that -1 = 5(4)+b.  That is, -1 equals 20+b. The only b satisfying
that is b = -21. So the equation is y = 5x + (-21) , or simply
y = 5x - 21.

That's all there is to it.  Pretty brill!  I hope this helps.

-Doctor Mike,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Basic Algebra
High School Equations, Graphs, Translations
Middle School Algebra
Middle School Graphing Equations

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