Equations of a Line given Two Points

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

Please can you help my children?  Many thanks in advance.

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., 

The answer to (a) is yes, the answer given is correct.  

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/