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Slope-Intercept Formula


Date: 06/05/2001 at 11:39:18
From: Vittorio Smith
Subject: Algebra

How come y = mx + b has to be that way?


Date: 06/05/2001 at 19:02:56
From: Doctor Ian
Subject: Re: Algebra

Hi Vittorio,

You can write any equation in a lot of different ways. What's nice 
about the form 

  y = mx + b

is that it allows you to sketch the graph of the corresponding line 
very quickly. 

How do you do that? Well, you know that the line (if it's not 
vertical) has to cross the y-axis somewhere... in particular, where 
x = 0. Since you know x = 0, 

  y = m(0) + b

    = b

which tells you that (0,b) is a point on the line. You also know that 
the line (if it's not horizontal) has to cross the x-axis... in 
particular, where y = 0. Since you know y = 0,

      0 = mx + b

     -b = mx

   -b/m = 0

which tells you that (-b/m,0) is a point on the line. So just by 
looking at the equation

  y = mx + b

you can graph two points:

  (0,b) and (-b/m, 0)

And once you've graphed these two points, you can fill in the rest of 
the graph with a ruler.  

A line can also be written so that it looks like this:

  ax + by + c = 0

But in order to graph the line, you have to do a lot of work to find 
values of x and y that fall on the line. So it's not very good for 
graphing, although it's a good description for some other situations. 

Another nice thing about the form

  y = mx + b

is that you can tell pretty quickly whether two lines are parallel or
perpendicular.  If they have the same slope, they are parallel. If the
slopes can be multiplied to get -1, they are perpendicular.

For example, the following two lines are parallel:

  y = 2x + 3    

  y = 2x + 14

And the following two lines are perpendicular:

  y = 2x + 3

  y = (-1/2)x - 6

It's important to realize that a lot of the things that you learn in 
math class are shortcuts like these, and that the choice of whether to 
write something in one form or another usually depends on the 
shortcuts that each form makes available. If you keep that in mind, 
you'll see that the things they're teaching you aren't completely 
random. There is a method to the mathness, so to speak.  :^D

I hope this helps. Write back if you'd like to talk about this some 
more, or if you have any other questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Basic Algebra
High School Equations, Graphs, Translations
Middle School Algebra
Middle School Graphing Equations

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