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Standard Form in Linear Equations

Date: 11/21/2001 at 02:42:46
From: Nikki
Subject: Standard form

Where is standard form used?

Date: 11/21/2001 at 10:41:07
From: Doctor Ian
Subject: Re: Standard form

Hi Nikki,

Standard form for what? I assume you mean standard form for a linear 
equation. Let me know if that's not correct. 

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. 

Later on, you'll learn that a number of common curves - circles,
ellipses, parabolas, hyperbolas, and lines - can all be created by
intersecting a cone with a plane at various angles. Every such curve
will have an equation of the form

  ax^2 + bxy + cy^2 + dx + ey + f = 0 

although each curve can also be written in other forms that are more
useful for various kinds of computations.  For example, writing the
equation of a parabola as 

  y = (x - h)^2 + k

tells us right away that the vertex of the parabola is at the point
(h,k), which lets us sketch it very quickly. 

But writing the equations in the canonical form allows them to be
ordered uniquely, which makes them easier to find in tables or lists
of equations - in much the same way that putting words in
alphabetical order in a dictionary makes them easier to look up. Note
that when a, b, and c are all zero, this canonical form reduces to

  0x^2 + 0xy + 0y^2 + dx + ey + f = 0 

                      dx + ey + f = 0

So this form, while not so good for graphing the equation of a line,
makes the line easier to find in a table of equations.
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   
Associated Topics:
High School Equations, Graphs, Translations
Middle School Graphing Equations

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