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Standard Form of a Line

Date: 04/23/2002 at 22:48:05
From: Tricia Swanson
Subject: Standard Form

What is standard form? What does Ax+by = c mean and what do the 
letters stand for?

Date: 04/24/2002 at 23:05:27
From: Doctor Peterson
Subject: Re: Standard Form

Hi, Tricia.

This is what is sometimes called the "standard form" for a line. (Some 
people think of slope-intercept as the standard form.) It is most 
commonly used when we write systems of linear equations.

Although the slope-intercept form is the most familiar form, because 
it expresses y directly in terms of x, not all lines can be written 
that way, because vertical lines have no slope, and have to be written 
as x = k. But all lines can be written in standard form, because it is 
symmetrical (treating x and y in the same way). In particular, the 

    y = mx + b

can be written as

    mx - y = -b

which is in your "standard form" - the sum of a multiple of x and a 
multiple of y is equal to a constant.

And the line

    x = k

is already in "standard form," with A = 1, B = 0, and C = k.

This form doesn't tell you what y is for a given x, because that can't 
be done for all lines. What it does is to say that any point (x,y) for 
which Ax + By has a particular value is on this line.

So the value of standard form is that it can be applied to all lines, 
including vertical lines, which can't be written in slope-intercept 
form. And that, in turn, is because it is "unbiased": it treats x and 
y in the same way, rather than putting y in a special position.

What's awkward about "standard form" is that the parameters A, B, and 
C do not stand for something obvious, as in the other forms. In fact, 
they are not fixed; if you double them all, you still have the same 
line. But, rewriting the standard form in slope-intercept form,

    Ax + By = C

    y = -A/B x + C/B

and finding the x-intercept

    y=0 => x = C/A

you can determine that the slope is

    m = -A/B

and x- and y- intercepts are

    a = C/A
    b = C/B

So the ratios of parameters in this form tell you about the line. Note 
that if A is zero, there is no x-intercept, and if B is zero, there is 
neither an x-intercept nor a slope. And that makes sense if you look 
at the lines.

Notice also that the slope is independent of C; if you change C, you 
make a line parallel to the original line.

Converting between standard form and slope-intercept form shows how 
they are related. This also serves to explain an important fact about 
the standard form, namely that although A, B, and C have no direct 
graphical interpretation, they are related to the visible concepts 
of slope and intercept. That mitigates the objection that the standard 
form is not helpful in graphing.

I can add a couple of additional reasons why the standard form is worth 

  1. It fits well with a vector formulation of geometry that you
     may see in the future; using vectors, it can be written as
     a "dot product":

     (A,B).(x,y) = C

  2. It relates the equation of a line to what is called "linear
     combination" in more advanced mathematics, namely the sum of
     multiples Ax + By. This has use far beyond graphing lines.

But mostly it is important just because it is a "standard" that can 
be used for all lines, not a special form that makes it easier to 
visualize those lines for which it can be used, as with slope-intercept.

You will find a variation on standard form, called General Form, in 
the Dr. Math FAQ (select Two Dimensions: Lines):

   Analytic Geometry Formulas

- Doctor Peterson, The Math Forum

Date: 04/24/2002 at 23:09:15
From: Tricia Swanson
Subject: Standard Form

Thank you so much for the help! This site is wonderful!

Date: 03/02/2014 at 20:52:06
From: Quinn
Subject: Your article about Why Use Standard Form

I read the above article. Just wanted to add that the standard form does a
good job of relating an equation to the meaning of a real-world scenario
(in a bivariate relationship). For example,

   Joe is selling t-shirts for $10 and jeans for $15. 
   What combinations of t-shirts and jeans can he sell to make $1000?

   10t + 15j = 100

If this was in slope-intercept form, it would not be so clear how t and j
relate to the prices of the items for sale, or to the total goal of

Date: 03/02/2014 at 21:54:14
From: Doctor Peterson
Subject: Re: Your article about Why Use Standard Form

Hi, Quinn.

To fill out what you said in a way consistent with the original
discussion, let's define variables this way:

   x = number of t-shirts sold
   y = number of pairs of jeans sold

Now the question can be modeled by this equation:

   10x + 15y = 1000

Of course, this is more a way to come up with an equation (and to show how
it arose) than a way to display the relationship. The equation could then
be simplified, e.g., by dividing everything by 5:

   2x + 3y = 200

What is being displayed here is more of a proportional relationship: that
the way that the weights of x and y combine observes the ratio 2:3. This,
again, is an interesting relationship to see, and, as you say, more
enlightening than the sort of functional relationship that slope-intercept
form displays. As I said, it's an "unbiased" relationship, treating the
two variables as equals.

This is actually an example of linear combination, which I mentioned as
one of the places where standard form is natural. As your example shows
clearly, it is not only in "advanced mathematics" that this occurs, as I
said, but in any kind of problem where, rather than one variable being
presented as a function of the other, we know the value of the function
and are instead interested in a linear function of BOTH variables -- in
your case, the number of t-shirts and of jeans. This kind of situation
also arises often in problems that lead to systems of equations.

Thanks for the ideas!

- Doctor Peterson, The Math Forum

Date: 03/03/2014 at 10:43:26
From: Quinn
Subject: Thank you (Your article about Why Use Standard Form)

Dear Dr. Math,

Thank you for a most interesting and detailed answer. 

For some time now, I have felt it a little odd that a bias is always given
to the dependent variable, and did not know of "linear combinations" (as I
have not yet studied linear algebra). I especially enjoyed your
description of the standard form equation showing the weighting of x and y
so that the way they combine forms a ratio of 2:3. 

Thank you once again.

Quinn Hechtkopf
Teacher at Q752, NYC
Associated Topics:
High School Linear Equations

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