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Standard Form Equation of a Line

Date: 01/05/2007 at 05:52:18
From: Pao
Subject: why is it that..

Why is it that in an equation of a line, the x can never be a 
negative?  For example, -2x - y + 3 = 0 must always be 2x + y - 3 = 0.
Why do we need to make the x term positive?  Thanks in advance...



Date: 01/05/2007 at 22:46:20
From: Doctor Peterson
Subject: Re: why is it that..

Hi, Pao.

It isn't really true: -2x - y + 3 = 0 is a perfectly good equation of
a line.  You don't HAVE to make the coefficient of x positive.

But it makes some sense to define the STANDARD FORM of a line with
that requirement; a standard form is sort of like a police mug shot,
where they make everyone stand facing the same way so all the pictures
can be easily compared.  If nothing else, it makes it really easy for
you or a teacher to check your answer by comparing it to the one
correct answer.  It doesn't really matter whether they face right or
left, or whether we write the equation as Ax + By + C = 0 with A
positive, or as C + By + Ax = 0 with C required to be positive.  We
just choose one form as the "standard" (which, by the way, differs a
bit from book to book) and stick with that.

In the same way, some texts require that, IF the coefficients are all
rational, you should always multiply by the LCD so that the
coefficients are all integers and have no common factor.  That is
equivalent to asking students to write all fractions in lowest terms-- 
other forms have the same value, but this is easy to work with and 
easy to compare.

In both cases, the ultimate reason is that you can multiply an 
equation by any constant, and the result will be an equivalent
equation.  In order to ensure that there is only one "correct" form,
we make a restriction that can always be met by doing such a
multiplication.  (By the way, if any coefficients are irrational, or
if the coefficient of x is zero, these rules CAN'T be followed, so you
need some extra rules to cover those cases!)

See this page for more thoughts:

  Standard Form of a Line (Ax + By = C)
    http://mathforum.org/library/drmath/view/68284.html 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Linear Equations

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