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Standard Form of a Line (Ax + By = C)

Date: 10/07/2005 at 14:38:29
From: Kelly
Subject: Ax + By = C form of a line

In the standard form of a linear equation, Ax + By = C, what 
conditions do A, B, and C have to meet?  Do you always have to have a 
positive integer for A?  Do you have to have all the common factors of 
A, B, and C factored out?

We know that A, B, and C should be integers according to our textbook 
definition.  The textbook also gives all answers beginning with a 
positive A.  Is this a standard way to write it, or just a preference 
of our book editors?  

We had a problem that worked out to .6x - y = 2.4.  One student
multiplied by 10 to get 6x - 10y = 24.  Another used the fraction for
.6 (3/5) and multiplied by 5 to get 3x - 5y = 12.  I know they're
equivalent equations, but are they both considered correct?

Why doesn't our book say that you must have factored out the terms so
you have the lowest values of A, B, and C?  I hate to think I have to
consider the infinite possibilities each time I check a paper.   

Date: 10/07/2005 at 16:17:16
From: Doctor Rick
Subject: Re: Ax + By = C form of a line

Hi, Kelly.
If your textbook says that A, B, and C must be integers, it is wrong!  
If that were so, the standard form Ax + By = C could not represent any 
line with irrational slope, or with an x- or y-intercept 
incommensurate with the slope.  How would you put this equation into 
that form?

  y = pi*x - sqrt(2)

The advantage of the standard form is that, unlike slope-intercept 
form for example, it can be used to represent *any* line.  I wouldn't 
want to limit it by requiring A, B, and C to be integers!

What I've said so far is math.  The rest is pedagogy.  Your last 
sentence says it all: "I hate to think I have to consider the 
infinite possibilities each time I check a paper."  The convenience 
of the teacher (or perhaps another reader of a paper) is the only 
reason to put restrictions on A, B, and C. 

In practice most, if not all, of the problems presented to the 
students will involve rational numbers, so it will in fact be possible 
to make A, B, and C integers.  I think it is quite reasonable to 
require this, and in addition that A be positive and that A, B, and C 
be relatively prime.  This reduces the infinite possibilities to 
exactly one, and indeed makes your work easier.  I'm all for that!

I only ask that you make clear to the students that this is to help 
you, and not because math requires it.  Maybe you could even drive 
this home to them by removing the restrictions for one assignment-- 
and having them grade one another's papers!

- Doctor Rick, The Math Forum 
Associated Topics:
High School Linear Equations

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