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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
http://mathforum.org/dr.math/
```
Associated Topics:
High School Linear Equations

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