I am teaching a class on “intermediate algebra” and I noticed that there are two definitions for the standard form of a linear equation in two variables $x$ and $y$ in the textbook that I am using.
$1.$ A linear equation in two variables (here $x$ and $y$) can be written in the form $$Ax + By =C$$ where $A$, $B$ and $C$ are real numbers and $A$ and $B$ are not both $0$.
$2.$ $Ax+By = C$ where $A, B$ and $C$ are integers, $A \geq 0$.
The first definition makes sense to me in that it is merely a non-trivial real-linear combination of $x$ and $y$ to create $C$ but the second definition seems to be awfully specific. It requires an integer-linear combination to form an integer $C$ and it also requires that $A$ be non-negative.
Now, perhaps insisting that $A, B$ and $C$ be integers follows from the fact that the book won’t be handling any real numbers that aren’t rational but I’m not at all sure why there is an insistence on $A$ being non-negative. Is there some form of advantage to the requirement that $A \geq 0$ or is it perhaps just to make sure that a student is repeating the exact same process over and over again without needing to think it over or some other reason entirely?
Note: I know essentially nothing about math education and so this question may be very silly.
If I should add or remove a tag then feel free to let me know.
Thanks in advance!
I can think of two reasons. One is that by requiring $A$ to be positive, slightly less ink is used in the equation, or at least on the left side, and so this might be considered "simpler."
The other is that often students are asked to put answers in a certain form just to make it easier for the teacher to grade. (Or nowadays, with the online homework servers, so that the computer doesn't mark a correct answer wrong.)