Which is the right notation to write the solutions of a single equation with two variables?

Question

I've recently encounter the problem of solving these two equations in $\mathbb{N}$ and $\mathbb{Z}$ for answer a question in SE.

$$ax-by=0$$ and$$ax+by=0$$ (Please note: are two single equations not a system) I would say that the solutions are an infinite set of $(x,y)$ tuples like this for the first: $$A=\{(b,a),(2b,2a),(3b,3a)...(nb,na)\}$$ I don't know how to formalize it with standard notation... For the second would be much harder to write the solutions in this way. Which is the standard notation for write the solutions of these equations?

Answer 1

The way that you've written your set $A$, it would be interpreted as a finite set -- containing exactly $n$ pairs. Instead, you can write $$ A=\{(b,a),(2b,2a),(3b,3a),\ldots\}. $$ Even a bit more formally: $$ A=\{(kb,ka)\mid k\in\mathbb{Z}\} $$ ("the set of all pairs $(kb,ka)$ where $k$ is an integer").

(Please note, here, that I've only addressed your notation -- not your solution.)

Answer 2

If $\gcd(a,b)=d$ and $a=da',b=db'$ then the full solution of $ax-by=0$ is the set of pairs $(b't,a't)$ where $t \in \mathbb{Z}$, and if you want positive solutions $t$ must be appropriately restricted. Essentially the same idea works for $ax+by=0$ only you use pairs $(b't,-a't).$