The general equation of a straight line is given as :
$$Ax + By + C = 0$$
In a book I was reading it said that " note that $x$ and $y$ are not the unknowns. In fact these are the coordinates of any point on the line and are known as current coordinates. Thus to determine a line we need two conditions to determine the two unknowns $A$ and $B$." My question is why are $A$ and $B$ considered unknowns and not $x$ and $y.$ Could anyone make this clear?
On
X and Y are the solutions of $Ax+By+c$ . They depend on A and B, and a set of different values of A and B results in a different line. So, X and Y are not considered as unknowns
On
A, B, and C are considered constants while X and Y are considered variables. X and Y take on an infinite number of values along the line $Ax+By+C=0$. Now, there are special cases in which A or B equal zero in which X or Y only takes on 1 value. However, A and B cannot both be zero in order for $Ax+By+C=0$ to still represent the equation of a line.
A straight line is the collection of all points $(x,y)$ which satisfy $Ax+By+C = 0$. Therefore $x$ and $y$ are intended to be arbitrary rather than unknown: once we know everything there is to know about the line, we still cannot even in principle know $x$ or $y$, because the line is a collection of different $x$ and $y$. It makes as much sense to ask what $x$ is as it does to ask what $n$ is in the statement $\mathbb{N} = \{n : n \in \mathbb{R}^{>0}, n = \lfloor n \rfloor\}$.
On the other hand, we can't know which line is intended without knowing $A, B$ and $C$, so they are "unknown".