Let's say we have an equation $x+y=0$, Now the solutions to this equation can be $(x=-1,y=1)$, $(x=2,y=-2)$.
Now are the solutions to the equation vectors?
Because when we represent them in coordinate axes we do it like there is a difference between $x=1$ and $y=1$, they are considered with a specific direction which I do not think is actually the case, is the graph just a relation between $x$ and $y$, not actual dimensions? is that the case?
Because when I was reading Elementary Linear Algebra by Howard Anton, the book claims that we are actually plotting in the $x-y$ coordinate system by which I interpret he is meaning the $x-y$ dimensions.
Also, it had said that if the solution is in $2\text{D}$ or $3\text{D}$ space? Aren't $x$ and $y$ just real numbers and not dimensions of space. Could somebody clear this up for me?
I think it's helpful to think about the equation $x+y = 0$ as the equation for a line in two-dimensional space (which line?). If you then have a system of two linear equations, you would be finding the intersection between two lines.
With this interpretation, you can also interpret visually what happens when a system of equations has no solutions, exactly one solution, or infinitely many solutions. Indeed, two lines in the plane are either parallel or not. If they are not parallel, they have a unique intersection point, and if they are not, then they are either on top of each other, or they don't intersect at all.
You can also think of a point in the plane as a vector from the origin to that point, that way you can also think about vectors.
NB. If there are three variables, then an equation like $x + y = 0$ would be the equation for a plane. Two planes generally intersect in a line. This is what happens when you have one free parameter in the solution to the system of linear equations.