Why is the graph of a constant, a line?

Question

Why is the graph for x=3 (Or C where C is a constant) as a vertical line on the xy coordinate plane or Y = some C as a horizontal line? Shouldn’t they be points?

Answer 1

This comes to the interpretation of Cartesian equations. The graph $(C)$ of equation $f(x, y)=0$ is by definition the set $$ \{ (x,y)\in\mathbb{R}^2| f(x,y)=0\} $$

In particular, the graph of $x=c$ is the set $\{(x, y)|x=c\}=\{(c, y)|y\in\mathbb{R}\}$ , The case of $y=c$ is analogous.

On a more intuitive level, a point in 2D space (the plane) has 2 degrees of freedom (corresponding to its two coordinates), fixing one of them only takes one degree of freedom which means that the resulting space should be 1-dimensional rather than 0-dimensional.