Why do the equations $Ax + By + Cz= D$ represent planes in $\Bbb R^3$ and not lines? I was wondering why $$Ax+By+Cz=D$$ represents a plane in $\mathbb{R}^3$, because from my perspective it seems that it could simply also represent a line in  $\mathbb{R}^3$. And, yes, I do acknowledge that lines aren't three-dimensional, but surely we can have lines in three-dimensional space.
If you know about dot product, the above equation can be written as (assume $C\neq 0$)

$$ \langle A, B, C\rangle \cdot \langle x, y, z - \frac{D}{C}\rangle = 0$$

Thus $(x, y, z)$ satisifies $Ax+By+Cz = D$ if and only if the vector from $(0,0,\frac{D}{C})$ to $(x, y, z)$ is perpencidular to $\langle A, B, C\rangle$. The set of all such $(x, y, z)$ will be a plane passing through $(0, 0, \frac DC)$ with normal $\langle A, B, C\rangle$.
If we fix a value of x and y, we get only one value of z. i.e x and y can take any real value but then z will be fixed.This is the algebraic way of stating a plane.