Equation of all points in space

Question

Does $0=0$ (or any equation with the same number on both sides) represent all possible points in 3D or 2D space, as per the context? It seems so, even if you can't really 'input' numbers to test if they satisfy the condition, because the condition is true no matter what point you take (you can even write it out as $0x+0y+0z=0$ if you need to try points out to make it rigorous).

I ask because I can't find this mentioned anywhere as a way to represent $\mathbb{R^3}$ or $\mathbb{R^2}$ ; there seems to be nothing online and neither WolframAlpha nor Geogebra graphs the whole plane for '$0=0$' or anything similar.

Answer 1

Echoing User's answer that $0=0$ corresponds to the entire Euclidean space, as per the context:

E.g., in $2D$ space, the equation $y=2x+3$ is true on precisely the line representing the equation. On the other hand, equation $7=7,$ being a (logical) validity, is true throughout the space.

Answer 2

Yes, as per context, the equation $0=0$ is satisfied by any couple or triple of coordinates.

As you noticed, to make the context clear we could represent any point in the plane for example by the equivalent $x+y=x+y$ and $x+y+z=x+y+z$ in the space. Note that these forms are indeed recognized by wolfram as solutions for all the couples/triples.