Are we not always working in an infinite number of dimensions? My question basically boils down to the fact that when dealing with 2 dimensions, $x$ and $y$, are we not technically also dealing with every other dimension too, just leaving them blank as there are infinite solutions, like with points on a polar plane, we might leave something in the form of $(5,\frac{3\pi}{2}+2\pi)$ if we wanted to represent the infinite solutions.
So, are we not basically dealing with a situation where if we are looking at a point, it's really an infinitely dimensional plane where we are only focusing on a specific set of planes. For example, if we were to be dealing with a 3 dimensional point, like $(3,4,5)$ are we not really dealing with something in the form of $(3,4,5,w,...)$ where "$...$" represents all other dimensions, and we're only narrowing down on the specific location of the infinite planes on 3 axis?
And, if that is true, then when we define a "line" on the $(x,y)$ axis, if we were to define it as something in the form of $y=x+2$, how are we not just creating a new plane or dimension to represent that. How is it that it is known that it is related to the initial $y$ plane.
I hope I'm making sense but this was kind of hard to explain my thought process.
Thanks for any clarification!
 A: To put it concisely, no, not really.
I think the source of your confusion is the convention to define a subset of $\Bbb{R}^2$ or $\Bbb{R}^3$ (i.e. $2$ or $3$ dimensional space) in terms of an equation in terms of $x$, $y$, and $z$. For example, the line in $\Bbb{R}^2$, going through the origin, of slope $-1/2$ is sometimes presented as an equation
$$x + 2y = 0.$$
However, this is something of an abuse of notation. An equation isn't the same thing as the set of points it represents; it's just a statement, depending on unknown variables $x$ and $y$, that is either true or false (depending on the value of $x$ and $y$).
That said, we often use an equation as a kind of shorthand to refer to the set of points (in a space understood by context) which satisfy this equation. If we are talking about $\Bbb{R}^2$, i.e. the Cartesian plane, then the above equation is understood to reference the set:
$$\{(x, y) \in \Bbb{R}^2 : x + 2y = 0\}.$$
However, if we mix up the context a bit, and we were talking about $\Bbb{R}^3$, the same equation could refer to the (vertical) plane:
$$\{(x, y, z) \in \Bbb{R}^3 : x + 2y = 0\}.$$
This is a different set, in a different space, but both are referred to by the same equation. If it is not clear which space we are talking about, and hence which of the above sets is being referred to, then simply writing the equation is ambiguous. The author should make it crystal clear which space they're working in.
Also note the assumption that $x$ refers to the first coordinate of a point, and $y$ refers to the second. This need not necessarily be the case, but it is assumed by convention (which is a kind of context). Without this convention, our equation could just as easily be referring to
$$\{(y, x) \in \Bbb{R}^2 : x + 2y = 0\},$$
or in other words,
$$\{(x, y) \in \Bbb{R}^2 : 2x + y = 0\},$$
which is a different line altogether!
So, no, we are not always working in infinite dimensions. You will typically be working in finite dimensions (probably $\Bbb{R}^2$ or $\Bbb{R}^3$ if you're looking at equations involving $x$, $y$, and/or $z$). Equations should be understood to refer to sets in a space that is clear from context, and if the space is unclear, then so is the equation.
