How exactly do you read the logic of an equation such as this.. I'm a computer science-type, and trying to get a deeper grip of "math-speak."
The use of the equal sign is not clear to me in terms of when it's being used as "assignment" vs. "equality."
Where $y=x$, it's intuitive that as $x$ varies over $\mathbb{R} $ (although not sure how "varies" is to be understood — traverse or crawls time, or space or what) it maps its value to $y$, and then you have pairs of $(x,y)$ points that form a continuous line through the origin of the Cartesian plane.  This seems like a function to me, not an equation — because $y$ just seems synonymous with $f(x)$ — it's simply a "conversion" of x to a "new x"."
But with:
$$x^2 + y^2 = r^2$$
Equation of a circle at the origin—clearly a different beast. Intuition on how to read this is gone here. Now the "dependent variable is on the same side as (what was) the independent" and the sense of assignment or "function" is entirely lost, and $r$ is called a constant, but in reality acts as some kind of user-defined variable (not in the sense of "varies over," but as something you can manually change or "slide" yourself technically over $\mathbb{R} $).
So how do you intuitively describe this with the same intuitive clarity as "the line" above?  Are there two independent variables, and the "$r$" constant is acting as a "dependent" on the right side?  My intuition wants to write it as:
$$y  = \sqrt{r^2-x^2}$$
To keep the sense of "assignment" from "one side" to the other intact.  But this doesn't work — it graphs as a half-circle and only when $r$ is parameterized.
Any help is appreciated.  Thank you!
 A: You are correct to be confused about this; it's legitimately confusing!
In math-speak using the equation "$x^2 + y^2 = r^2$" is shorthand. Formally one should be using set-builder notation to describe the set of points on the circle as
$$C_r = \{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 = r^2 \}$$
where $r$ is fixed. Neither $x$ nor $y$ is "dependent" or "independent" here and we have not introduced any kind of "time" variable (although we could if we wanted to).
If you like, you can think of the equation $x^2 + y^2 = r^2$ as defining a Boolean function called isAPointOnTheCircle(x, y) which returns true if $x^2 + y^2 = r^2$ (that is, if $(x, y)$ are the coordinates of a point on the circle of radius $r$ centered at the origin) and false otherwise. The set-builder notation picks out the subset of $\mathbb{R}^2$ consisting of the points on the circle, which is to say, the points satisfying this condition.
Here I'm thinking of $r$ as being something like a global variable we defined earlier but of course we can also consider a Boolean function isAPointOnTheCircle(x, y, r) taking three inputs. It depends on whether we're planning on saying something about different values of $r$ or not.
