One way to describe a set of points in the plane is by an equation or inequality in two variables, say $x$ and $y$. A solution of an equation in $x$ and $y$ is point $(x_0, y_0)$ in the plane for which the equation is true.
My questions: How do we describe a set points in the plane by an equation? Is there only single point which satisfies the equation? How do we know that the equation is true at a certain point? A circle is the locus of an equation: $(x - h)^2 + (y - k)^2 = r^2$ is this an equation in $x$ and $y$, I mean this seems to be an equation in change in x squared + change in y squared and $r^2$?