What should the graph of $x=n$ or $y=n$ for any constant real $n$ look like?

Question

Online calculators draw the graphs either as a vertical line in cases where there is a univariate linear equation in the variable $x$ and a horizontal line when the variable is $y$.

I don't understand how this is true, after all any univariate linear equation only provides data for a single coordinate ($x$ or $y$, but not both.)

Shouldn't the graph be a single point on either $x$ or $y$ axis? Is it a convention in maths to assume the coordinate for which no data is supplied by the equation may take any real value hence the vertical or horizontal line graphs?

Answer 1

Effectively, yes. There's essentially two criteria implied by an equation. You are looking at the full "universe" of the system you are working in that satisfies the criterion set up by the equation. The combination of the criteria implies more than just the single point.

For example. $y=x$ means all ordered pairs satisfying the criterion, i.e. $E_1=\{(x,y) \in R \times R|y=x\}$

In two dimensions $x^2+y^2=1$ means $E_2=\{ (x,y) \in R \times R| x^2+y^2=1\}$, a circle. A circle in the xy plane.

In three dimensions, the same equation means $E_3=\{(x,y,z)\in R \times R \times R| x^2+y^2=1\}$. Since the criterion only specifies a relationship between $x$ and $y$, then $z$ can be anything to satisfy it. So this is a cylinder and $E_2 \subset E_3$.

Answer 2

$x=n$ is equivalent to $x=n.(y^0)$. The equation $x=n$ defines a line passing through all the points of the form $(n,y)$. It does give information about the other co-ordinate $y$ that it is not bounded (above & below) or simply $sup(y) = \infty,inf(y) = -\infty$.