Why does the equation $x = 1$ represent a line in a 2-dimensional coordinate system?

Question

I'm posting a question because I was curious about something while studying linear algebra.

As we all know, $x = 1$ is a point in a one-dimensional coordinate system. I understand this part.

But why does $x = 1$ represent a line in two dimensions?

Is it simply defined that way?

If anyone knows anything about this, please help.

It may seem trivial, but I feel really uncomfortable and think about it every day.

Thank you.

Answer 1

Your problem probably comes from the fact that you don't really know what a two-dimensional coordinate system is and how a line in a two-dimensional coordinate system is mathematically defined.

It is linear algebra closely associated with elementary geometry.

First we define $$E=\mathbb R \times \mathbb R=\{(x,y): x\in \mathbb\ R,y\in \mathbb R \}$$ We multiply an element $\lambda$ of $\mathbb R$ by an element $(x,y)$ of E like this $$\lambda(x,y):=(\lambda x,\lambda y)$$ For example $$5(0,1)=(5\times 0,5\times 1)=(0,5)$$ We're adding two elements of $E$ like this $$(\color{green}1,0)+(\color{green}0,5):=(\color{green}{1+0},0+5)$$

You can think of an element of E as a point or a vector whichever makes the most sense to you.

Then, we define à line in $E$ by a part of $E$ of the form $$l=a+\mathbb R u$$ $\mathbb R u$ is called the direction of $l$.

Here, with $a=(1,0)$ and $u=(0,1)$, you obtain $$l=(1,0)+\mathbb R (0,1)=\{(x,y)\in \mathbb R \times \mathbb R: \exists \lambda \in \mathbb R, (x,y)=(1,0)+\lambda (0,1)\}$$ $$=\color{cyan}{\{(1,\lambda): \lambda \in \mathbb R\}}$$

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Note that you asked yourself a good question to study linear algebra further.

Answer 2

The graph is all the points $(x,y)$ such that $x=1$. There is no constraint on $y$, so we plot all the points that have $1$ as the first coordinate. This results in the graph you show.

Answer 3

As we all know, $x = 1$ is a point in a one-dimensional coordinate system. I understand this part.

Is it though? What if I write $\{a\in\mathbb{R}\ |\ x=1\}$. Is that a point? No, in fact it doesn't even make much sense.

What I'm trying to say, is that the equation "$x=1$" requires a context, on its own it is meaningless. So

$$\{(x,y)\in\mathbb{R}^2\ |\ x=1\}$$

indeed is a line, because $x$ is fixed, and only $y$ varies. So for example $(1,0)$ belongs to it, so does $(1,5)$ and $(1,-\pi)$, but not $(0,0)$.

However in say 3d space

$$\{(x,y,z)\in\mathbb{R}^3\ |\ x=1\}$$

it won't be a line, it will be a plane. Since now $x$ is fixed and $y,z$ varies.

Context. It changes everything.