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I have an equation $x=x$.

  1. Is that a valid algebraic equation?
  2. Can $x=x$ be simplified to $1=1$? Can it be simplified down to True?
  3. Can $x=x$, $1=1$, or True be graphed on a regular $(x,y)$ plot graph? Could it be graphed at all?
  4. If it is graph-able what would it look like?
Xander Henderson
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savvamadar
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  • How do you graph the algebraic equation $x^2=1$? Do you plot the solutions $\pm1$ on a number line? In that case all real numbers satisfy $x=x$, so the graph would be the entire number line... – Shubham Johri Oct 22 '20 at 21:12
  • I would plot x^2=1 as a straight vertical line where x=1. In my head I see a cartesian coordinate graph where the horizontal axis is x and the vertical axis is y. – savvamadar Oct 22 '20 at 21:14
  • There is no second variable $y$ so why would you plot it in $\Bbb R^2$ or $xy$ plane? – Shubham Johri Oct 22 '20 at 21:16
  • This is a good question. I think that realistically there is no reason to plot it at all. But shouldn't it not matter where I try to plot it so long as x is a variable of the graph? – savvamadar Oct 22 '20 at 21:17
  • You can "plot" it in $\Bbb R^n$ alright. In which case since all $x\in\Bbb R$ satisfy $x=x$, the "graph" would be the entire plane. – Shubham Johri Oct 22 '20 at 21:19
  • Yes, that answers all my questions! Thanks! If you would like to leave an answer I would gladly mark you correct. – savvamadar Oct 22 '20 at 21:20
  • Ok. Wait a sec... – Shubham Johri Oct 22 '20 at 21:21

2 Answers2

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  1. Any kind of $f(x)=g(x)$ formed things are an equation.
  2. It can be simplified into $1=1$ by subtracting $x-1$ to each side.
  3. Probably, it might cover the whole graph.
  4. Any kind of graph, either $x$ plot or $x,y$ plot or $x,y,z$ plot must be filled and will have no space left.
user
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  1. Yes, it is a valid algebraic equation.
  2. Yes, when $x\ne0,x=x$ can be simplified to $1=1$ by division. $x=x$ is always True but True is not an algebraic equation so it can't be "simplified" to True.
  3. Since all real numbers satisfy $x=x$, the graph would be the entire number line in $\Bbb R$ and the entire plane in $\Bbb R^2$ if you choose to introduce another variable $y~(x+0y=x)$. Note that $y$ takes any value for any $x$ and is not dependent on $x$ as expected.

Contrast this to the equation $x+y=1$ where $y$ only takes specific values for given $x$.

Shubham Johri
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    To add: rather than saying that the equation can be simplified to True, a better way to express the solutions to $x=x$ is to say that $x \in \mathbb{R}$. Of course, this assumes that $x$ is a real number. If $x$ is a complex number, for instance, then the solutions to $x=x$ are $x \in \mathbb{C}$. This highlights the importance of stating which system of numbers you are working in. Usually this is clear from context, but sometimes it is not. – Joe Oct 22 '20 at 21:42