Do Equations in More Than Three Variables Represent Graphs in Higher Dimensions?

[Liger20]

Hello, I posted this several weeks ago in another forum, but I never really got a good answer. Could someone please take a look at this an tell me if it's mathematically valid?
Thanks!





Do Equations in More Than Three Variables Represent Graphs in Higher Dimensions?

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Hey, first of all, I'd like to apologize if I'm posting this in the wrong forum. I wasn't sure whether I should post it here or in the mathematics forum. Recently I was going through an Algebra book, and I saw a chapter on solving linear equations in three variables. The book explained how these sets of equations can be solved using elimination, matrices, Cramer's rule, etc. Anyway, I worked several of these problems. When I was finished, just out of curiosity, I wondered what it would be like if I set up five sets of equations with five variables, and solved them. I created one, and using my preferred technique, elimination, I went about solving it and it went MUCH more smoothly than I expected it would. When I was finished, I had the values that I had started out with, and it had all worked out fine. I used five variables X, Y, Z, K, and J. When I had finished, I went back to the book I was using and saw that linear sentences in two variables represented graphs in two dimensions, and linear equations in three variables represent graphs in three dimensions. So here's my question: Do sets of equations with more than three variables represent graphs in higher dimensions?


P.S: Here are the equations I worked. Feel free to point out anything that I may have done wrong.


2x+4y-1z-6k+2j=-1
3x+6y-3z-8k+4j=1
1x+3y+4z-2k+4j=47
4x-2y-2z+6k+2j=28
5x+3y+4z+3k+3j=69

(The values are X=2, Y=3, Z=5, K=4, J=6).

 

[HallsofIvy]

Hello, I posted this several weeks ago in another forum, but I never really got a good answer. Could someone please take a look at this an tell me if it's mathematically valid?
Thanks!





Do Equations in More Than Three Variables Represent Graphs in Higher Dimensions?

--------------------------------------------------------------------------------

Hey, first of all, I'd like to apologize if I'm posting this in the wrong forum. I wasn't sure whether I should post it here or in the mathematics forum. Recently I was going through an Algebra book, and I saw a chapter on solving linear equations in three variables. The book explained how these sets of equations can be solved using elimination, matrices, Cramer's rule, etc. Anyway, I worked several of these problems. When I was finished, just out of curiosity, I wondered what it would be like if I set up five sets of equations with five variables, and solved them. I created one, and using my preferred technique, elimination, I went about solving it and it went MUCH more smoothly than I expected it would. When I was finished, I had the values that I had started out with, and it had all worked out fine. I used five variables X, Y, Z, K, and J. When I had finished, I went back to the book I was using and saw that linear sentences in two variables represented graphs in two dimensions, and linear equations in three variables represent graphs in three dimensions. So here's my question: Do sets of equations with more than three variables represent graphs in higher dimensions?


P.S: Here are the equations I worked. Feel free to point out anything that I may have done wrong.


2x+4y-1z-6k+2j=-1
3x+6y-3z-8k+4j=1
1x+3y+4z-2k+4j=47
4x-2y-2z+6k+2j=28
5x+3y+4z+3k+3j=69

(The values are X=2, Y=3, Z=5, K=4, J=6).

Actually, I would have put it the other way around: a graph represents an equation. Equations don't necessarily "represent" graphs or anything else. An equation is itself and, if it was derived from some application, represents whatever the application is about.

However, it certainly is possible to associate an equation in several variables with a graph. In this case, you have 5 linear equations in 5 variables. It would be possible to solve each of those equations for anyone of the variables "in terms of" the other 4. Given anyone of those 4 values, the 5th is determined. Yes, we could set up a "5 dimensional" coordinate system and each equation would "represent" (or be represented by) a "hyper-plane" in 5 dimensions.