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

In summary, it is possible to associate an equation in several variables with a graph, and in this case, 5 linear equations in 5 variables represent graphs in 5 dimensions.
  • #1
Liger20
65
0
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).
 
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  • #2
Liger20 said:
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.
 
  • #3



Hi there! Your question is a very interesting one and it shows that you have a strong understanding of solving equations with multiple variables. To answer your question, yes, equations in more than three variables do represent graphs in higher dimensions. In fact, equations in n variables represent graphs in n-dimensional space. This means that your five equations with five variables represent a graph in five-dimensional space.

The reason for this is because each variable represents a different axis or direction in the graph. For example, in a two-variable equation, one variable represents the x-axis and the other represents the y-axis. In a three-variable equation, one variable represents the x-axis, one represents the y-axis, and the third represents the z-axis. Similarly, in a five-variable equation, each variable represents a different axis in a five-dimensional graph.

Your equations are mathematically valid and you have solved them correctly using elimination. It's great that you experimented with more variables and saw that the same techniques can be used to solve them. This shows that the methods for solving equations are applicable in higher dimensions as well.

I hope this helps answer your question. Keep exploring and experimenting with different equations and variables - it's a great way to deepen your understanding of mathematics. Keep up the good work!
 

1. How are equations in more than three variables graphed in higher dimensions?

Equations in more than three variables are graphed in higher dimensions using a coordinate system with more than three axes. Each axis represents a different variable, and the point where the axes intersect represents the solution to the equation.

2. Can equations in more than three variables be represented graphically?

Yes, equations in more than three variables can be represented graphically in higher dimensions. However, it may be difficult to visualize and interpret the graph due to the increased complexity.

3. Are there any specific techniques for graphing equations in more than three variables?

Yes, there are several techniques for graphing equations in more than three variables, such as using computer software or plotting points to create a 3D or 4D graph. Another technique is to use contour plots, where the equation is graphed as a series of curves on a 2D plane.

4. How do equations in more than three variables differ from those in three variables?

Equations in more than three variables differ from those in three variables in that they have more variables and therefore can have more complex shapes and solutions. Three variable equations can be graphed in 3D, but equations with more variables require higher dimensions to be graphed.

5. What is the significance of studying equations in more than three variables?

Studying equations in more than three variables allows us to solve more complex problems and understand relationships between multiple variables. It also has practical applications in fields such as physics, economics, and engineering, where equations with more than three variables are commonly used to model real-world systems.

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