How to use matrix method on this system of equations

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Homework Help Overview

The original poster presents a system of equations involving variables x, y, and z, specifically 2x + 2yz = 0, 2y + 2xz = 0, and 2z + 2xy = 0. They express a desire to use matrix methods to solve this system, although they acknowledge that the critical points can be identified more easily through inspection.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Some participants question the applicability of the matrix method, suggesting that the system may not be linear. There are discussions about using algebraic methods, including substitution, to find solutions. The original poster expresses concern about the complexity of substitution compared to inspection.

Discussion Status

Participants are exploring different methods to approach the problem, with some suggesting algebraic techniques and others emphasizing the challenges of systematic substitution. There is no explicit consensus on the best method, but various lines of reasoning are being examined.

Contextual Notes

Participants note that the terms "critical points" and "solutions" may be used interchangeably, indicating a potential misunderstanding of terminology. The original poster has provided a list of solutions, which may influence the discussion on how to approach the problem.

Typhon4ever
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Say that I have 2x+2yz=0 2y+2xz=0 and 2z+2xy=0 how would I use matrix methods to solve this system of equations? I know you can just look at it and easily figure out what the critical points are but I want to do it the safe way. Or is using the matrix method not the easiest way here?

For extra information the crit points are (0,0,0) (-1,1,1) (1,-1,1) (1,1,-1) (-1,-1,-1)

Thanks!
 
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There is no 'matrix method' to use. It's not a system of linear equations. And they aren't really called 'critical points'. They are called 'solutions'. Just use algebra and logic. Which I think you are already doing.
 
Dick said:
There is no 'matrix method' to use. It's not a system of linear equations. And they aren't really called 'critical points'. They are called 'solutions'. Just use algebra and logic. Which I think you are already doing.

What's the "algebraic" way of doing this? Multiple rounds of substitution?
 
Typhon4ever said:
What's the "algebraic" way of doing this? Multiple rounds of substitution?

Yes, basically. If e.g. x+yz=0 then either y=(-x/z) or z=0, subsitute that into the other equations. Explain how you got your solutions.
 
Last edited:
Dick said:
Yes, basically. If e.g. x+yz=0 then either y=(-x/z) or z=0, subsitute that into the other equations. Explain how you got your solutions.

I just looked at this and thought of good numbers that would satisfy the equations. That's why I wanted a more systematic way of doing it. The substitution looks like a real headache to do.
 
Typhon4ever said:
I just looked at this and thought of good numbers that would satisfy the equations. That's why I wanted a more systematic way of doing it. The substitution looks like a real headache to do.

OK, you just solved by inspection. Doing it systematically isn't that much of headache. Just do it. If I substitute y=(-x/z) into the second relation I get x(1-z^2)=0 after a little algebra. What does that tell you?
 
Well I get that x=0 and z=+-1

If I plug into the third one I get x=+-z and z=+-x. Where should I start plugging things in next?
 
Last edited:
Dick said:
OK, you just solved by inspection. Doing it systematically isn't that much of headache. Just do it. If I substitute y=(-x/z) into the second relation I get x(1-z^2)=0 after a little algebra. What does that tell you?

Well I get that x=0 and z=+-1

If I plug into the third one I get x=+-z and z=+-x. Where should I start plugging things in next?
 
Typhon4ever said:
Well I get that x=0 and z=+-1

That's actually an or, not an and that you want
 
  • #10
Office_Shredder said:
That's actually an or, not an and that you want

Oh ok. Thank you. So now that I know x=0 or z=+-1 what should my next step be? Should I plug 1 and -1 into the first equation?
 
  • #11
Typhon4ever said:
Oh ok. Thank you. So now that I know x=0 or z=+-1 what should my next step be? Should I plug 1 and -1 into the first equation?

You know you are going to get 5 different solutions. Take those 'or' cases separately. In solving the first equation you had the z=0 OR x=(-y/z). First supposing z=0 what can you conclude about x and y? Then after solving the second equation consider the cases x=0, z=1 and z=(-1) separately. Substitute anywhere you like.
 

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