A problem in algebric geometry

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Homework Help Overview

The original poster is attempting to solve a problem in algebraic geometry involving three unknown variables, x, y, and z. The problem is framed around a set of equations that relate these variables to a constant A, which is defined in terms of x, y, and z. The equations presented include both linear and quadratic forms, and the poster seeks guidance on effective methods to determine the values of the variables.

Discussion Character

  • Exploratory, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • Some participants suggest treating A as a variable and using linear equations to eliminate variables from the set {x, y, z, A}. Others express difficulty in this approach, noting challenges in eliminating variables and often reverting to equations involving one variable in terms of the others. There are also mentions of matrix methods and the potential redundancy of the linear equations.

Discussion Status

The discussion is ongoing, with participants providing hints and suggestions for methods to approach the problem. There is an acknowledgment of the challenges faced in variable elimination and the exploration of different strategies, including the use of matrix methods. No consensus has been reached, and various interpretations of the problem are still being explored.

Contextual Notes

Participants are navigating the complexity of the equations, with some noting that the linear equations may not be independent, leading to potential simplifications. The original poster is also seeking additional resources or examples to aid in their understanding of similar problems.

mathman99
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Homework Statement



Hello,
I am trying to solve a problem in algebraic geometry where three unknown variables x,y and z needs to be determined to further progress in this problem.
The equations involving the variables x,y and z are 10y+28z-20=A;
18z-30x+60=A;-84x-18y+204=A.
Where A=3(x^2+y^2+z^2)=3(5x+5y+11z-20)
also 3x+y+z=8.


Could you suggest an effective method to find the possible values of x,y and z.

If you have any ideas, anything will be appreciated. Thanks

Homework Equations



3x+y+z=8.

The Attempt at a Solution

 
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Treat A as another variable. Use the linear equations to eliminate as many variables as you can from the set {x,y,z,A}. Substitute the results into the quadratic.
 
Dick said:
Treat A as another variable. Use the linear equations to eliminate as many variables as you can from the set {x,y,z,A}. Substitute the results into the quadratic.

Thanks for the hint Dick. Could you give me some elaboration on this method. Its difficult to eliminate any variable from the set {x,y,z,A}. I always get into equation of one variable in terms of other two.
Or it will be helpful to give me some e-links where I can find similar problems (Topics) discussed.
 
I don't see what the problem is. Start with 3x+y+z=8, so z=8-3x-y. Put that into all the other linear equations. Boom. z eliminated and one less equation, at least. Your linear equations aren't independent, so you'll find some of them turning into the same thing. There's also matrix type methods.
http://en.wikipedia.org/wiki/System_of_linear_equations
 
As I tried to solve by using
10y+28z-20=18z-30x+60
18z-30x+60=-84x-18y+204
10y+28z-20=-84x-18y+204

All of the equation turn into 3x+y+z=8 since

30x+10y+10z=80 /: 10
54x+18y+18z=144 /: 18
84x+28y+28z=224 /: 28

or the same equation 3x+y+z=8

Now the only remaining equations are
3(x^2+y^2+z^2)=3(5x+5y+11z-20)

and 3x+y+z=8
 

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