# Solve a Set of Equations with Constraints and Interchangeable Varaibles

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MarshallSc el 10 de Jun. de 2021
Hello, is there any way to solve the equations below using interchangeability algorithm (function of X in 4 dimensions-i,j,k,l)? I’m not a computer science guy and apparently interchangeability algorithm can solve the equations and render multiple solutions with the constraints that are within the equations (The Rs and Hs are known).
R_i-(X_l*X_j) * H_k=0;
R_j-(X_k*X_i) * H_l=0;
R_k-(X_j*X_l) * H_i=0;
R_l-(X_i*X_k) * H_j=0;
The (I ; j) pair and (k ; l) pair are interchangeable.
Each equation apparently is quadratic (I don’t understand this part about it, but the Xs correspond to each other).
I would really appreciate if someone can at least make a suggestion or give their opinion as to what approach can be used.
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Adam Danz el 23 de Jun. de 2021
This highlights the importance of investigating the question before investigating the answer. Often times the process of dissecting and clearly framing a question leads to the answer before the question is even asked. Unfortunately, it's common that the real problem, and therefore, the solution, is found at the end of a windy, loopy trail bounded by investigations in the wrong directions.
“If I had only one hour to save the world, I would spend fifty-five minutes defining the problem, and only five minutes finding the solution.”
-Albert Einstein
Rik el 24 de Jun. de 2021
I fully agree with Adam. That is also the reasons why Mathworks decided to hide the delete button when an answer has been posted. (and editing away your question is both rude and pointless, as there is now a capture of this thread)

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### Respuestas (1)

John D'Errico el 10 de Jun. de 2021
I think you are worrying about some magic algorithm, when simple linear algebra will suffice. Yes. LINEAR algebra.
Each equation is of the same form
Ri = X_i*X_j*H_k
TAKE THE LOG. Then you have
log(R_i) - log(H_k) = log(X_i) + log(X_j)
So your equations are LINEAR in the log of the unknowns. Solve the linear system for the unknowns. Again, this is linear algebra.
You will find the linear system is singular, at least as you wrote the equations, so no solution will exist unless something rare happens with the constant terms. And if that does happen, then infinitely many solutions will eist.
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Walter Roberson el 11 de Jun. de 2021
The equations divide by Li*Lj so for stability analysis, it is nice to know that the result is "about 2" instead of having to worry about covering many order of magnitudes. It means the code does not have to be as careful as might otherwise be the case.
However, beyond that, I do not think it helps much.
(My tracking number: T0098914 )
MarshallSc el 12 de Jun. de 2021
Thanks a lot again Walter, really appreicate your kind responses. I still haven't figured out how I can solve the equations and what method I can use. I guess that's the beauty of research. But I appreciate you very much.

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