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Is there a way to include the inequalities of variables in calculating family of equations while solving under determined simultaneous linear equations?

For Eg: Lets say x + y = 4, y + z = 4.

But I also have the additional information that x and y are unequal and y and z are unequal. How can I make use of this extra information? Is there a way to include this information in the matrix while calculating reduced Echelon form to family of solutions.

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  • $\begingroup$ Introduce further variables that make up the difference e.g. x + y = 4 and x + 2<y is now x + y = 4 with x+2+s = y. You will possibly need conditions on x, y, like they are always positive for example, to get something useful. $\endgroup$ – Paul Mar 8 at 12:44
  • $\begingroup$ The thing is the only information I have is that they are not equal, I don't know which one is greater, or what exactly is the difference between them. IS there a way to formally include this information in to matrix? $\endgroup$ – VARUN.N RAO Mar 8 at 12:47
  • $\begingroup$ If you have a set of linear equations then your solutions will form a vector space e.g a point, line or plane if you have 3 variables. You can simply ignore such solutions with equal values. $\endgroup$ – Paul Mar 9 at 8:44
  • $\begingroup$ @Paul no, the three variable equation was jut for example, I have 18 equations, and 81 variables, now I also have some extra information that some of those variables are not equal to other variables. I can't manually do it, its for a computer program, So is there an algebraic way to solve this? $\endgroup$ – VARUN.N RAO Mar 9 at 14:00

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