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I would like to know if there is a method to check the existence of the solution for a given set of linear equations composed with both equalities and inequalities? I'm not interested in the solution, but whether there exist at least one solution or not.

Edit1: Mixed systems of linear equations and inequalities,

$a_{1,1} x_1 + \dots + a_{1,n} x_n = b_1$

$a_{2,1} x_1 + \dots + a_{2,n} x_n = b_2$


$a_{m-1,1} x_1 + \dots + a_{m-1,n} x_n \geq b_{m-1}$

$a_{m,1} x_1 + \dots + a_{m,n} x_n \geq b_m$

The number of equations $m$ might be less than, equal to or greater than $n$ (the number of unknowns).

Edit2: Example1,

$ \begin{cases} 2x-y \geq -3 \\ -4x-y \geq -5 \\ -x+y=4 \end{cases} $

There is no solution for above system. I'm looking for a systematic (algorithmic) way to determine if such systems of equations have any solution or not.

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Yes. You can easily find out the set of solutions given certain equations or inequations. The various cases that you come across are :

1) You get unique solution.

2) You get infinitely many solutions.

3) You get no solution.

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Yes. You can determine whether a given inequality has a solution or not, it all depends on the equation. It all depends on the restrictions you pose on the solution.

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