Mixed systems of linear equations and inequalities, a 1 , 1 x 1 + ⋯ + a 1 , n x n = b 1 a 2 , 1 x 1 + ⋯ + a 2 , n x n = b 2 … a m − 1 , 1 x 1 + ⋯ + a m − 1 , n x n ≥ b m − 1 a m , 1 x 1 + ⋯ + a m , n x n ≥ b m The number of equations m might be less than, equal to or greater than n (the number of unknowns).Example1, { 2 x − y ≥ − 3 − 4 x − y ≥ − 5 − x + y = 4 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.

Question

Mixed systems of linear equations and inequalities,      a          1      ,      1            x    1    +  ⋯  +      a          1      ,      n            x    n    =      b    1        a          2      ,      1            x    1    +  ⋯  +      a          2      ,      n            x    n    =      b    2    …      a          m      −      1      ,      1            x    1    +  ⋯  +      a          m      −      1      ,      n            x    n    ≥      b          m      −      1            a          m      ,      1            x    1    +  ⋯  +      a          m      ,      n            x    n    ≥      b    m  The number of equations   m might be less than, equal to or greater than   n (the number of unknowns).Example1,      {                            2          x          −          y          ≥          −          3                                      −          4          x          −          y          ≥          −          5                                      −          x          +          y          =          4                        There is no solution for above system. I&#039;m looking for a systematic (algorithmic) way to determine if such systems of equations have any solution or not.

Hailee Henderson · Accepted Answer

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.

Kozankaht2 · Accepted Answer

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.

Mixed systems of linear equations and inequalities, a 1 , 1 </mr

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