Checking the existence of a solution for a set of linear equality and ineaulity equations 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,

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

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

$\dots$

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

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

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.
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.
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.