Suppose I have the simple linear program

$$Ax \geq 0, \quad x \geq 0$$

We know that this system has a solution (for example, $x=0$). But, what if we made this rule for this system?

$$Ax \geq 0, \quad e^T x > 0$$

where $e = (1, 0, \dots, 0)$. Rewriting,

$$Ax \geq 0, \quad x_1 > 0$$

Is there a way to decide whether this system has solutions or not before using some algorithms to solve it?

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