Let me preface this by saying I only include it since a beginner knowing row reduction could apply the method.
The constraints are two equations in four unknowns. I set up the augmented matrix and row reduced it. The first two terms in rows 1 and 2 make up the 2x2 identity matrix, so we let $x_3 = s$ and $x_4$ = t and express all four variables in terms of $s$ and $t$.
$x_1 = -5/8+(5/8)s-(11/8)t$
For a nonempty feasible region we need the first two variables to be nonnegative provided $s,t$ are nonnegative. But a graph (using $s,t$ axes) shows the inequalities on the first two variables give a pair of lines intersecting in the fourth quadrant, with the "shaded part" going off away from quadrant 1. So we have an empty feasible region, as already noted by copper.hat and D. Nehme.