inequalities = {Y[5], Y[7], Y[9], Y[10],
 155 + 27 Y[5] - 13 Y[7] - 65 Y[9] - 13 Y[10], 162 - 27 Y[5],
 45 - 9 Y[7] + 43 Y[9] + 12 Y[10], 374 + 105 Y[9] - 51 Y[10],
 1288 + 23 Y[7] - 759 Y[9] + 3 Y[10], 1 + Y[7] - Y[9] + Y[10],
 3 + 3 Y[7] + 4 Y[9] - 2 Y[10]}

ToRules[
 Reduce[# >= 0 & /@ inequalities, Variables[inequalities], Integers]]