Ok so we have the following linear programming problem :
maximize $ x_1+2x_2 - 3x_3 +x_4$
subject to $x_1+x_2+x_3\le4 \\x_2-x_3+x_4\le3\\x_1-2x_2+2x_4\ge1 \\x_1,x_3,x_4\ge0\\x_2\ge2$
We are given a solution where $x_1=1,x_2=2,x_3=1,x_4=2 $. Is there a way to prove that its an optimal solution without using simplex?? I am thinking about using the graph to find all the corner points,then confirm that the given values are indeed a corner point, and then prove that for these given values, the function is indeed maximized. But i am not sure.