Hi just as a test I'm trying to solve the following LP with Lagrange multipliers.
$x_2 \leq 1 - x_1$
$x_1, x_2 \geq 0 $
I add slack variables to have a equality constrained LP
$x_2 +x_1 +s_1 -1 =0$
$x_1 -s_2 = 0 $
$x_2 - s_3=0$
I then look at the gradients and find the following system
$-1 + \lambda_1 + \lambda_2 =0 $
$\lambda_1 + \lambda_3 =0 $
$\lambda_1 =0 $
$-\lambda_2 =0 $
$-\lambda_3 =0 $
This system is not consistent though, since it says $-1=0$
If this happens, that is there is no $\lambda_i$ such that the system is satisfied then the problem is unbounded. That is not the case though, since we see that the optimal of $x_1=1,x_2=0$ is optimal with objective value $-1$. I've messed something up here basic. Some feedback would be much appreciated.