Consider the LP:

$\{\begin{array}{c}\text{:}3x_1+6x_2+6x_3+6x_{4}150\end{array}$

Someone claims a final dictionary has

$z=540-x_2-3x_4-4x_6-3x_7$

Explain what does this imply to the dual optimal solution and why there must have been an error in the final dictionary. You must not solve this by doing the simplex method iterations yourself; you will get zero mark if you do so$.$ Instead, use relevant theorems and deduce your conclusion from them.