[lp_intro] update notations

Dear John @jstac,

This pull request finishes the issue #469 by

- add the notation for $z$, $A_{eq},b_{eq}, A_{ub},b_{ub}$.

- change the orange to black to correspond to the changes in the visualization plot.

Best,
Longye

Author: longye-tian

lectures/lp_intro.md

@@ -76,7 +76,7 @@ Required per unit material and labor  inputs and  revenues  are shown in table b
 
 A firm's problem is to construct a  production plan that uses its  30 units of materials and 20 units of labor to maximize its revenue.
 
-Let $x_i$ denote the quantity of Product $i$ that the firm produces.
+Let $x_i$ denote the quantity of Product $i$ that the firm produces and $z$ denote the total revenue.
 
 This problem can be formulated as:
 
@@ -132,11 +132,11 @@ plt.show()
 
 The blue region is the feasible set within which all constraints are satisfied.
 
-Parallel orange lines are iso-revenue lines.
+Parallel black lines are iso-revenue lines.
 
-The firm's objective is to find the  parallel orange lines to the upper boundary of the feasible set.
+The firm's objective is to find the  parallel black lines to the upper boundary of the feasible set.
 
-The intersection of the feasible set and the highest orange line delineates the optimal set.
+The intersection of the feasible set and the highest black line delineates the optimal set.
 
 In this example, the optimal set is the point $(2.5, 5)$.
 
@@ -451,6 +451,8 @@ $$
 \end{aligned}
 $$
 
+$A_{eq}, b_{eq}$ denote the equality constraint matrix and vector, and $A_{ub}, b_{ub}$ denote the inequality constraint matrix and vector.
+
 ```{note}
 By default $l = 0$ and $u = \text{None}$ unless explicitly specified with the argument `bounds`.
 ```