1. Polygon of constraints L.O. graph the polygon of constraints
This is the THIRD part of the optimization problem.
When you graph the inequalities the part you shade will form a POLYGON (shape).
The goal in this part is to find the VERTICES (corners) of the shape.
Some vertices you will be able to read off the graph but others you will have to solve ALGEBRAICALLY .
We solve for the vertices algebraically by solving the SYSTEM OF EQUATIONS of the two lines.

2. Polygon of constraints L.O. graph the polygon of constraints
Example:
Find the vertices of the following polygon of constraints
𝑥≥ 𝑦≥0
20𝑥+𝑦≤500
𝑦<300
Step 1: Graph the system of inequalities
L1: 20𝑥+𝑦≤500
𝑥=0
𝑦=500
𝑦=0
20𝑥=500 𝑥=
𝑥=25
𝑇𝑒𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 0,0
0+0≤500
0≤500
True x y
500 25

3. Polygon of constraints L.O. graph the polygon of constraints
Example:
Find the vertices of the following polygon of constraints
𝑥≥ 𝑦≥0
20𝑥+𝑦≤500
𝑦<300
Step 1: Graph the system of inequalities x y
300 20
L2: 𝑦<300
𝑦=300
Horizontal line
𝑇𝑒𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 0,0
0<300
𝑇𝑟𝑢𝑒

4. Polygon of constraints L.O. graph the polygon of constraints

5. Polygon of constraints L.O. graph the polygon of constraints
Step 2: Identify the vertices and write the coordinates by either reading it off the graph or solve algebraically
𝐴 0, 𝐵 0, 𝐷 25,0
Vertex C looks like it is 10,300 . We will solve it algebraically to make sure we are correct.
𝐿1:20𝑥+𝑦=500
𝐿2:𝑦=300
Use Substitution Method:
20𝑥+300=500
20𝑥=500−300
20𝑥=200 𝑥=
𝑥=10
→𝐶 10,300

6. Polygon of constraints L.O. graph the polygon of constraints
Step 3: Answer
𝐴 0, 𝐵 0,300
𝐷 25, 𝐶 10,300