Step-by-Step Solution

Topic: Graphical Constraints & Optimization

(I) TRIVIAL CONSTRAINTS

In most Linear Programming Problems involving physical quantities or number theory, the variables cannot be negative.

The trivial constraints are:

(II) NON-TRIVIAL CONSTRAINTS

Identify the Lines:

• • Line 1 (Points D & E): Intercepts (100,0) & (0,200) ⇒ 2x + y = 200
• • Line 2 (Points A & E): Intercepts (100,0) & (0,50) ⇒ x + 2y = 100 (Given)
• • Line 3 (Points O, B, C): Origin to (50,100) ⇒ y = 2x ⇒ 2x – y = 0

(a) For Region R₁ (Quadrilateral ABCD):

R₁ is below Line 1, above Line 2, and to the left of Line 3 (towards y-axis).

• 1. Below 2x + y = 200 ⇒ 2x + y ≤ 200
• 2. Left of y = 2x ⇒ y ≥ 2x (or 2x – y ≤ 0)

(b) For Region R₂ (Triangle BCE):

R₂ is below Line 1, above Line 2, and to the right of Line 3 (towards x-axis).

• 1. Below 2x + y = 200 ⇒ 2x + y ≤ 200
• 2. Right of y = 2x ⇒ y ≤ 2x (or 2x – y ≥ 0)

(III) MAXIMIZE z = 5x + 2y (On Region R1)

We evaluate Z at the corner points of R₁: A, B, C, D.