Linear Programming

Step 1: Convert inequalities to equations

• Line 1:
• Line 2:
• Line 3:

Step 2: Find Intercepts

• Line 1: ,
• Line 2: ,
• Line 3: ,

Step 3: Graph and Feasible Region

Plot all three lines and shade the region that satisfies all constraints including and .

Step 4: Find Corner Points (Intersections)

1. Line 1 & Line 2:
   Solve: Multiply (i) by 2: Subtract from (ii): So point A:
2. Line 2 & Line 3:
   Substitute: So point B:
3. Line 1 & Line 3:
   Substitute: So point C:

Step 5: Evaluate at Each Corner Point

✅ Final Answer:

Maximum value of is at two corner points:

Step 1: Let the variables be Let = number of Toy A units produced
Let = number of Toy B units produced 
Step 2: Write the Objective Function We want to maximize profit:  
Step 3: Translate constraints from machine limits

• M1:
• M2:
• M3:
• Non-negativity:

Step 4: Draw the Feasible Region 

Step 5: Find Corner Points of Feasible Region We solve the equations of intersecting lines to find vertices of the region: 
(i) Intersection of and : (ii) Intersection of and : (iii) Intersection of and :  
Step 6: Evaluate Objective Function at Each Corner Point  
Step 7: Select Optimal Value
Final Answer: