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Graphical method max:Z=7x₁+10x₂

Short answer

Plot constraints 3x1+x2≤9, x1+2x2≤8 with non-negativity. Corner points: (0,0), (3,0), (2,3), (0,4). Z=7x1+10x2 maximum 44 at (2,3).

Graphically maximize Z = 7x₁ + 10x₂ subject to the constraints 3x₁ + x₂ ≤ 9, x₁ + 2x₂ ≤ 8, x₁, x₂ ≥ 0.

Step 1: Draw the constraint lines. Replace each inequality by an equation, find the intercepts and decide which side of the line is allowed (test the origin).

ConstraintLineMeets x-axisMeets y-axisAllowed side
(1) 3x₁ + x₂ ≤ 93x₁ + x₂ = 9(3, 0)(0, 9)towards the origin
(2) x₁ + 2x₂ ≤ 8x₁ + 2x₂ = 8(8, 0)(0, 4)towards the origin

Also x₁ ≥ 0 and x₂ ≥ 0, so the region lies in the first quadrant.

Step 2: Feasible region. The region satisfying all constraints is the shaded polygon below (dots), with corner points marked A, B, C, ….

   5|                                            
    |                                            
    |                                            
    |D                                           
    |......                                      
    |.............                               
    |...................   C                     
    |.......................                     
    |........................                    
    |.........................                   
    |..........................                  
    |...........................                 
    |............................                
    |.............................               
    |...............................             
    |................................            
   0|A...............................B           
    +--------------------------------------------
     0                                          4

Step 3: Corner points and the objective value. The optimum of a linear function over a convex polygon occurs at a corner point.

PointCoordinatesLines meeting thereZ = 7x₁ + 10x₂
A(0, 0)x₁ = 0, x₂ = 00
B(3, 0)(1), x₂ = 021
C(2, 3)(1), (2)44
D(0, 4)(2), x₁ = 040

Step 4: Conclusion. The largest value of Z is 44, at C(2, 3).

Answer: Maximum Z = 44 at x₁ = 2, x₂ = 3.

✓ Answer checked by Claude Opus 5.5 (high effort)

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