Maximize value Z = 15x + 10y subject to the constraints 3x + 2y ¡Ü
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12, 2x + 3y ¡Ü
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15, x ¡Ý
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0, y ¡Ý
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To solve this linear programming problem and maximize the value Z = 15x + 10y, subject to the given constraints, follow these steps:
Step 1: Graph the feasible region:
Start by graphing the constraints on a coordinate plane.
Constraint 1: 3x + 2y <= 12
To graph this constraint, rewrite the inequality as an equation: 3x + 2y = 12
Plot the line 3x + 2y = 12 on the coordinate plane, either by finding the x- and y-intercepts or choosing arbitrary values for x and calculating corresponding y.
Constraint 2: 2x + 3y <= 15
Similarly, rewrite the inequality as an equation: 2x + 3y = 15
Plot the line 2x + 3y = 15 on the coordinate plane.
Constraint 3: x >= 0
This constraint represents the vertical axis (x = 0) and all values to the right of it.
Constraint 4: y >= 0
This constraint represents the horizontal axis (y = 0) and all values above it.
The feasible region is the overlapping area of all the shaded regions on the graph.
Step 2: Identify the corner points of the feasible region:
The corner points of the feasible region are the vertices where the constraint lines intersect. Use the graph to determine these points.
Step 3: Evaluate Z at each corner point:
Substitute the coordinates of each corner point into the equation Z = 15x + 10y to find the corresponding Z value.
Step 4: Select the corner point with the highest Z value:
Compare the Z values obtained from step 3 and choose the corner point with the highest Z value. This corner point represents the optimal solution that maximizes Z.
Therefore, by following these steps, you can solve the linear programming problem and find the maximum value of Z = 15x + 10y that satisfies the given constraints.
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