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The maximum value of Z = 3x + 4y subject to the constraints x ≥ 0, y ≥ 0 and x + y ≤ 1 is
Constraints in linear programming are conditions or limitations that restrict the possible solutions of a problem. They define the feasible region where the solution must lie. Constraints can be expressed as linear inequalities or equations that limit the values of variables in optimization problems, such as resource limitations or requirements.
Linear Programming is a method used to optimize a linear objective function subject to linear constraints. The solution to a problem lies within the feasible region defined by these constraints. The optimal solution is typically found at the vertices of the feasible region. Linear Programming has applications in business economics and resource management.
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To solve for the maximum value of Z = 3x + 4y subject to the constraints:
1. x ≥ 0
2. y ≥ 0
3. x + y ≤ 1
We plot the constraints first and determine the feasible region.
The feasible region is bounded by the points:
– (0, 0) where x = 0 and y = 0
– (1, 0) where x + y = 1 and y = 0
– (0, 1) where x + y = 1 and x = 0
Then we calculate Z = 3x + 4y at all the corner points:
– At (0, 0), Z = 3(0) + 4(0) = 0
– At (1, 0), Z = 3(1) + 4(0) = 3
– At (0, 1), Z = 3(0) + 4(1) = 4
The maximum value for Z is 4 and it occurs at (0, 1).
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