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Solve the linear programming problem graphically:
Maximise Z = 4x + y subject to the constraints:
x + y $$\le$$ 50
3x + y $$\le$$ 90
x $$\ge$$ 0, y $$\ge$$ 0
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## QuestionMathsClass 12

Solve the linear programming problem graphically:
Maximise Z = 4x + y subject to the constraints:
x + y $$\le$$ 50
3x + y $$\le$$ 90
x $$\ge$$ 0, y $$\ge$$ 0

To Maximize Z = 4x + y ......(i)
subject to the constraints:
x + y $$\le$$ 50  .....(ii)
3x + y $$\le$$ 90 ......(iii)
x $$\ge$$ 0, y $$\ge$$ 0 .....(iv)
The shaded region in a figure is the feasible region determined by the system of constraints (ii) to (iv). We observe that the feasible region OABC is bounded. So, we now use Corner Point Method to determine the maximum value of Z.
The coordinates of the corner points O, A, B and C are (0, 0), (30, 0), (20, 30) and (0, 50) respectively. Now we evaluate Z at each corner point.
 Corner Point Corresponding value of Z (0, 0) 0 (30, 0) 120 (20, 30) 110 (0, 50) 50

Hence, maximum value of Z is 120 at the point (30, 0).