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Linear Programming
Solve the following Linear Programming Problems graphically:
Minimise Z = 5x + 3y
subject to the constraints: 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0.
We have to maximise
Z = 5x + 3 y
subject to the constraints
3x + 5 y ≤ 15
5x + 2 y ≤ 10
x ≥ 0, y ≥ 0
Consider a set of rectangular cartesian axes OXY in the plane.
It is clear that any point which satisfies x ≥ 0, y ≥ 0 lies in the first quadrant.
Let us draw the graph of 3x + 5 y= 15
For x = 0, 5 y = 15 or y = 3
For y = 0, 3x = 15 or x = 5
∴ line meets OX in A(5, 0) and OY in L(0, 3).
Again we draw the graph of 5x + 2 y = 10
For x = 0, 2 y = 10 or y = 5
For y = 0, 5x = 10 or x = 2
∴ line meets OX in B(2, 0) and OY in M(0, 5).
Since feasible region is the region which satisfies all the constraints.
∴ OBCL is the feasible region and corner points are O(0, 0), B(2, 0),
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Some More Questions From Linear Programming Chapter
Minimize z = 2x + 3y, such that 1 ≤ x + 2y ≤ 10, x ≥ 0, y ≥ 0.
Solve the following linear programming problem graphically:
Minimise Z = 200x + 500y
subject to the constraints x + 2y ≥ 10, 3x + 4 y ≤ 24, x ≥ 0, y ≥ 0
Solve the following problem graphically:
Minimise and Maximise Z = 3x + 9y
subject to the constraints:
x + 3y ≤ 60
x + y ≥ 10
x ≤ y
x ≥ 0, y ≥ 0
Minimise and Maximise Z = 3x + 9y
subject to the constraints:
x + 3y ≤ 60
x + y ≥ 10
x ≤ y
x ≥ 0, y ≥ 0
Show that the minimum of Z occurs at more than two points.
Minimise and Maximise Z = 5x + 10y subject to constraints x + 2y ≤ 120, x + y ≥ 60, x - 2 y ≥ 0, x, y ≥ 0.
Minimize z = 5x + 7y such that 2x + y ≥ 8, x + 2y ≥ 10, x, y ≥ 0.
Show that the minimum of Z occurs at more than two points.
Minimise and Maximise Z = x + 2y subject to constraints x + 2y ≥ 100, 2x - y ≤ 0, 2x + y ≤ 200, x, y ≥ 0
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