Page 74 - Mathematics_Form_3

Linear programming

Example 3.5
Solve graphically the following linear programming problem.
Minimize f (x, y) = 2x + 5y
Subject to: 2x + 4y ≤ 8
FOR ONLINE READING ONLY
3x + 2y ≤ 6
x +≥ 1
y
x ≥ 0, y ≥ 0
Solution
The graph of the given constraints is as follows.

y
x=0
3

2 C(0, 2)
D 1, 3
x+y=1
2
1
B(0, 1) Feasible region 2x+4y=8
A(1, 0) E(2, 0) y=0
-2 -1 0 0 1 2 3 4 x

-1 3x+2y=6

The values of the objective function at each of the extreme points are shown
in the following table.

Corner points of the Values of the objective function
feasible region f (, )xy = 2x + 5y
A(1, 0) f = 2(1) + 5(0) = 2

B(0, 1) f = 2(0) + 5(1) = 5 Mathematics for Secondary Schools
C(0, 2) f = 2(0) + 5(2) =10
3
⎛ 3 ⎞  1
( ) 5+
D 1, ⎟ f = 21  = 9
⎜
2
⎝ 2 ⎠  2
E(2, 0) f = 2(2) + 5(0) = 4
From the table, the objective function has the smallest value at point A (1, 0).
Therefore, the minimum value is 2.

Student\s Book Form Three 67

18/09/2025 09:59:10
MATHEMATIC F3 SB.indd 67 18/09/2025 09:59:10
MATHEMATIC F3 SB.indd 67