Linear programming involves solving problems related to linear inequalities and equations. There are four main steps: (1) write the linear equations and inequalities that describe the situation, (2) graph the equations and shade the feasible region, (3) determine and graph the objective function, (4) determine the optimum value of the objective function graphically. Key skills include identifying and shading feasible regions, writing linear equations and inequalities, and solving problems with integer values of variables within given constraints.

1. sbpisb LINEAR PROGRAMMING
THE CONCEPT OF LINEAR PROGRAMMING
The problems related to linear programming can be solved by the following steps:
(i) Write linear inequalities and equations that describe a situation.
SKILL 2
SKILL 1 (ii) Draw the graphs and shade the region where the points in the region Identifying and shading the
are feasible solutions. region in which every point
Drawing a graph of
satisfies a linear inequality.
a straight line. (iii) Determine and draw the object function.
(iv) Determine graphically the optimum value of the objective function.
EXAMPLES
EXAMPLES SKILL 3 1) x ≤ 4 x ≥4
1) x = 4 y Writing linear inequality or equation that describe a situation.
EXAMPLES
Description Mathematical
representation 4 4
0 x
4 1) y is more or equal to x. y ≥x
y 2) y is at least x. y≥x 2) y ≤ 7 y ≥7
2) y = 7
3) y is at most two times x y≤2x
7 4) y is at not more than x. y≤x 7 7
5) x + y has a maximum value of 10 x + y ≤ 10
0 x 6) y is at least 20 more than x. y- x ≥ 20
7) The minimum value of the total of x x + y ≥ 40
and y is 40.
3) y ≤ x + 3 y ≥x +3
3) y = x + 3 y
SKILL 4 3
x 0 1 X (1,4) y 3
y 3 4 3 Given (x,y) € R ,x and y are integers:
(0,3) (1,4) 8
x 3) 2x +3 y < 6 2x +3 y > 6
0 7 Maximum value of x = 6
Maximum value of y = 8
3) 2x +3 y = 6 6 Minimum value of x + y =
y 2 3
5 Maximum value of 2x + 3y = 34
x 0 3 R
2 3 Minimum value of y when x=2 :
y 2 0 (0,2) (3,0) Ans y = 2 3 3
2
0 x Maximum value of 2x + y when y
3 1 = 4 : Ans: 14
1 2 3 4 5 6
x +y <  below +y >  above
2x +3 y > 6 0
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