Cambridge International Mathematics

In many problems involving inequalities, the only points which are possible are those with integer coordinates.

We therefore consider only these points within the region

Example 5 Self Tutor

x> 2 , y> 3 , x+y 610 , x+2y 614.

a On grid paper, graph the region

b

c y=^12 x+3.

d Find the maximum value of 5 x+4y for the 15 feasible points in

a

b We look for all points

There are 15 points with integer coordinates, as shown.

c The line y=^12 x+3 passes through(2,4)and(4,5)in

d The vertices all have integer coordinates, so we need only consider these points:

Points 5 x+4y

(2,3) 10 + 12 = 22

(2,6) 10 + 24 = 34

(6,4) 30 + 16 = 46

(7,3) 35 + 12 = 47

5 x+4y has maximum value 47

when x=7, y=3.

C INTEGER POINTS IN REGIONS [7.7]

R.

R.

R.

R.

inRwhere the grid lines meet.

Find all points inRwith integer coordinates that lie on the line

How many points inRhave integer coordinates?

O

y

y¡=¡3

x

xy¡+¡ ¡=¡10

yx¡=¡¡Qw_\¡+¡3

()2 ¡3,

()7 ¡3,

()6 ¡4,

()2 ¡6,

x¡=¡2

55 1010 14141515

55

77

R

1010

xy¡+¡2 ¡=¡14

A regionRis defined by

644 Inequalities (Chapter 32)

We are often asked to find the minimum or maximum value of a function in the region. If all of the

constraints are linear, the minimum and maximum occur at a vertex or corner point of the region.

If all of the vertices have integer coordinates, we need only consider these points when finding the

maximum or minimum. However, if we are considering only integer points and the vertices are not all

integer points, we need to be more careful.

must

R

R

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Y:\HAESE\IGCSE01\IG01_32\644IGCSE01_32.CDR Friday, 31 October 2008 9:31:30 AM PETER