9. y x x
x + y = 5

10. POSITIVE

16. Note – Each shaded region is the region NOT required 16 14 12 10 8 6 4 2 x y 18
2x + y = 20
x ≤ 0
Note – Each shaded region is the region NOT required
2x + y ≥ 20
x + 2y ≥ 12
x + 2y = 12 a
x + 3y ≥ 15 b
Feasible Region c
x + 3y = 15 d
y ≤ 0

17. Note – Each shaded region is the region NOT required 16 14 12 10 8 6 4 2 x y 18
2x + y = 20
You need to add a Profit Line.
As 5 and 12 can’t be simplified the Lowest Common Multiple of 5 and 12 is 60.
x ≤ 0
Note – Each shaded region is the region NOT required
2x + y ≥ 20
x + 2y ≥ 12
So 5x = 60, then x = 12 and
12y = 60, then y = 5 then join these two point to establish the profit line
x + 2y = 12 a
x + 3y ≥ 15 b
Feasible Region c
Use a ruler and slide it parallel to the profit line to establish the highest (or lowest) corner in the Feasible region, which is at b
Profit Line
x + 3y = 15 d
y ≤ 0

19. Profit Line P = 5x + 7y LCM of 5 and 7 = 35 So 5x = 35, x = 7 and 16 14 12 10 8 6 4 2 x y 18
x ≤ 0
2x + 3y ≥ 30
Profit Line
P = 5x + 7y
x + y = 12
x + y ≥ 12
LCM of 5 and 7 = 35
So 5x = 35, x = 7 and
Feasible Region
2x + 3y = 30
7y = 35, y = 5
Profit Line
Plot these points
y ≤ 0

23. Profit Line P = 5x + 6y LCM of 5 and 6 = 30 So 5x = 30, x = 6 and
6y = 30, y = 5
Plot these points 32 28 24 20 16 12 8 4 x y 36
3x + 2y ≥ 48
x + 2y ≥ 36
x + y ≥ 20
3x + 2y = 48
x ≤ 0
F. R.
x + 2y = 36
F. R. = Feasible Region
P. L.
x + y = 20
P. L. = Profit Line
y ≤ 0

27. As x would = 8 and y = 4, I do not need to simplify, giving the co-ordinates I need
Feasible Region
y ≤ 0
5x + 2y = 10
Cost Line
x + y = 4
x + 3y = 6
x ≤ 0

29. y x 2x + y ≥ 16 y ≤ 0 2x + y = 16 x + 3y ≥ 20 Feasible Region 16 14 12 10 8 6 4 2 x y 18
2x + y ≥ 16
y ≤ 0
2x + y = 16
x + 3y ≥ 20
Feasible Region
x + 3y = 20
Profit Line
x ≤ 0

30. x + y occurs at (5,5) or (6,4) = 10
4x + 5y occurs at (5,5) = 45

31. Exam Technique
If you have to FORMULATE the problem, then you need to establish
What are the unknowns?
What are the constraints?
What is the profit/cost to be maximised or minimised?
If you have to GRAPH the problem, then
Draw each constraint (including x = 0 and y = 0) and SHADE the REGION you DON’T require
Maximise the PROFIT or minimise the COST by working out the value at EACH vertex in the FEASIBLE region
Also prove this by using and showing a PROFIT/COST LINE
Understand that some problems produce a FINITE feasible region, whilst others don’t