1.	Find the graph of the equation 3x + 5y = 30.

2.	Find the graph of the equation 4x + 2y = 12.

3.	Find the graph of the equation 4x + 6y = 18.

4.	Find the graph of the equation 5x + 2y = 15.

5.	Find the graph of the inequality 3x + 4y £ 12.

6.	Find the graph of the inequality 4x + 5y £ 40.

7.	Find the graph of the inequality 6x + 4y £ 48.

8.	Find the graph of the inequality 3x + 7y £ 21.

9.	Find the point of intersection of the lines whose equations are 2x + 3y = 12 and 1x + 5y = 13.	
	A)  (2, 3)    B)  (3, 2)    C)  (6, 0)    D)  (–2, 3)	

10.	Find the point of intersection of the lines whose equations are 4x + 2y = 12 and 3x + 9y = 39.	
	A)  (5, –4)    B)  (10, 1)    C)  (1, 4)    D)  (2, 2)	

11.	Find the point of intersection of the lines whose equations are 3x + 2y = 21 and 2x + 1y = 13.	
	A)  (5, 3)    B)  (29, 45)    C)  (8, –3)    D)  (3, 5)	

12.	Find the point of intersection of the lines whose equations are 2x + 5y = 6 and 3x + 2y = 9.	
	A)  (3, 0)    B)  (2, 1)    C)  (–3, 0)    D)  (1, 2)	

13.	Graph the constraint inequalities for a linear programming problem shown below.  Which feasible region shown is correct? 
	
2 x + 3y £ 12	

x ³ 0, y ³ 0	

 	

14.	Graph the constraint inequalities for a linear programming problem shown below.  Which feasible region shown is correct?
	
4 x + 3y £ 24	

x ³ 0, y ³ 0	

 	

15.	Graph the constraint inequalities for a linear programming problem shown below.  Which feasible region shown is correct?
	
6 x + 4y £ 12	

x ³ 0, y ³ 0	

 	

16.	Graph the constraint inequalities for a linear programming problem shown below.  Which feasible region shown is correct?
	
1 x + 4y £ 8	

x ³ 0, y ³ 0	

 	

17.	Write a resource constraint for this situation:  A lawn service company has 40 hours of worker time available.  Mowing a lawn (x) takes 3 hours and trimming (y) takes 2 hours.  The profit from mowing is $15 and the profit from trimming is $10.	
	A)  3x + 2y £ 40    B)  (40/3)x + 10y £ 40    C)  15x + 10y £ 40    D)  5x + 5y £ 40	

18.	Write a resource constraint for this situation:  Producing a plastic ruler (x) requires 10 grams of plastic while producing a pencil box (y) requires 30 grams of plastic.  There are 2000 grams of plastic available.	
A)	200x + (2000/30)y £ 2000	C)	10x + 30y £ 2000	
B)	30x + 10y £ 2000	D)	x + y £ 2000	

19.	Write the constraint inequalities for this situation:  Kim and Lynn produce pottery vases and bowls.  A vase requires 35 oz. of clay and 5 oz. of glaze.  A bowl requires 20 oz. of clay and 10 oz. of glaze.  There are 500 oz. of clay available and 200 oz. of glaze available.  The profit on one vase is $5 and the profit on one bowl is $4.	
A)	35x + 5y £ 5, 20 x + 10y £ 4, x ³ 0, y ³ 0	
B)	35x + 5y £ 500, 20 x + 10y £ 200, x ³ 0, y ³ 0	
C)	35x + 20y £ 500, 5 x + 10y £ 200, x ³ 0, y ³ 0	
D)	35x + 20y £ $5, 5 x + 10y £ $4, x ³ 0, y ³ 0	

20.	Write the constraint inequalities for this situation:  A cheeseburger requires 5 oz. of meat and 0.7 oz. of cheese while a superburger requires 7 oz. of meat and 0.6 oz. of cheese.  The burger stand has 350 oz. of meat and 42 oz. of cheese available.  The profit on a cheeseburger is 10 cents and the profit on a superburger is 40 cents.	
A)	5x + 7y £ 350, 0.7 x + 0.6y £ 42, x ³ 0, y ³ 0	
B)	5x + 0.7y £ 10, 7 x + 0.6y £ 40, x ³ 0, y ³ 0	
C)	5x + 7y £ 10, 0.7 x + 0.6y £ 40, x ³ 0, y ³ 0	
D)	70x + 50y £ 350, 60 x + 70y £ 42, x ³ 0, y ³ 0	

21.	Write the resource constraints for this situation:  A small stereo manufacturer makes a receiver and a CD player.  Each receiver takes 8 hours to assemble and 1 hour to test and ship.  Each CD player takes 15 hours to assemble and 2 hours to test and ship.  The profit on each receiver is $30 and the profit on each CD player is $50.  There are 160 hours available in the assembly department and 22 hours available in the testing and shipping department.	
A)	8x + 1y £ 30, 15 x + 2y £ 50, x ³ 0, y ³ 0	
B)	8x + 1y £ 160, 15 x + 2y £ 22, x ³ 0, y ³ 0	
C)	8x + 15y £ 30, 1 x + 2y £ 50, x ³ 0, y ³ 0	
D)	8x + 15y £ 160, 1 x + 2y £ 22, x ³ 0, y ³ 0	

22.	Write the resource constraints for this situation:  Kim and Lynn produce tables and chairs.  Each piece is assembled, sanded, and stained.  A table requires 2 hours to assemble, 3 hours to sand, and 3 hours to stain.  A chair requires 4 hours to assemble, 2 hours to sand, and 3 hours to stain.  The profit earned on each table is $20 and on each chair is $12.  Together Kim and Lynn spend at most 16 hours assembling, 10 hours sanding, and 13 hours staining.	
A)	2x + 4y £ 16, 3 x + 2y £ 10, 3x + 3y £ 13, x ³ 0, y ³ 0	
B)	2x + 3y + 3z £ 20, 4x + 2y + 3z £ 12, x ³ 0, y ³ 0, z ³ 0	
C)	16x + 10y + 13z £ 0, 2x + 3y + 3z £ 20, 4x+ 2y + 3z £ 12, x ³ 0, y ³ 0, z ³ 0	
D)	8x + 4y £ 16, (10/3) x + 5y £ 10, (13/3)x + (13/3)y £ 13, x ³ 0, y ³ 0	

23.	Write the resource constraints for this situation:  A company manufacturers patio chairs and rockers.  Each piece is made of wood, plastic, and aluminum.  A chair requires 1 unit of wood, 1 unit of plastic, and 2 units of aluminum.  A rocker requires 1 unit of wood, 2 units of plastic, and 5 units of aluminum.  The company's profit on a chair is $7 and on a rocker is $12.  The company has available 400 units of wood, 500 units of plastic, and 1450 units of aluminum.	
A)	1x + 1y + 2z £ 7, 1x + 2y + 5z £ 12, x ³ 0, y ³ 0, z ³ 0	
B)	1x + 1y £ 400, 1 x + 2y £ 500, 2x + 5y £ 1450, x ³ 0, y ³ 0	
C)	400x + 500y + 1450z £ 0, 1x + 1y + 2z £ 7, 1x+ 2y + 5z £ 12, x ³ 0, y ³ 0,  z ³ 0	
D)	7x + 12y £ 400, 2 x + 5y £ 1450, x ³ 0, y ³ 0	

24.	Graph the feasible region identified by the inequalities:
	
2 x + 3y £ 12	

1 x + 5y £ 10	

x ³ 0, y ³ 0	

 	

25.	Graph the feasible region identified by the inequalities:
	
4 x + 1y £ 12	

2 x + 7y £ 28	

x ³ 0, y ³ 0	

 	

26.	Graph the feasible region identified by the inequalities: 
	
5 x + 1y £ 10	

3 x + 3y £ 18	

x ³ 0, y ³ 0	

 	

27.	Graph the feasible region identified by the inequalities: 
	
4 x + 3y £ 12 	

3 x + 3y £ 18 	

x ³ 0, y ³ 0	

 	

28.	Given below is the sketch of the feasible region in a linear programming problem.  Which point is not in the feasible region?

	A)  (0, 8)    B)  (12, 0)    C)  (6, 4)    D)  (2, 2)	

29.	Given below is the sketch of the feasible region in a linear programming problem.  Which point is not in the feasible region?

	A)  (0, 4)    B)  (4, 0)    C)  (6, 0)    D)  (1, 2)	

30.	Given below is the sketch of the feasible region in a linear programming problem.  Which point is not in the feasible region?

	A)  (6, 4)    B)  (0, 10)    C)  (2, 6)    D)  (0, 8)	

31.	Given below is the sketch of the feasible region in a linear programming problem.  Which point is not in the feasible region?

	A)  (0, 6)    B)  (4, 0)    C)  (4, 2)    D)  (6, 0)	

32.	Given below is the sketch of the feasible region in a linear programming problem.  Which point is not in the feasible region?

	A)  (0, 32)    B)  (0, 24)    C)  (8, 16)    D)  (20, 12)	

33.	Write a profit formula for this mixture problem:  Kim and Lynn produce pottery vases and bowls.  A vase requires 35 oz. of clay and 5 oz. of glaze.  A bowl requires 20 oz. of clay and 10 oz. of glaze.  There are 500 oz. of clay available and 200 oz. of glaze available.  The profit on one vase is $5 and the profit on one bowl is $4.	
	A)  P = 500x + 200y    B)  P = 35x + 20y    C)  P = 5x + 4y    D)  P = 5x + 10y	

34.	Write a profit formula for this mixture problem:  A small stereo manufacturer makes a receiver and a CD player.  Each receiver takes 8 hours to assemble, 1 hour to test and ship, and earns a profit of $30.  Each CD player takes 15 hours to assemble, 2 hours to test and ship, and earns a profit of $50.  There are 160 hours available in the assembly department and 22 hours available in the testing and shipping department.	
	A)  P = 8x + 1y    B)  P = 160x + 22y    C)  P = 15x + 2y    D)  P = 30x + 50y	

35.	Write a profit formula for this mixture problem:  Kim and Lynn produce tables and chairs.  Each piece is assembled, sanded, and stained.  A table requires 2 hours to assemble, 3 hours to sand, and 3 hours to stain.  A chair requires 4 hours to assemble, 2 hours to sand, and 3 hours to stain.  The profit earned on each table is $20 and on each chair is $12.  Together Kim and Lynn spend at most 16 hours assembling, 10 hours sanding, and 13 hours staining.	
A)	P = 20x + 12y	C)	P = 16x + 10y + 13z	
B)	P = 2x + 3y + 3z	D)	P = 8x + 9y	

36.	Write a profit formula for this mixture problem:  A company manufacturers patio chairs and rockers.  Each piece is made of wood, plastic, and aluminum.  A chair requires 1 unit of wood, 1 unit of plastic, and 2 units of aluminum.  A rocker requires 1 unit of wood, 2 units of plastic, and 5 units of aluminum.  The company's profit on a chair is $7 and on a rocker is $12.  The company has available 400 units of wood, 500 units of plastic, and 1450 units of aluminum.	
A)	P = 400x + 500y + 1450z	C)	P = 7x + 12y	
B)	P = 4x + 8y	D)	P = 1x + 2y + 5z	

37.	The graph of the feasible region for a mixture problem is shown below.  Find the point that maximizes the profit function P = 2x + y. 

	A)  (0, 2)    B)  (2, 4)    C)  (4, 1)    D)  (3, 0)	

38.	The graph of the feasible region for a mixture problem is shown below.  Find the point that maximizes the profit function P = x + 4y. 

	A)  (0, 9)    B)  (6, 7)    C)  (7, 3)    D)  (6, 0)	

39.	The graph of the feasible region for a mixture problem is shown below.  Find the point that maximizes the profit function P = 2x + 5y. 

	A)  (0, 5)    B)  (3, 4)    C)  (7, 2)    D)  (9, 0)	

40.	The graph of the feasible region for a mixture problem is shown below.  Find the point that maximizes the profit function P = 3x + 6y. 

	A)  (0, 4)    B)  (3, 3)    C)  (5, 1)    D)  (6, 0)	

41.	The graph of the feasible region for a mixture problem is shown below.  Find the point that maximizes the profit function P = 3x + 6y. 

	A)  (0, 8)    B)  (6, 6)    C)  (10, 2)    D)  (12, 0)	

42.	The graph of the feasible region for a mixture problem is shown below.  Find the point that maximizes the profit function P = 3x + y. 

	A)  (0, 2)    B)  (2, 4)    C)  (4, 1)    D)  (5, 0)	

43.	The simplex algorithm always gives optimal solutions to linear programming problems.	
	A)  True    B)  False	

44.	An optimal solution for a linear programming problem will always occur at a corner point of the feasible region.	
	A)  True    B)  False	

45.	Any linear programming problem has at most two products.	
	A)  True    B)  False	

46.	An optimal production policy for a linear programming mixture problem may eliminate one product.	
	A)  True    B)  False	

47.	The graph of the inequality 2x + 7y £ 10 is a straight line.	
	A)  True    B)  False	

48.	The ordered pair (200, 400) satisfies the inequality x + 2y £ 1500.	
	A)  True    B)  False	

49.	The feasible region for a linear programming mixture problem may have holes in it.	
	A)  True    B)  False	

50.	The feasible region for a linear programming mixture problem with two products is in the first quadrant of the Cartesian plane.	
	A)  True    B)  False	

51.	Suppose the feasible region has four corners, at these points: (0, 0), (5, 0), (0, 4), and (2, 3). If the profit formula is $2x + $3y, what is the maximum profit possible?	
	A)  $12    B)  $13    C)  $14    D)  $15	

52.	Suppose the feasible region has four corners, at these points: (0, 0), (8, 0), (0, 12), and (4, 8). If the profit formula is $2x + $4y, what is the maximum profit possible?	
	A)  $16    B)  $40    C)  $48    D)  $54	

53.	Suppose the feasible region has five corners, at these points: (1, 1), (1, 7), (5, 7), (5, 5), and (4, 3). If the profit formula is $10x + $5y, which point maximizes the profit?	
	A)  (1, 7)    B)  (5, 7)    C)  (5, 5)    D)  (4, 3)	

54.	Suppose the feasible region has five corners, at these points: (1, 1), (1, 7), (5, 7), (5, 5), and (4, 3). If the profit formula is $5x - $2y, which point maximizes the profit?	
	A)  (1, 7)    B)  (5, 7)    C)  (5, 5)    D)  (4, 3)	

55.	Find the graph of the equation 3x + 2y = 6.	
A)		C)		
B)		D)		

56.	Find the graph of the inequality 2x + 6y £ 18.	
A)		C)		
B)		D)		

57.	Find the point of intersection of the lines whose equations are x + 3y = 18 and 2x + y = 11.	
	A)  (3, 5)    B)  (5, 3)    C)  (2, 3)    D)  (3, 2)	

58.	Suppose the feasible region has four corners, at these points: (0, 0), (5, 0), (0, 4), and (2, 3). For which of these profit formulae is the profit maximized, producing a mix of products?	
	A)  $4x + $3y    B)  $3x + $4y    C)  $x – $y    D)  $2x – $y	

59.	Suppose the feasible region has four corners, at these points: (0, 0), (8, 0), (0, 12), and (4, 8). For which of these profit formulae is the profit maximized, producing a mix of products?	
	A)  $5x + $2y    B)  $2x + $5y    C)  $x – $y    D)  $2x – $y	

60.	Consider the feasible region identified by the inequalities below. 
 
  x ³ 0; y ³ 0; x + y £ 4; x + 3y £  6    
 
Which point is not a corner of the region?	
	A)  (0, 2)    B)  (0, 4)    C)  (3, 1)    D)  (4, 0)	

61.	Which of these methods for the transportation problem produces a feasible solution?	
A)	Stepping Stone Method (SSM)	C)	both SSM and NCR	
B)	Northwest Corner Rule (NCR)	D)	neither SSM nor NCR	

62.	Which of these methods for the transportation problem produces an improved solution?	
A)	Stepping Stone Method (SSM)	C)	both SSM and NCR	
B)	Northwest Corner Rule (NCR)	D)	neither SSM nor NCR	

Answer Key

1.	c	
2.	a	
3.	b	
4.	b	
5.	d	
6.	a	
7.	c	
8.	d	
9.	B	
10.	C	
11.	A	
12.	A	
13.	a	
14.	d	
15.	b	
16.	a	
17.	A	
18.	C	
19.	C	
20.	A	
21.	D	
22.	A	
23.	B	
24.	b	
25.	a	
26.	b	
27.	d	
28.	B	
29.	C	
30.	B	
31.	D	
32.	A	
33.	C	
34.	D	
35.	A	
36.	C	
37.	C	
38.	A	
39.	B	
40.	B	
41.	B	
42.	D	
43.	A	
44.	A	
45.	B	
46.	A	
47.	B	
48.	A	
49.	B	
50.	A	
51.	B	
52.	C	
53.	B	
54.	C	
55.	A	
56.	C	
57.	A	
58.	B	
59.	A	
60.	B	
61.	C	
62.	A