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

2.	Sketch the graph of the equation 4x + 3y = 24.	

3.	Sketch the graph of the equation x + 3y = 9.	

4.	Sketch the graph of the inequality 2x + 4y £ 12.	

5.	Sketch the graph of the inequality 5x + y £ 15.	

6.	Sketch the graph of the inequality 4x + 6y £ 12.	

7.	Find the point of intersection for the lines represented by the equations 2x + 4y = 12 and 3x + y = 13. 	

8.	Find the point of intersection for the lines represented by the equations x + 5y = 28 and 4x + 3y = 27. 	

9.	Find the point of intersection for the lines represented by the equations 3x + 2y = 14 and 4x + 5y = 28. 	

10.	Find the point of intersection for the lines represented by the equations 2x + 7y = 61 and 3x + 4y = 46. 	

11.	With the given the constraints for the following linear programming mixture problem, graph the feasible region.
	
2 x + 3y £ 1800	

x ³ 0	

y ³ 0	

12.	With the given the constraints for the following linear programming mixture problem, graph the feasible region.
	
x + 6y £ 110	

x ³ 0	

y ³ 0	

13.	With the given the constraints for the following linear programming mixture problem, graph the feasible region.
	
2 x + 3y £ 180	

5 x + 2y £ 230	

x ³ 0	

y ³ 0	

14.	With the given the constraints for the following linear programming mixture problem, graph the feasible region.
	
x + 6y £ 1100	

4 x + y £ 2100	

x ³ 0	

y ³ 0	

15.	With the given the constraints for the following linear programming mixture problem, graph the feasible region.
	
x + 3y £ 23	

3 x + 7y £ 50	

x ³ 2	

y ³ 4	

16.	Find the constraint inequalities and the profit formula for this linear programming mixture problem:  Toni has a small business producing dried floral wreaths and table arrangements.  Each wreath takes 7 hours to produce, uses 12 stems of flowers, and earns a profit of $23.  Each table arrangement takes 5 hours to produce, uses 20 stems of flowers, and earns a profit of $26.  Toni can work no more than 30 hours per week and has a steady supply of 100 stems of dried flowers per week.  What should Toni's production schedule be to optimize profit?	

17.	Find the constraint inequalities and the profit formula for this linear programming mixture problem:  The Acme construction company builds two types of houses.  Plan A requires 200 man-hours for rough construction and 70 man-hours for finish work.  Plan B requires 300 man-hours for rough construction and 50 man-hours for finish work.  Acme has carpenters available to provide up to 900 hours of rough construction per month and 260 hours of finish work per month.  If Acme clears $7000 profit on a plan A house and $8000 profit on a plan B house, how should they schedule production to maximize profit?	

18.	Write the constraint inequalities and the profit formula for this linear programming mixture problem:  The "Dig-The-Pig" ham shop sells regular and special ham and cheese sandwiches.  The regular is made of 5 oz. of meat, 0.7 oz. of cheese, and requires 3 minutes of preparation time.  The special is made of 7 oz. of meat, 0.6 oz. of cheese, and requires 11 minutes of preparation time.  The profit on a regular sandwich is 10 cents while the profit on a special sandwich is 40 cents.  If "Dig-The-Pig" has 350 oz. of meat, 42 oz. of cheese and 330 minutes of preparation time available each lunch period, how many of each type sandwich should they try to sell to maximize profit?	

19.	Write the constraint inequalities and the profit formula for this linear programming mixture problem:  Amazin' Raisin Baking Co. makes both raisin cake and raisin pie.  A batch of raisin cakes requires 5 lbs. of flour, 2 lbs. of sugar, and 1 lb. of raisins.  A batch of raisin pies requires 2 lbs. of flour, 3 lbs. of sugar, and 4 lbs. of raisins.  There are 165 lbs. of flour, 110 lbs. of sugar, and 120 lbs. of raisins available each week.  Standing orders require at least five batches of raisin cakes and eight batches of raisin pies per week.  If profit on a batch of raisin cakes is $35 and profit on a batch of raisin pies is $40, how many batches of each should be made per week to maximize profit?	

20.	Solve this linear programming mixture problem:  Kim and Lynn produce pottery vases and bowls.  A vase requires 25 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 160 oz. of glaze available.  The profit on one vase is $5 and the profit on one bowl is $3.	

21.	Solve this linear programming 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 26 hours available in the testing and shipping department.  What should the production schedule be to maximize profit?	

22.	Explain what the real world implications are if the optimal production policy for a linear programming mixture problem is represented by a point on the x axis of the Cartesian plane. 	

23.	Describe the shape of the feasible region for linear programming mixture problems with two products. 	

24.	Name two alternatives to the graphical approach for solving linear programming problems. 	

25.	Explain why a linear programming mixture problem might have minimum constraints other than zero. 	

26.	Explain why the feasible region for a linear programming mixture problem must be in the first quadrant of the Cartesian plane. 	

27.	Graph the feasible region identified by the inequalities:

     x + y £ 5    
     x + 2y £  8    
     x ³ 0, y ³ 0	

28.	Graph the feasible region identified by the inequalities:
 
     2x + 5y £ 70    
     5x + y £ 60    
     x ³ 0, y ³ 0	

29.	Find the maximum value of P, where P = 10x + 100y subject to the constraints:
 
 x ³ 0; y ³ 0 ; 2x + 5y £ 20; 2x + y £ 12 	

30.	Find the maximum value of P, where P = 3x + 4y subject to the constraints:
 
 x ³ 0; y ³ 0 ; x + 2y £ 8; x + y £ 5 	

31.	Find the point of intersection of the lines whose equations are 6x + 5y = 80 and 2x + y = 20. 	

32.	What type of solution does the Northwest Corner Rule produce for the transportation problem? 	

33.	What type of solution does the Stepping Stone Method produce for the transportation problem? 	

34.	A solution for the transportation problem is optimal when the empty cells have what property? 	

Answer Key

1.	
	
2.	
	
3.	
	
4.	
	
5.	
	
6.	
	
7.	(4,1)	
8.	(3,5)	
9.	(2,4)	
10.	(6,7)	
11.	
	
12.	
	
13.	
	
14.	
	
15.	
	
16.	x is the number of wreaths produced and y is the number of table arrangements produced.
Constraint inequalities:	
7x + 5y £ 30	

2x + 20y £ 100	

x ³ 0	

y ³ 0	

Profit formula: 	
23x + 26y = P	

17.	x is the number of Plan A homes produced and y is the number of Plan B homes.
Constraint inequalities:	
200x + 300y £ 900	

70x + 50y £ 260	

x ³ 0	

y ³ 0	

Profit formula:	
P = 7000x + 8000y	

18.	x is the number of regular sandwiches made and y is the number of special sandwiches made.
Constraint inequalities:	
5x + 7y £ 350	

.7x + .6y £ 42	

3x + 11y £ 330	

x ³ 0	

y ³ 0	

Profit formula:	
P = 10x + 40y	

19.	x is the number of cakes and y is the number of pies.
Constraint inequalities:	
5x + 2y £ 165	

2x + 3y £ 110	

1x + 4y £ 120	

x ³ 5	

y ³ 8	

Profit formula:	
P = 35x + 40y	

20.	x is the number of vases and y is the number of bowls.
Constraint inequalities:	
25x + 20y £ 500	

5x + 10y £ 160	

x ³ 0	

y ³ 0	

Profit formula:	
5x + 3y = P 	

21.	x is the number of receivers and y is the number of CD players.
Constraint inequalities:	
8x + 15y £ 160	

1x + 2y  £ 26	

x ³ 0	

y ³ 0	

Profit formula:	
P = 30x + 50y	

 
 
Maximum profit occurs at (20,0).  Make 20 receivers and no CD players.	
22.	If the optimal production policy for a linear programming problem is represented by a point on the x axis, it means that profit is optimized by making only one type of product.  In order to maximize profit, the company needs to drop from its product line the item represented by the variable y.	
23.	A linear programming mixture problem with two products has a feasible region that is represented by a convex polygon in Quadrant I of the Cartesian plane.	
24.	Two alternatives to the graphical approach are the Simplex algorithm and Karmarkar's algorithm.	
25.	Answers may vary.  A linear programming problem might have minimum constraints other than 0 if the company had standing orders for some number of its products that must be filled and cannot be canceled.	
26.	In a linear programming mixture problem, x and y represent the number of two different products to be made.  Since it is impossible to produce negative numbers of products, the feasible region must be in Quadrant I.	
27.	
	
28.	
	
29.	The maximum occurs at the point (6, 0), where P = 600.	
30.	The maximum occurs at the point (2, 3), where P = 18.	
31.	(5, 10)	
32.	feasible, but probably not optimal	
33.	feasible, more efficient, but possibly not optimal	
34.	Every empty cell's indicator value is positive or zero