7.6 Linear Programming - OTHS Precaltompkinsprecal.weebly.com/uploads/3/7/9/2/37924299/... · A two-dimensional linear programming problem consists of a linear ... solutions x y z

Linear Programming: A Graphical ApproachMany applications in business and economics involve a process calledoptimization, in which you are asked to find the minimum or maximum of aquantity. In this section, you will study an optimization strategy called linearprogramming.

A two-dimensional linear programming problem consists of a linearobjective function and a system of linear inequalities called constraints. Theobjective function gives the quantity that is to be maximized (or minimized), andthe constraints determine the set of feasible solutions. For example, suppose youare asked to maximize the value of

Objective function

subject to a set of constraints that determines the shaded region in Figure 7.31.

FIGURE 7.31

Because every point in the shaded region satisfies each constraint, it is not clearhow you should find the point that yields a maximum value of Fortunately, itcan be shown that if there is an optimal solution, it must occur at one of thevertices. This means that you can find the maximum value of z by testing z at eachof the vertices.

Some guidelines for solving a linear programming problem in two variables arelisted at the top of the next page.

z.

Feasiblesolutions

x

y

z � ax � by

552 Chapter 7 Systems of Equations and Inequalities

What you should learn• Solve linear programming

problems.

• Use linear programming tomodel and solve real-life problems.

Why you should learn itLinear programming is oftenuseful in making real-life economic decisions. For example, Exercise 44 on page560 shows how you can deter-mine the optimal cost of a blendof gasoline and compare it withthe national average.

Linear Programming

Tim Boyle/Getty Images

7.6

Optimal Solution of a Linear Programming ProblemIf a linear programming problem has a solution, it must occur at a vertex ofthe set of feasible solutions. If there is more than one solution, at least oneof them must occur at such a vertex. In either case, the value of the objectivefunction is unique.

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Solving a Linear Programming Problem

Find the maximum value of

Objective function

subject to the following constraints.

Constraints

SolutionThe constraints form the region shown in Figure 7.32. At the four vertices of thisregion, the objective function has the following values.

At At At Maximum value of z

At

So, the maximum value of is 8, and this occurs when and

Now try Exercise 5.

In Example 1, try testing some of the interior points in the region. You willsee that the corresponding values of are less than 8. Here are some examples.

At At

To see why the maximum value of the objective function in Example 1 mustoccur at a vertex, consider writing the objective function in slope-intercept form

Family of lines

where is the -intercept of the objective function. This equation represents afamily of lines, each of slope Of these infinitely many lines, you want theone that has the largest -value while still intersecting the region determined bythe constraints. In other words, of all the lines whose slope is you want theone that has the largest -intercept and intersects the given region, as shown inFigure 7.33. From the graph you can see that such a line will pass through one(or more) of the vertices of the region.

y�

32,

z�

32.

yz�2

y � �3

2x �

z

2

z � 3�12� � 2�3

2� �92�1

2, 32�:z � 3�1� � 2�1� � 5�1, 1�:

z

y � 1.x � 2z

z � 3�0� � 2�2� � 4�0, 2�:z � 3�2� � 2�1� � 8�2, 1�:z � 3�1� � 2�0� � 3�1, 0�:z � 3�0� � 2�0� � 0�0, 0�:

x ≥ 0y ≥ 0

x � 2y ≤ 4x � y ≤ 1

�z � 3x � 2y

Section 7.6 Linear Programming 553

It is important that students immediatelybecome familiar with the terminology ofthis section. Emphasize that linearprogramming is powerful in that itidentifies quickly those few points outof many that can be easily tested to findthe point that gives the maximum orminimum value.

2 3

1

3

2

4

(0, 2)

(2, 1)

(1, 0)(0, 0)

x + 2y = 4

x − y = 1

x = 0

y = 0

x

y

FIGURE 7.32

2 31

1

2

3

4

x

y

z = 2

z = 4

z = 6

z = 8

FIGURE 7.33

Solving a Linear Programming Problem1. Sketch the region corresponding to the system of constraints. (The points

inside or on the boundary of the region are feasible solutions.)

2. Find the vertices of the region.

3. Test the objective function at each of the vertices and select the values ofthe variables that optimize the objective function. For a bounded region,both a minimum and a maximum value will exist. (For an unboundedregion, if an optimal solution exists, it will occur at a vertex.)

Example 1

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The next example shows that the same basic procedure can be used to solvea problem in which the objective function is to be minimized.

Minimizing an Objective Function

Find the minimum value of

Objective function

where and subject to the following constraints.

Constraints

SolutionThe region bounded by the constraints is shown in Figure 7.34. By testing theobjective function at each vertex, you obtain the following.

At Minimum value of z

At

At

At

At

At

So, the minimum value of is 14, and this occurs when and

Now try Exercise 13.

Maximizing an Objective Function


Objective function


Constraints

SolutionThis linear programming problem is identical to that given in Example 2 above,except that the objective function is maximized instead of minimized. Using thevalues of at the vertices shown above, you can conclude that the maximumvalue of is

and occurs when and


y � 3.x � 6

z � 5�6� � 7�3� � 51

zz

2x3x

�x2x

� 3y ≥ 6� y ≤ 15� y ≤ 4� 5y ≤ 27

�y ≥ 0,x ≥ 0

z � 5x � 7y

y � 2.x � 0z

z � 5�3� � 7�0� � 15�3, 0�:

z � 5�5� � 7�0� � 25�5, 0�:

z � 5�6� � 7�3� � 51�6, 3�:

z � 5�1� � 7�5� � 40�1, 5�:

z � 5�0� � 7�4� � 28�0, 4�:

z � 5�0� � 7�2� � 14�0, 2�:

2x3x

�x2x

� 3y ≥� y ≤� y ≤� 5y ≤

6154

27

�y ≥ 0,x ≥ 0

z � 5x � 7y


1 2 3 4 5 6

1

2

3

4

5

(0, 2)

(0, 4)

(1, 5)

(6, 3)

(3, 0) (5, 0)

y

x

FIGURE 7.34

Example 2

Example 3

Historical NoteGeorge Dantzig (1914– ) was the first to propose thesimplex method, or linearprogramming, in 1947. Thistechnique defined the stepsneeded to find the optimalsolution to a complex multivariable problem.

Edw

ard

W.S

ou

za/N

ews

Serv

ice/

Stan

ford

Un

iver

sity

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It is possible for the maximum (or minimum) value in a linear programmingproblem to occur at two different vertices. For instance, at the vertices of theregion shown in Figure 7.35, the objective function

Objective function

has the following values.

At At At Maximum value of z

At Maximum value of z

At

In this case, you can conclude that the objective function has a maximum valuenot only at the vertices and it also has a maximum value (of 12) atany point on the line segment connecting these two vertices. Note that theobjective function in slope-intercept form has the same slope asthe line through the vertices and

Some linear programming problems have no optimal solutions. This canoccur if the region determined by the constraints is unbounded. Example 4illustrates such a problem.

An Unbounded Region


Objective function


Constraints

SolutionThe region determined by the constraints is shown in Figure 7.36. For thisunbounded region, there is no maximum value of To see this, note that thepoint lies in the region for all values of Substituting this point intothe objective function, you get

By choosing to be large, you can obtain values of that are as large as youwant. So, there is no maximum value of However, there is a minimum value of

At At Minimum value of z

At

So, the minimum value of is 10, and this occurs when and


y � 1.x � 2z

z � 4�4� � 2�0� � 16�4, 0�:z � 4�2� � 2�1� � 10�2, 1�:z � 4�1� � 2�4� � 12�1, 4�:

z.z.

zx

z � 4�x� � 2�0� � 4x.

x ≥ 4.�x, 0�z.

x � 2y ≥ 43x � y ≥ 7

�x � 2y ≤ 7 �

y ≥ 0,x ≥ 0

z � 4x � 2y

�5, 1�.�2, 4�y � �x �

12 z

�5, 1�;�2, 4�

z � 2�5� � 2�0� � 10�5, 0�:z � 2�5� � 2�1� � 12�5, 1�:z � 2�2� � 2�4� � 12�2, 4�:z � 2�0� � 2�4� � 18�0, 4�:z � 2�0� � 2�0� � 10�0, 0�:

z � 2x � 2y


1 2 3 4 5

1

2

3

4(0, 4)

(2, 4)

(0, 0) (5, 0)

(5, 1)

z =12 forany pointalong thislinesegment.

x

y

FIGURE 7.35

1 2 3 4 5

1

2

3

4

5

(1, 4)

(2, 1)

(4, 0)x

y

FIGURE 7.36

Example 4

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ApplicationsExample 5 shows how linear programming can be used to find the maximumprofit in a business application.

Optimal Profit

A candy manufacturer wants to maximize the profit for two types of boxedchocolates. A box of chocolate covered creams yields a profit of $1.50 per box,and a box of chocolate covered nuts yields a profit of $2.00 per box. Market testsand available resources have indicated the following constraints.

1. The combined production level should not exceed 1200 boxes per month.

2. The demand for a box of chocolate covered nuts is no more than half thedemand for a box of chocolate covered creams.

3. The production level for chocolate covered creams should be less than or equalto 600 boxes plus three times the production level for chocolate covered nuts.

SolutionLet be the number of boxes of chocolate covered creams and let be thenumber of boxes of chocolate covered nuts. So, the objective function (for thecombined profit) is given by

Objective function

The three constraints translate into the following linear inequalities.

1.

2.

3.

Because neither nor can be negative, you also have the two additionalconstraints of and Figure 7.37 shows the region determined by theconstraints. To find the maximum profit, test the values of at the vertices of theregion.

Maximum profit

So, the maximum profit is $2000, and it occurs when the monthly productionconsists of 800 boxes of chocolate covered creams and 400 boxes of chocolatecovered nuts.


In Example 5, if the manufacturer improved the production of chocolatecovered creams so that they yielded a profit of $2.50 per unit, the maximum profitcould then be found using the objective function By testing thevalues of at the vertices of the region, you would find that the maximum profit was $2925 and that it occurred when and y � 150.x � 1050

PP � 2.5x � 2y.

At �600, 0�: P � 1.5�600� � 2�0� � 900 At �1050, 150�: P � 1.5�1050� � 2�150� � 1875 At �800, 400�: P � 1.5�800� � 2�400� � 2000

At �0, 0�: P � 1.5�0� � 2�0� � 0

Py ≥ 0.x ≥ 0yx

x � 3y ≤ 600 x ≤ 600 � 3y

�x � 2y ≤ 0 y ≤ 12x

x � y ≤ 1200 x � y ≤ 1200

P � 1.5x � 2y.

yx


Example 5

Activities

1. Sketch the region determined bythe constraints and

Then find the minimumand maximum values of the objectivefunction and wherethey occur, subject to the constraints.Answer: Minimum: 0 at (0, 0);

Maximum: 15 at (0, 3)

2. Find the maximum value of theobjective function ,and where it occurs, subject to theconstraints

and

Answer: Maximum: 37 at (2, 5)

3. Find the maximum value, and whereit occurs, of where

and subject to the constraints and

Answer: There is no maximum valueof z.

� x � y ≤ 10.3x � y ≥ 14,x � 2y ≥ 8,

y ≥ 0x ≥ 0z � 2x � 5y

y ≥ 0.x ≥ 0,x � 4y ≤ 22,3x � 2y ≤ 16,

z � 6x � 5y

x

y

1−1 2 3 4

2

1

−1

3

4

(0, 0) (2, 0)

(0, 3)

z � 4x � 5y,

3x � 2y ≤ 6.y ≥ 0,x ≥ 0,

Boxes of chocolatecovered creams

Box

es o

f ch

ocol

ate

cove

red

nuts

x

y

1200800400

100

200

300

400(800, 400)

(1050, 150)

(600, 0)(0, 0)

Maximum Profit

FIGURE 7.37

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Optimal Cost

The liquid portion of a diet is to provide at least 300 calories, 36 units of vitaminA, and 90 units of vitamin C. A cup of dietary drink X costs $0.12 and provides60 calories, 12 units of vitamin A, and 10 units of vitamin C. A cup of dietarydrink Y costs $0.15 and provides 60 calories, 6 units of vitamin A, and 30 unitsof vitamin C. How many cups of each drink should be consumed each day toobtain an optimal cost and still meet the daily requirements?

SolutionAs in Example 9 in Section 7.5, let be the number of cups of dietary drink Xand let be the number of cups of dietary drink Y.

Constraints

The cost is given by Objective function

The graph of the region corresponding to the constraints is shown in Figure 7.38.Because you want to incur as little cost as possible, you want to determine theminimum cost. To determine the minimum cost, test at each vertex of theregion.

At

At

At Minimum value of C

At

So, the minimum cost is $0.66 per day, and this occurs when 3 cups of drink Xand 2 cups of drink Y are consumed each day.


C � 0.12�9� � 0.15�0� � 1.08�9, 0�:

C � 0.12�3� � 0.15�2� � 0.66�3, 2�:

C � 0.12�1� � 0.15�4� � 0.72�1, 4�:

C � 0.12�0� � 0.15�6� � 0.90�0, 6�:

C

C � 0.12x � 0.15y.C

For calories:For vitamin A:For vitamin C:

60x � 60y ≥ 30012x � 6y ≥ 3610x � 30y ≥ 90

x ≥ 0y ≥ 0

�y

x


Example 6

Alternative Writing AboutMathematics

Explain what difficulties you mightencounter with each set of linearprogramming constraints.

a.Constraints

b.

Constraints�2x

xx

�

�

yy

y

> 11> 0< 4< 0

�x � y <3x � y >

�4x � y >

09

�2

2 4 6 8 10

2

4

6

8

(0, 6)

(1, 4)

(3, 2)

(9, 0)

y

x

FIGURE 7.38

W RITING ABOUT MATHEMATICS

Creating a Linear Programming Problem Sketch the region determined by thefollowing constraints.

Constraints

Find, if possible, an objective function of the form that has amaximum at each indicated vertex of the region.

a. b. c. d.

Explain how you found each objective function.

�0, 0��5, 0��2, 3��0, 4�

z � ax � by

�x � 2y ≤ 8

x � y ≤ 5

x ≥ 0

y ≥ 0

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In Exercises 1–12, find the minimum and maximum valuesof the objective function and where they occur, subject tothe indicated constraints. (For each exercise, the graph ofthe region determined by the constraints is provided.)

1. Objective function: 2. Objective function:

Constraints: Constraints:


Constraints: Constraints:(See Exercise 1.) (See Exercise 2.)



FIGURE FOR 5 FIGURE FOR 6







z � 15x � 20yz � 25x � 30y

400

400

800 (0, 800)(0, 400)

(900, 0)

(450, 0)x

y

(0, 45)(30, 45)

(60, 20)

(60, 0)(0, 0)

20 40 60

20

40

60

x

y

8x � 9y ≥ 3600

8x � 9y ≤ 72005x � 6y ≤ 420

y ≥ 0 0 ≤ y ≤ 45

x ≥ 0 0 ≤ x ≤ 60

z � 25x � 35yz � 10x � 7y

z � 2x � yz � 5x � 0.5y

(4, 3)(0, 4)

1 2 53 4

2

1

3

4

5

(0, 2)

(3, 0)x

y

(3, 4)(0, 5)

1 2 3 4 5

2

1

3

4

5

(0, 0)(4, 0)

x

y

x � 4y ≤ 16 4x � y ≤ 16

3x � y ≤ 9 x � 3y ≤ 15

2x � 3y ≥ 6 y ≥ 0

x ≥ 0 x ≥ 0

z � 4x � 5yz � 3x � 2y

z � 7x � 3yz � 3x � 8y

(0, 0) (2, 0)

(0, 4)

−1 1 2 3

2

3

4

y

x1 2 3 4 5 6

123456

(0, 5)

(5, 0)(0, 0)x

y

2x � y ≤ 4 x � y ≤ 5

y ≥ 0x y ≥ 0

x ≥ 0 x ≥ 0

z � 2x � 8yz � 4x � 3y


Exercises 7.6

VOCABULARY CHECK: Fill in the blanks.

1. In the process called ________, you are asked to find the maximum or minimum value of a quantity.

2. One type of optimization strategy is called ________ ________.

3. The ________ function of a linear programming problem gives the quantity that is to be maximized or minimized.

4. The ________ of a linear programming problem determine the set of ________ ________.

5. If a linear programming problem has a solution, it must occur at a ________ of the set of feasible solutions.

PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.

Exercise with no feasible solution: 36Exercises with unbounded regions: 17, 19, 34, 44

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In Exercises 13–20, sketch the region determined by theconstraints. Then find the minimum and maximum valuesof the objective function and where they occur, subject tothe indicated constraints.









In Exercises 21–24, use a graphing utility to graph theregion determined by the constraints. Then find theminimum and maximum values of the objective functionand where they occur, subject to the constraints.





In Exercises 25–28, find the maximum value of the objectivefunction and where it occurs, subject to the constraints

and

25.

26.

27.

28.

In Exercises 29–32, find the maximum value of the objectivefunction and where it occurs, subject to theconstraints and

29.

30.

31.

32.

In Exercises 33–38, the linear programming problem has anunusual characteristic. Sketch a graph of the solutionregion for the problem and describe the unusual charac-teristic. Find the maximum value of the objective functionand where it occurs.







2x � 2y ≤ 4 2x � y ≤ 4

x � 2y ≤ 4 x � y ≤ 1

y ≥ 0 y ≥ 0

x ≥ 0 x ≥ 0

z � x � 2yz � 3x � 4y

�3x � y ≥ 3x � y ≤ 7

�x � y ≤ 0 x ≤ 10

y ≥ 0 y ≥ 0

x ≥ 0 x ≥ 0

z � x � yz � �x � 2y

�x � 2y ≤ 4 5x � 2y ≤ 10

�x � y ≤ 1 3x � 5y ≤ 15

y ≥ 0 y ≥ 0

x ≥ 0 x ≥ 0

z � x � yz � 2.5x � y

z � 4x � y

z � 4x � 5y

z � 2x � 4y

z � x � 5y

2x � 2y ≤ 21.x � y ≤ 18,x � 4y ≤ 20,y ≥ 0,x ≥ 0,

z � 3x � y

z � x � y

z � 5x � y

z � 2x � y

4x � 3y ≤ 30.3x � y ≤ 15,y ≥ 0,x ≥ 0,

z � yz � x � 4y

4x � y ≤ 48

2x � y ≤ 28 2x � 3y ≥ 72

2x � 3y ≤ 60 x � 2y ≤ 40

y ≥ 0 y ≥ 0

x ≥ 0 x ≥ 0

z � xz � 4x � y

z � 2x � yz � 2x � 7y

x � 2y ≤ 6 3x � 5y ≥ 30

2x � 2y ≤ 10 x � 5y ≥ 8

y ≥ 0 y ≥ 0

x ≥ 0 x ≥ 0

z � 4x � 5yz � 4x � 5y

z � 7x � 2yz � 9x � 24y

x �12 y ≤ 4 2x � 5y ≤ 10

x y ≥ 0x y ≥ 0

x x ≥ 0x x ≥ 0

z � 7x � 8yz � 6x � 10y

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39. Optimal Profit A manufacturer produces two models ofbicycles. The times (in hours) required for assembling,painting, and packaging each model are shown in the table.

The total times available for assembling, painting, andpackaging are 4000 hours, 4800 hours, and 1500 hours,respectively. The profits per unit are $45 for model A and$50 for model B. What is the optimal production level foreach model? What is the optimal profit?

40. Optimal Profit A manufacturer produces two models ofbicycles. The times (in hours) required for assembling,painting, and packaging each model are shown in the table.

The total times available for assembling, painting, andpackaging are 4000 hours, 2500 hours, and 1500 hours,respectively. The profits per unit are $50 for model A and$52 for model B. What is the optimal production level foreach model? What is the optimal profit?

41. Optimal Profit A merchant plans to sell two models ofMP3 players at costs of $250 and $300. The $250 modelyields a profit of $25 per unit and the $300 model yields aprofit of $40 per unit. The merchant estimates that the totalmonthly demand will not exceed 250 units. The merchantdoes not want to invest more than $65,000 in inventory forthese products. What is the optimal inventory level for eachmodel? What is the optimal profit?

42. Optimal Profit A fruit grower has 150 acres of landavailable to raise two crops, A and B. It takes 1 day to triman acre of crop A and 2 days to trim an acre of crop B, andthere are 240 days per year available for trimming. It takes0.3 day to pick an acre of crop A and 0.1 day to pick anacre of crop B, and there are 30 days available for picking.The profit is $140 per acre for crop A and $235 per acre forcrop B. What is the optimal acreage for each fruit? What isthe optimal profit?

43. Optimal Cost A farming cooperative mixes two brandsof cattle feed. Brand X costs $25 per bag and contains twounits of nutritional element A, two units of element B, andtwo units of element C. Brand Y costs $20 per bag andcontains one unit of nutritional element A, nine units ofelement B, and three units of element C. The minimumrequirements of nutrients A, B, and C are 12 units, 36 units,and 24 units, respectively. What is the optimal number ofbags of each brand that should be mixed? What is theoptimal cost?

45. Optimal Revenue An accounting firm has 900 hours ofstaff time and 155 hours of reviewing time available eachweek. The firm charges $2500 for an audit and $350 for atax return. Each audit requires 75 hours of staff time and 10 hours of review time. Each tax return requires 12.5 hours of staff time and 2.5 hours of review time. Whatnumbers of audits and tax returns will yield an optimal rev-enue? What is the optimal revenue?

46. Optimal Revenue The accounting firm in Exercise 45lowers its charge for an audit to $2000. What numbers ofaudits and tax returns will yield an optimal revenue? Whatis the optimal revenue?


Process Hours, Hours,model A model B

Assembling 2 2.5

Painting 4 1

Packaging 1 0.75

44. Optimal Cost According to AAA (AutomobileAssociation of America), on January 24, 2005, thenational average price per gallon for regular unleaded(87-octane) gasoline was $1.84, and the price forpremium unleaded (93-octane) gasoline was $2.03.

(a) Write an objective function that models the cost ofthe blend of mid-grade unleaded gasoline (89-octane).

(b) Determine the constraints for the objective functionin part (a).

(c) Sketch a graph of the region determined by theconstraints from part (b).

(d) Determine the blend of regular and premiumunleaded gasoline that results in an optimal cost ofmid-grade unleaded gasoline.

(e) What is the optimal cost?

(f) Is the cost lower than the national average of $1.96per gallon for mid-grade unleaded gasoline?

Model It

Process Hours, Hours,model A model B

Assembling 2.5 3

Painting 2 1

Packaging 0.75 1.25

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47. Investment Portfolio An investor has up to $250,000 toinvest in two types of investments. Type A pays 8%annually and type B pays 10% annually. To have a well-balanced portfolio, the investor imposes the followingconditions. At least one-fourth of the total portfolio is to beallocated to type A investments and at least one-fourth ofthe portfolio is to be allocated to type B investments. Whatis the optimal amount that should be invested in each typeof investment? What is the optimal return?

48. Investment Portfolio An investor has up to $450,000 toinvest in two types of investments. Type A pays 6%annually and type B pays 10% annually. To have a well-balanced portfolio, the investor imposes the followingconditions. At least one-half of the total portfolio is to beallocated to type A investments and at least one-fourth ofthe portfolio is to be allocated to type B investments. Whatis the optimal amount that should be invested in each typeof investment? What is the optimal return?

Synthesis

True or False? In Exercises 49 and 50, determine whetherthe statement is true or false. Justify your answer.

49. If an objective function has a maximum value at thevertices and you can conclude that it also hasa maximum value at the points and

50. When solving a linear programming problem, if theobjective function has a maximum value at more than onevertex, you can assume that there are an infinite number ofpoints that will produce the maximum value.

In Exercises 51 and 52, determine values of t such that theobjective function has maximum values at the indicatedvertices.

51. Objective function: Constraints:

(a)

(b)

52. Objective function: Constraints:

(a)

(b)

Think About It In Exercises 53–56, find an objectivefunction that has a maximum or minimum value at theindicated vertex of the constraint region shown below.(There are many correct answers.)

53. The maximum occurs at vertex



56. The minimum occurs at vertex

Skills Review

In Exercises 57–60, simplify the complex fraction.

57. 58.

59. 60.

In Exercises 61–66, solve the equation algebraically. Roundthe result to three decimal places.

61.

62.

63.

64.

65.

66.

In Exercises 67 and 68, solve the system of linear equationsand check any solution algebraically.

67.

68.

�7x � 3y � 5z � �284x � 4z � �167x � 2y � z � 0

��x � 2y � 3z � �232x � 6y � z � 17

5y � z � 8

ln�x � 9�2 � 2

7 ln 3x � 12

150e �x � 4

� 75

8�62 � e x�4� � 192

e 2x � 10e x � 24 � 0

e 2x � 2e x � 15 � 0

� 1

x � 1�

1

2�� 3

2x 2 � 4x � 2�� 4

x 2 � 9�

2

x � 2�� 1

x � 3�

1

x � 3�

�1 �2

x��x �

4

x��9

x��6

x� 2�

C.

C.

B.

A.

−1 1 6432

123

56

A(0, 4) B(4, 3)

C(5, 0)x

y

�0, 2��2, 1�

x � y ≤ 1

x � 2y ≤ 4

y ≥ 0

x ≥ 0z � 3x � t y

�3, 4��0, 5�

4x � y ≤ 16

x � 3y ≤ 15

y ≥ 0

x ≥ 0z � 3x � t y

�7.8, 3.2�.�4.5, 6.5��8, 3�,�4, 7�

333202_0706.qxd 12/5/05 9:46 AM Page 561

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7.6 Linear Programming - OTHS Precaltompkinsprecal.weebly.com/uploads/3/7/9/2/37924299/... · A two-dimensional linear programming problem consists of a linear ... solutions x y z