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Unit 3: Systems of Equations and Linear Programming

Solving Systems of Equations

1. Graphing

2. Substitution

3. Elimination

Graphing

Substitution

Elimination

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Systems with Three Variables

3 Patterns for a solution:

1.

2.

3.

**Use elimination or substitution methods to get a system with two variables and then follow the rules for two

variable systems

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Systems of Equations Homework

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17) Critical Thinking Question:

Write your own system of equations with the solution (2, 1, 0)

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1. Two small pitchers and one large pitcher can hold 8 cups of water. One large pitcher minus one small pitcher

constitutes 2 cups of water. How many cups of water can each pitcher hold?

2. A test has twenty questions worth 100 points. The test consists of True/False questions worth 3 points each and

multiple choice questions worth 11 points each. How many multiple choice questions are on the test?

3. Margie is responsible for buying a week's supply of food and medication for the dogs and cats at a local shelter.

The food and medication for each dog costs twice as much as those supplies for a cat. She needs to feed 164 cats and 24

dogs. Her budget is $4240. How much can Margie spend on each dog for food and medication?

4. Bill and Steve decide to spend the afternoon at an amusement park enjoying their favorite activities, the water

slide and the gigantic Ferris wheel. Their tickets are stamped each time they slide or ride. At the end of the afternoon

they have the following tickets:

How much does it cost to ride the Ferris Wheel?

How much does it cost to slide on the Water Slide?

5. The equations 5x + 2y = 48 and 3x + 2y = 32 represent the money collected from school concert tickets sales

during two class periods. If x represents the cost for each adult ticket and y represents the cost for each student ticket,

what is the cost for each adult ticket?

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6. Jane is asked to buy some chickens, dogs and ducks for her farm. The total number of animals she needs to buy

is 50. She has a budget of $1500 to spend on $20/chicken, $50/dog, and $30/duck. Additionally, the number of chickens

should be equal to that of ducks.

How many of each animal should she buy? Write a system of equations to help you solve this problem.

7. Tommy has to buy some pens, pencils and notebooks for the upcoming semester. He has $102 to spend on $5

pens, $3 pencils, and $9 notebooks. He would like to spend the same amount of money on pens as on pencils. He also

wants the combined number of pens and pencils to be equal to that of notebooks.

How many of each item should he buy? Write a system of equations to help you solve this problem.

8. Rooney’s company decides to buy some keyboards, mouse cursors and PC Cameras. They have a budget of

$1500 to spend on $30 keyboards, $20 mice, and $50 cameras. Additionally, the number of cursors should be equal to

that of keyboards and twice the number of cameras.

How many of each item should he buy? Write a system of equations to help you solve this problem.

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Break- Even Point

You are currently looking to join a gym. After talking with some of your friends you have narrowed it down to Silver’s

Gym, Moon Fitness, and The Companionship Center. You have visited all with friends and you like the facilities all the

same. If Silver’s Gym charges a monthly fee of $25 and additional $2 for each monthly class you sign up to participate in,

Moon Fitness charges a monthly fee of $10 and additional $5 for each monthly class, and The Companionship Center

doesn’t have a monthly fee at all but charges $8 for each monthly class you sign up to participate in. Use this

information to complete the following.

1. Write an equation to model the cost of a monthly membership for all three gyms.

2. Graph all three equations from #1 on a piece of graph paper. Compare the graphs and discuss what information

they give us regarding monthly gym memberships.

3. If you think that you would sign up for 3 classes a month which gym would be the best for you? Why?

a. If you are a class taking machine and think you would sign up for 8 classes which gym would be the best for

you? Why?

4. If you are signing up for 5 classes a month which gym would be best for you? Why?

5. Would your answers change to questions 3-5 if Silver’s Gym ups the per class fee to $6 and The Companionship

Center Drops to $5 per class? Write new equations, graph these new equations on a new piece of paper, and answer

questions 3-5 with the new prices.

6. Reflective Writing: What did you learn from this problem other than the concept of break-even point?

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Break Even Point Practice Name:___________________________

1. Candy bars are sold in a local store for 50 cents each. The factory has $1000 in fixed costs plus 10 cents of

addtional expense for each candy bar made. Assuming all candy bars manufactured can be sold, find the break-even

point.

2. Start up machinery for a company costs $10,000. The cost for the company to produce 1 item is $5.25. If the

company sells their items for $10.50 each, find the break-even point.

3. Find the break-even quantity for a company that makes X number of computer monitors at a cost C given by C =

870 + 70x and receives revenue R given by R = 105x.

4. Imagine your class is going to try to raise $400 by making school T-shirts. There is a $150 set-up charge for the T-

shirt design that you have designed. Once the design is set, it costs $4 for each T-shirt. You feel it is possible to charge

$10 for each T-shirt. How many T-shirts do you have to sell before you break even, i.e., make enough money to cover

your costs?

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5. You decide to market your own custom computer software. You must invest $3255 for computer hardware, and

spend $2.90 to buy and package each disk. If each program sells for $13.75, how many copies must you sell to break

even?

6. Mike and Kim invest $12,000 in equipment to print yearbooks for schools. Each yearbook costs $5 to print and

sells for $15. How many yearbooks must they sell before their business breaks even?

7. Mike and Kim invest $12,000 in equipment to print yearbooks for schools. Each yearbook costs $5 to print and

sells for $30. How many yearbooks must they sell before their business breaks even?

8. At the local ballpark, the team charges $5 for each ticket and expects to make $1,300 in concessions. The team

must pay its players $1,800 and pay all other workers $1,500. Each fan gets a free bat that costs the team $3 per bat.

How many tickets must be sold to break even?

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9. Your little cousin Kate decided to sell Lemonade at a lemonade stand. Your Aunt purchases all the ingredients

and tells your cousin she has to pay her back and can keep anything above the expense cost. You help your little cousin

calculate the cost per cup and find that for each cup it is going to cost $1.75. She decides she wants to sell her cups of

lemonade for $2.50. How many cups of lemonade must she sell to pay her mother back the $32.64 she spent on

supplies?

10. A department store has an abundant amount of non-purchased Halloween costumes left. Since Halloween is

over, they are interested in clearing out the costumes to make room for the Christmas items. The store manager tells

you that he isn’t worried about making a profit and he just wants to break even. If there are 118 costumes left to get rid

of and the purchase cost of the costumes was $2,456.92. What should you sell each costume for?

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Linear Programming

Minimize � = 4� + 2�

� � + � ≥ 74� + 3� ≥ 240 ≤ � ≤ 10,0 ≤ � ≤ 10

Minimize � = 5� + 4�

� 3� + 4� ≥ 32� + 4� ≥ 240 ≤ � ≤ 12,0 ≤ � ≤ 15

Minimize � = 4� + 3�

� 2� + � ≥ 82� + 3� ≥ 16� + 3� ≥ 11

� ≤ 20, � ≤ 20

Maximize � = 6� + 5�

�2� + 3� ≤ 277� + 3� ≤ 42� > 0, � > 0

Maximize � = 6� + 7�

� � + 2� ≤ 165� + 3� ≤ 45� > 0, � > 0

Maximize � = 3� + 4�

� 2� + � ≤ 102� + 3� ≤ 18� − � ≤ 2� ≥ 0, � ≥ 0

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AMT Linear Programming Example

Roger is making hand-tooled leather belts and wallets to sell at a craft fair. He can bring no more than 20 items to the

fair. Each belt takes 8 hours to make and sells for $45. Each wallet takes 2 hours to make and sells for $15. He has up

to 70 hours to spend working on the leather items. He assumes that he will sell all the items that he makes. How many

belts and wallets should he make in order to make the most money?

1. USE WORDS TO STATE WHAT EACH VARIABLE REPRESENTS:

x = ____________________________________

y = ____________________________________

2. WRITE A SYSTEM OF CONSTRAINTS (INEQUALITIES)

3. WRITE A LINEAR COMBINATION TO MAXIMIZE OR MINIMIZE

4. GRAPH CONSTRAINTS AND SHADE THE FEASIBLE REGION...USE GRAPH PAPER!

5. IDENTIFY THE VERTICES OF THE FEASIBLE REGION

6. SUBSTITUTE ALL VERTEX VALUES INTO THE LINEAR COMBINATION

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7. IDENTIFY THE MAXIMUM OR MINIMUM VALUE AS APPROPRIATE

8. WRITE YOUR FINAL ANSWER IN SENTENCE FORM.

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1. You are the assistant manager of an appliance store. Next month you will order two types of stereo systems.

How many of each model should you order to minimize your cost?

• Model A: Your cost is $300 and your profit is $40

• Model B: Your cost is $400 and your profit is $60

• You expect a profit of at least $4800

• You expect to sell at least 100 units

2. A furniture manufacturer can make from 30 to 60 tables a day and from 40 to 100 chairs a day. It can make at

most 120 units in one day. The profit on a table is $150 and the profit on a chair is $65. How many tables and chairs

should they make per day to maximize their profit? What is their maximum profit?

3. Your school has contracted with a professional magician to perform at the school. The school has guaranteed an

attendance of at least 1000 and total ticket receipts of at least $4800. The tickets for students are $4 for students and

$6 for non-students, of which the magician receives $2.50 and $4.50 profit respectively. What is the minimum amount

of money the magician could receive?

4. A t-shirt company makes t-shirts and hoodies. They can make between 80 and 100 t-shirts in one day. They can

produce between 50 and 80 hoodies in one day. They can make, at most, 160 total units in one day. If the profit on

each t-shirt is $6 and the profit on each hoodie is $10, how many of each kind do they need to make a maximum profit?

What will this maximum profit be?

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5. A candy manufacturer has 130 pounds of chocolate-covered cherries and 170 pounds of chocolate-covered

mints in stock. He decides to sell them in the form of two different mixtures. One mixture will contain half cherries and

half mints by weight and will sell for $2.00 per pound. The other mixture will contain one-third cherries and two-thirds

mints by weight and will sell for $1.25 per pound. How many pounds of each mixture should the candy manufacturer

prepare in order to maximize his sales revenue?

6. The Osgood County refuse department runs two recycling centers. Center 1 costs $40 to run for an eight hour

day. In a typical day 140 pounds of glass and 60 pounds of aluminum are deposited at Center 1. Center 2 costs $50 for an

eight-hour day, with 100 pounds of glass and 180 pounds of aluminum deposited per day. The county has a commitment

to deliver at least 1540 pounds of glass and 1440 pounds of aluminum per week to encourage a recycler to open up a

plant in town. How many days per week should the county open each center to minimize its cost and still meet the

recycler’s needs?

Linear Programming

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1. Bob builds tool sheds. He uses 10 sheets of dry wall and 15 studs for a small shed and 15 sheets of dry wall and

45 studs for a large shed. He has available 60 sheets of dry wall and 135 studs. If Bob makes $390 profit on a

small shed and $520 on a large shed, how many of each type of building should Bob build to maximize his profit?

2. A company makes a product in two different factories. At factory X it takes 30 hours to produce the product and

at factory Y it takes 20 hours. The costs of producing these items are $50 at factory X and $60 at factory Y. The

company’s labor force can provide 6000 hours of labor each week and resources are $12,000 each week. How

should the company allocate its labor and resources to maximize the number of products produced?

3. Josie is on a diet. Daily, she needs three dietary supplements, A, B and C as follows: at least 16 units of A, 5 units

of B, and 20 units of C. These can be found in either of two marketed products Squabb I and Squabb II. The

Squabb I pill cost $10 and the Squabb II pill costs $20. How many of each pill should Josie buy to satisfy her

dietary needs and at the same time minimize costs?

4. An agriculture company has 80 tons of type I fertilizer and 120 tons of type II fertilizer. The company mixes these

fertilizer into two products. Product X requires 2 parts of type I and 1 part of type II fertilizers. Product Y requires

1 part of type I and 3 parts of type II fertilizers. If each product sells for $2000, what is the maximum revenue

and how many of each product should be made and sold to maximize revenue?

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5. A dietician formulates a special breakfast cereal by mixing Oat Flakes and Crunchy O’s. The cereals each provide

protein and carbohydrates. The cost is 38 cents for 1 cup of Oat Flakes and 32 cents for 1 cup of Crunchy O’s.

How many cups of each cereal will satisfy the constraints and minimize the cost? What is the minimum cost?

6. A tent manufacturer makes a two person tent and a family tent. Each type of tent requires time in the cutting

room and time in the assembly room as indicated below.

Two person Tent: 2 hours in cutting room and 2 hours in assembly room

Family Tent: 2 hours in cutting room and 4 hours in assembly room

The total number of hours available per week in the cutting room is 50. There are 80 hours available per week

in the assembly room. The manager requires that the number of two person tents manufactured be no more

than four times the number of family tents manufactured. The profit for the two person ten is $34 and the

profit for the family tent is $49. Assuming that all the tents produced can be sold, how many of each should be

manufactured per week to maximize the profit? What is the maximum profit.

7. A farmer is planning to raise wheat and barely. Each acre of wheat yields a profit of $50, and each acre of barley

yields a profit of $70. To sow the crop, two machines, a tractor and a tiller, are rented. The tractor is available

for 200 hours, and the tiller, is available for 100 hours. Sowing an acre of barley requires 3 hours of tractor time

and 2 hours of tilling. Sowing an acre of wheat requires 4 hours of tractor time and 2 hour of tilling. How many

acres of each crop should be planted to maximize the farmer’s profit?

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8. An ice cream supplier has two machines that produce vanilla and chocolate ice cream. To meet one of its

contractual obligations, the company must produce at least 60 gallons of vanilla ice cream and 100 gallons of

chocolate ice cream per day. One machine makes 4 gallons of vanilla and 5 gallons of chocolate ice cream per

hour. The second machine makes 3 gallons of vanilla and 10 gallons of chocolate ice cream per hour. It costs

$28 per hour to run machine 1 and $25 per hour to run machine 2. How many hours should each machine be

operated to fulfill the contract at the least expense?

9. A manufacturer makes two types of golf clubs:a starter model and a profession model. The start model requires

4 hours in the assembly room and 1 hour in the finishing room. The professional model requires 6 hours in the

assembly room and 1 hour in the finishing room. The total number of hours available in the assembly room is

108. There are 24 hours available in the finishing room. The profit for each starts model is $35 and the profit for

each professional model is $55. Assuming all the sets produced can be sold, find how many of each set should

be manufactured to maximize profit.

10. A company makes two types of telephone answering machines: the standard model and the deluxe model.

Each machine passes through three processes: P1, P2, P3. One standard answering machine requires 1 hour in P1

and 1 hour in P2 and 2 hours in P3. One deluxe answering machine requires 3 hours in P1, 1 hour in P2 and 1 hour

in P3. Because of employee work schedules P1 is available for 24 hours, P2 is available for 10 hours, and P3 is

available for 16 hours. If the profit is $25 for each standard model and $35 for each deluxe model, how many

units of each type should the company produce to maximize profit?