chapter 2: reasoning with systems of equations 2: reasoning with systems of ... by equations or...

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Chapter 2: Reasoning with Systems of Equations

Utah Core Standards for Mathematics Correlation:

A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, and exponential functions. A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

I CAN STATEMENTS: 2.1 I can find and explain solutions to systems of equations graphically. 2.2 I can Find and explain solutions to systems of equations algebraically. 2.3 I can write and solve systems of equations and justify the solving method used. 2.4 I can graph/solve/explain an inequality in two variables. I can

write/graph/solve/explain systems of inequalities. 2.5 I can write constraint equations, apply linear programming and explain solutions.

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2.0 (TASKS)Mixing Candies and Kimi & Jordan (from illustrativemathematics.org) 2.0 (task) Mixing Candies: A candy shop sells a box of chocolates for $30. It has $29 worth of chocolates plus $1 for the box. The box includes two kinds of candy: caramels and truffles. Lita knows how much the different types of candies cost per pound and how many pounds are in a box. She said,

If x is the number of pounds of caramels included in the box and y is the number of pounds of truffles in the box, then I can write the following equations based on what I know about one of these boxes:

x + y = 3 8x + 12y + 1 = 30

Assuming Lita used the information given and her other knowledge of the candies, use her equations to answer the following:

1. How many pounds of candy are in the box? 2. What is the price per pound of the caramels?

3. What does the term 12y in the second equation represent? 4. What does 8x + 12y + 1 in the second equation represent?

2.0 (task)Kimi and Jordan

Kimi and Jordan are each working during the summer to earn money in addition to their weekly allowance. Kimi earns $9 per hour at her job, and her allowance is $8 per week. Jordan earns $7.50 per hour, and his allowance is $16 per week.

a. Jordan wonders who will have more income in a week if they both work the same number of hours. Kimi says, "It depends." Explain what she means.

b. Is there a number of hours worked for which they will have the same income (in a week)? If so, find that number of hours. If not, why not?

c. What would happen to your answer to part (b) if Kimi were to get a raise in her hourly rate? Explain. d. What would happen to your answer to part (b) if Jordan were no longer to get an allowance? Explain.

__________________________

For problems above 1. In your notebook, record your solution to the problem. Explain your thinking using mathematical evidence

to support your conclusions.

2. PRESENTATION of thinking and work: Be prepared to explain your groups solution and the process you used to arrive at the solution. Think about how to present your results so the class can see and understand your work.

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2.1Find Solutions to Systems of Equations Graphically. Explain.

Use this blank page to compile the most important things you want to remember for cycle 2.1:

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2.1a (build)What is a Solution?

Find the solution to each equation below. 1. x + 7 = 10

2. 5y = 15

3. 4x 6 = 10

4. In your own words, describe what a solution

is (talk with your neighbor).

5. Refine your definition of a solution now that

we have discussed it as a class. A solution

is_________

6. Can there be more than one solution to an

equation? Why or why not?

Find a solution to each equation below (it is okay to guess).

7. x + y = 12 8. m n = 2 9. xy = 24 10. y = 2x + 1

Compare your solutions with your neighbor. 11. Is the definition for a solution the same if you

have two different variables in your equation

as opposed to only one variable?

12. How many total solutions are there for

equations with more than one variable?

Practice: Find four solutions to each equation. Write the solutions as ordered pairs.

13. y = 2x

x y (x, y)

14. x + y = 5

x y (x, y)

15. y = 3x

x y (x, y)

16. Why did you write solutions as ordered

pairs?

17. Is it possible to list every single solution to

these equations?

18. Finish the sentence: To show every solution

to an equation with two different variables

you______________.

19. Show the solution to the equation, y = x + 3

and describe in detail why it is the solution.

20. Why do you draw arrows on your graph?

21. Put a smiley on the graph where the solution

is negative.

22. Put a star on the graph where the solution is a

fraction.

23. Are solutions to the equation y = x + 3 just

integer points on the line (like the tables in

#13, 14) or are they continuous along the

line?

24. Research the words continuous and discrete.

What do they mean and how do they relate to

question #23? (entrance slip tomorrow)

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Solutions to Systems of Equations Previously, you have considered solutions to an equation with two variables. In questions below, you will consider two equations or a system of two equations. The solution for a system of equations is at the intersection of the two equation lines. Answer the questions related to the system of equations for the lines 2x y+ = and 4x y = .

25. What is the solution to this system?

26. A student states that the point (3, 4) is not a

solution to the system. What does this mean

in relationship to the graphs and in

relationship to the equations?

A system of two linear equations can have one solution, an infinite number of solutions, or no solution. Study the Concept Summary below and answer the questions.

Concept Summary Possible Solutions

Number of Solutions

exactly one infinite no solution

Terminology consistent & independent

consistent & dependent

inconsistent

Graph

27. What does it mean if a system of equations

has exactly one solution?

28. Explain the terminology consistent and

dependent as related to a system of

equations

29. If two equation lines are parallel, what does

that mean?

x + y = 2

x y = 4

3, -1

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2.1b (refine/task)Chickens and Pigs, What is a System of Equations?

A farmer saw some chickens and pigs in a field. He counted 60 heads and 176 legs. Problem solve with your group to find out exactly how many chickens and how many pigs he saw. Share and discuss methods for solving the problem. These methods might include a table, a graph, equations, etc.

__________________________________

3. In your notebook, record your solution to the problem. Explain your thinking using mathematical evidence

to support your conclusions.


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2.1c (refine/task)Where Does the Money Go?

Kim currently has $25 and decides to save $10 per week from her weekly babysitting job. Jenny currently has $160 and decides to spend $5 per week on entertainment.

1. Write an equation that describes Kims savings. 2. Write an equation that describes Jennys spending.

Graph the equations you wrote for Kim and Jenny on the graphing calculator. Then draw a sketch of the graph in your notebook. Just use a simple grid as below.

Conclusions:

3. Which axis represents the number of weeks? Label that axis. 4. Which axis represents the amount of money each girl will have? Label that axis. 5. How long will it take Jenny to run out of money at this rate if she has no additional income? 6. How long will it take Kim to have $150? 7. What does it mean when the two graphs intersect? 8. When will Kim and Jenny have the same amount of money? 9. Who has more money after 10 weeks? 10. Who has more money after 15 weeks?

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2.1d (refine)System of Equations Graphing Review

Solve the following systems by graphing without a graphing calculator. Then check them on the graphing calculator

1. 3

3

y

y x

=

=

2. 4

3 3 12

x y

x y

+ =

+ =

3. 2

2

x y

x y

=

+ =

4. 2 3

5

x y

x

+ =

=

5. 2 3 12

2 4

x y

x y

+ =

=

6. 2 4

2 3

x y

y x

+ =

+ =

7. 6

2

y x

y x

=

= +

8. 6 6

3 6

y x

y x

= +

= +

9. 2 8 6

4 3

x y

x y

=

=

Use a graphing calculator to solve each system of equations. Remember you might have to change your equations to slope-intercept form to enter them in the calculator. Record your graphs on graph paper.

10. 2 3

0.4 5

y x

y x

=

= +

11. 5.23 7.48

6.42 2.11

x y

x y

+ =

=

12. 2.32 6.12

4.5 6.05

x y

x y

=

+ =

13. The manager of a school store is ordering shirts printed with the school name and mascot. The cost is

$6.50 for each shirt plus a one-time design fee of $75 to create the stencil for the name and mascot. The

school store will be selling the shirts for $8.00. Write two equations below, one for cost and one for

revenue. Then use your graphing calculator to find the solution to the system. Interpret what the

solution means in context of the story. (Hint: You will have to change your viewing window on your

calculator)

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2.2Find Solutions to Systems of Equations Algebraically. Explain.


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2.2a (build/refine)Chickens and Pigs, Solving Systems Algebraically You can solve systems of equations using methods other than graphing. If you know these methods, you can choose the easiest method for each problem. To study these methods, revisit the chickens and pigs problem.

A farmer saw some chickens and pigs in a field. He counted 60 heads and 176 legs. Problem solve with your group to find out exactly how many chickens and how many pigs he saw.

1. When you first solved chickens and pigs, you probably used a trial and error method. Use trial and error to solve the following System equations. Show the trials and errors to get to the solution.

10

2

x y

x y

+ =

=

Now we will examine some algebraic methods to solve systems of equations 2. The substitution method.

a. Start with 2 equations in slope intercept form: Heads equation: __________________ Legs equation: ___________________

b. Solve one equation above for the variable.

c. Substitute from the 1st equation into the 2nd equation (so you only have one variable). Solve for that variable.

d. Knowing the value of one variable, substitute back in to find the value for the other variable. Show ALL work.

3. The elimination method.

Write the equations in this form Heads equation_________ = 60

Legs equation__________ = 176

Line the equations up one above the other like an addition problem. You are going to add the 2 equations together to arrive at 1 equation by eliminating a variable. (You may need to multiply one of the equations in order that a variable will be eliminated. See if you can figure out how to do this.)

Knowing the value of one variable, substitute back in to find the value for the other variable. Show ALL work.

Solve the following systems of equations. Use any of the following methods.

Graphing

Substitution

Elimination

Trial and error (pick 1 problemhint: pick this one first)

Explain why you used each method.

1. 2 15

5 21

x y

x y

+ =

+ =

Elimination because all I have to do is multiply the top equation by -2.

2. 2 3

4 5 9

x y

x y

= +

=

Substitution because I already have one equation in terms of x.

3.

2 3 9

5 3 12

x y

x y

+ =

=

4.

3 2

5 3 12

x y

x y

+

=

5.

2 3 4

2 8 19

x y

x y

+ =

+ =

6.

2

7

y x

y x

=

=

7.

2 5 1

3 7 3

x y

x y

=

+ =

8.

2 1

4 2 18

y x

x y

= +

+ =

9.

15

3

7

y x

y x

= +

= +

10.

5

2 10

x y

x y

=

=

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2.2b (refine/task)Prove the Elimination Method for Solving Systems The QUESTION: Why does the elimination method work in solving a system of equations? (You will present your reasoning to the class)

Finding the Answer to the question: (Study the sample problem below to help you get started.)

To use the elimination method, we actually create 2 extra equations (in bold below). The question is, why can we do that?

The Problem To eliminate the variable y, multiply the top equation by 2, and then add the equations.

Then solve for y.

1st equation: 3x + y = 12 2nd equation: x 2y = -3

1st equation: 3x + y = 12 (2) So 6x + 2y = 24 (3rd equation) + x 2y = -3 (2nd equation) 7x = 21 (4th equation) 3x =

2 3x y =

3 2 3

2 6

3

y

y

y

=

=

=

Suggestion: Use graphs as the basis for your reasoning. Use graphing technology to graph the equations (calculators, software or Geogebra). Here is a link to a good online graphing calculator: http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html )

Use the following steps to answer the question:

1. Thinking about the 3rd equation: Is it OK to multiply the 1st equation by 2? Why or why not?

2. Generalize: Does your answer to question 2 always apply (no matter the equations or the numbers you multiply or divide by)?

3. Thinking about adding the 3rd equation and the 2nd equation: Why can we add the two equations together? What does the 4th equation tell us about the solution? (hint: look at the graphs)

4. Generalize: Does your answer to question 3 always apply (no matter the system or whether you add or

subtract to eliminate a variable and solve for the solutions)?

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2.2c (apply)Find the Error Christina was solving the following problems, but was having some troubles. In each problem, she made a mistake. See if you can help her fix the mistakes.

1. Find the mistake. 2. Explain the mistake 3. Redo the problem correctly.

___________________________________

___________________________________

___________________________________

___________________________________

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2.3Write & Solve Systems of Equations. Justify Method


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2.3a (build)Translating Word Problems to Equations, Systems

Word Problem Variable

and what it represents

Equation (words)

Equation (math symbols)

The solution and what it means

Peter and Rick went to the music store. Peter bought some guitar picks for $6 and a package of guitar strings. Rick bought some new drumsticks for $8. Then Peter remembered he had a coupon for $5 off. The final bill after the coupon was $15. How much was the package of guitar strings?

g = prince of guitar string

cost of guitar picks + cost of guitar strings + cost of drumsticks coupon = final bill

Scott and Jake spent a total of $75 at the football game. It costs $10 for one person to get into the game. Each boy bought a team hat. Scott also bought a team magazine for $15. What is the cost of one hat?

h = cost of hat

total amount spent = entrance cost x number of people + cost of hat x number of hats purchased + cost of magazine

Jan noticed at the store that a sweater costs $3.95 more than a shirt. She bought 3 shirts and 2 sweaters and the total cost (before tax) was $71.65. What is the price of a shirt?

x = price of a sweater y = price of a shirt

1) 2)

1) 2)

1) The price of a sweater is? 2)

Luke purchased 2 pairs of pants and 3 pairs of shorts for $84. If a pair of shorts is 12 less than a pair of pants, determine the price of a pair of shorts.

1) 2)

1) 2)

1) 2)

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2.3b (refine)Writing & Solving Systems: Renting Cars Kurts Car company charges $25.20 for a daily rental plus $ .60 per mile. Rent-A-Wreck car company charges a flat rate of $42 for the same model car. 1. Fill in the following tables:

Kurts

Miles Calculations Cost K(x)

5

50

100

x

Rent-A-Wreck

Miles Calculations Cost R(x)

x

2. Write the equations for each company when

traveling x miles. K(x) = R(x) =

3. Graph. Then label the independent and dependent variables.

4. What is the slope (rate of change) for each

companys pricing plan?

5. What does the slope represent for each company? (Write this in a sentence.)

6. What is the y-intercept for each companys pricing plan? (Dont forget that the y-intercept should be written as a point.)

7. How many miles would you have traveled, renting from Kurts, if the final cost (before taxes) is $35.40?

8. If you travel 30 miles, what would be the cost renting from Kurts? Rent-A-Wreck?

9. For what situation is Kurts better?

10. For what situation is Rent-A-Wreck better?

11. After how many miles will the prices be the same?

12. Make up two price plans where the cars would be the same cost at 130 miles. Both plans must include a charge for mileage. Represent the two price plans both algebraically and graphically.

Writing assignment: You are the business consultant to the Kurts Car Company. Your job is to present the two price plans to the management team. Your presentation should include an explanation of how the two plans affect the customer and how the company might benefit or lose from the two plans. How might each plan affect sales? How might each plan affect overall profit? Extra for Experts: 1. Imagine that you have an odometer that can

record negative miles if the car is in reverse. Kurts charges the same fees for going in reverse as they do for a car going forward. On the same graph, plot the charges for a rented car that goes in reverse for 20 miles and another car that goes forward for twenty miles.

2. Imagine that Kurts charges the same $25.20 + $.60 per mile rate, except that the mileage is rounded so that renters pay $.60 for the first 0.5 1.4 miles and then an additional $.60 if they go 1.5-2.4 miles, and so on. Graph this situation for the first ten miles.

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2.3c (refine)Systems Tasks

Dimes and Quarters (from illustrativemathematics.org)

The only coins that Alexis has are dimes and quarters. Her coins have a total value of $5.80. She has a total of 40 coins.

1. Which of the following systems of equations can be used to find the number of dimes, d, and the number of

quarters, q, Alexis has? Explain your choice. Then solve the system.

a. 5.80

40 40 5.80

d q

d q

+ =

+ =

b. 40

0.25 0.10 5.80

d q

d q

+ =

+ =

c. 5.80

0.10 0.25 40

d q

d q

+ =

+ =

d. 40

0.10 0.25 5.80

d q

d q

+ =

+ =

Growing Coffee (from illustrativemathematics.org)

The coffee variety Arabica yields about 750 kg of coffee beans per hectare, while Robusta yields about 1200 kg

per hectare (reference). Suppose that a plantation has a hectares of Arabica and r hectares of Robusta.

a. Write an equation relating a and r if the plantation yields 1,000,000 kg of coffee. Then create the graph for the equation (use graphing calculator if desired). Online graphing calculator link: http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

b. On August 14, 2003, the world market price of coffee was $1.42 per kg of Arabica and $0.73 per kg of Robusta. Write an equation relating a and r if the plantation produces coffee worth $1,000,000. Then create the graph for the equation (use graphing calculator if desired)

c. To produce a million kg of coffee for a million dollars, how much of the plantation is planted in Arabica and how much in Robusta? Explain your solution.

_________________________

For Tasks above, be sure to follow these steps.

1. In your notebook, record your solution to the problem. Be sure to sketch the graphs created on graphing calculators. Explain your thinking using mathematical evidence to support your conclusions.


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2.3d (apply/justify)Writing & Solving Systems Practice

a. Write equations to solve the following problems.

b. Solve for the solution to the system.

c. Justify the method used to solve the solution.

1. Nadia and Peter visit the candy store. Nadia buys three candy bars and four fruit roll-ups for $2.84. Peter also buys three candy bars, but can only afford one additional fruit roll-up. His purchase costs $1.79. What is the cost of a candy bar and a fruit roll-up individually?

2. A small plane flies from Los Angeles to Denver with a tail wind (the wind blows in the same direction as the plane) and an air-traffic controller reads its ground-speed (speed measured relative to the ground) at 275 miles per hour. Another, identical plane, moving in the opposite direction has a ground-speed of 227 miles per hour. Assuming both planes are flying with identical air-speeds, calculate the speed of the wind.

3. An airport taxi firm charges a pick-up fee, plus an additional per-mile fee for any rides taken. If a 12-

mile journey costs $14.29 and a 17-mile journey costs $19.91, calculate: a. the pick-up fee b. the per-mile rate c. the cost of a seven mile trip

4. Calls from a call-box are charged per minute at one rate for the first five minutes, then a different rate

for each additional minute. If a 7-minute call costs $4.25 and a 12-minute call costs $5.50, find each rate.

5. A plumber and a builder were employed to fit a new bath, each working a different number of hours. The plumber earns $35 per hour, and the builder earns $28 per hour. Together they were paid $330.75, but the plumber earned $106.75 more than the builder. How many hours did each work?

6. Paul has a part time job selling computers at a local electronics store. He earns a fixed hourly wage, but can earn a bonus by selling warranties for the computers he sells. He works 20 hours per week. In his first week, he sold eight warranties and earned $220. In his second week, he managed to sell 13 warranties and earned $280. What is Pauls hourly rate, and how much extra does he get for selling each warranty?

7. Supplementary angles are two angles whose sum is 180. Angles A and B are supplementary angles. The measure of angle A is 18 less than twice the measure of angle B. Find the measure of each angle.

8. A 150-yard pipe is cut to provide drainage for two fields. If the length of one piece is three yards less than twice the length of the second piece, what are the lengths of the two pieces?

9. A baker sells plain cakes for $7 and decorated cakes for $11. On a busy Saturday the baker started with 120 cakes, and sold all but three. His takings for the day were $991. How many plain cakes did he sell that day, and how many were decorated before they were sold?

10. Twice Johns age plus five times Claires age is 204. Nine times Johns age minus three times Claires age is also 204. How old are John and Claire?

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2.3e (apply)Systems of Equations: Problems in Context Mixture Stories

1. Twenty pounds of dried fruit mix contained

prunes worth $2.90 a pound and apricots worth

$3.15 a pound. How many pounds of each did

the mix contain if the total value of the mix was

$59.75?

2. Thirty pounds of dried fruit contained apples

worth $3.15 a pound and pears worth $4.25 a

pound. How many pounds of each did the mix

contain if the value of the mix was $107.70?

3. Peanuts worth $2.90 a pound were mixed with

cashews worth $4.60 a pound to produce a

mixture worth $3.50 a pound. How many

pounds of each kind were used to produce 51

pounds of the mixture?

4. Coffee worth $2.95 a pound was mixed with

coffee worth $3.50 a pound to produce a blend

worth $3.30 a pound. How much of each kind

of coffee was used to produce 44 pounds of

blended coffee?

5. I have a total of 13 dimes and quarters which

equals $2.05. How many quarters and dimes do

I have?

6. A change purse has 100 nickels and dimes in it.

The coins total $7. How many of each coin are

there?

7. The bank has a large display of pennies and

nickels. The value of the coins in the display is

$24 and the coins total 800. How many of each

coin is in the display?

8. Henry has $4.25 in change consisting of dimes

and quarters. He has a total of 26 coins. Find

the number of dimes and quarters.

Percent stories: Read the story. Define the givens and write 2 equations. Solve use substitution or elimination. 1. Mr. Hill had a part of his $5000 savings in an

account that earned 8% interest and the rest in

an account that earned 12% interest. How much

did he have in each account if his annual income

from the total investment was $514.80?

2. Mrs. Pine had a savings of $9000, part of which

was invested at 7% interest and the rest at 9%

interest. How much did she have invested at

each rate if her annual income from the

investments was $741.60?

3. Mr. Toya had a part of his $5000 savings in an

account that earned 7% interest and the rest in

an account that earned 9% interest. How much

did he have in each account if his annual income


4. Marc invested part of his $1590 in account with

a 13% interest rate and the rest in an account

with a 15% interest rate. How much did he

invested in each account if his annual income


5. A 9% solution of sulfuric acid was mixed with a

30% solution of sulfuric acid to produce an 18%

solution. How much 9% solution and how

much 30% solution were used to make 21 liters

of 18% solution?

6. Milk that was 3% butterfat was mixed with

cream with 36% butterfat to produce 33 liters of

half-and-half with 18% butterfat content. How

much of each was used?

7. Copper that was 63% pure was melted together

with copper that was 90% pure to make 18

kilograms of an alloy that was 75% pure. How

many kilograms of each kind were used?

8. A 45% salt solution was mixed with a 75% salt

solution to produce 15 kilograms of solution

that was 67% salt. How much of each solution

was used?

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Rate and Perimeter Problems Problem solving steps:

a. Write givens and define our variables-----

you should have 2 things you are trying to

find so you should have 2 variables

b. Write 2 equations----since you have 2

variables you need 2 equations to solve.

Each equation should have one of each

variable in them

c. Solve for your variables---use graphing (not

recommended), substitution, or elimination--

-use your judgment as to which way would

be best (remember what we talked about

earlier)

d. Check---either do it a different way or

substitute both answers in to both equations

and see if it is true

1. In Prairie Dog Creek, Geri can row 60 km

downstream in 4 hours or she can row 36 km

upstream in the same amount of time. Find the

rate she rows in still water and the rate of the

current.

2. A jet liner flying east with the wind traveled

3600 km in 6 hours. The return trip, flying

against the wind, took 8 hours. Find the rate at

which the jet flew in still air and the rate of the

wind.

3. The perimeter of a rectangle is 68 cm. The

length is four cm more than the width. Find the

dimensions of the rectangle.

4. The perimeter of a rectangle is 132 meters. The

length is 2 m more than three times the width.

Find the dimensions of the rectangle.

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2.3f (apply)Systems of Equations Problems: Extra Practice

Divide into groups to work on the following problems and arrive at a solution. Create tables, graphs, and equations to justify your solutions or conclusions. Be prepared to show your work and explain all aspects of your problem and conclusions to the larger group. 1. Pick the best cell phone plan.

ChitChat cell phone company charges a monthly fee of $35 plus $0.05 per text that you send or receive.

Number of Texts

0 50 100 150 200 250 300 350 400 450 500

Total Bill

TalkAway cell phone company charges a monthly fee of $20 plus $0.10 per text that you send or receive. Number of Texts

0 50 100 150 200 250 300 350 400 450 500

Total Bill

2. Board-Game Battles

Your mission in the board-game battle is to intersect the enemy lanes. The enemy shipping lanes are represented by the following equations. At what points do you hope to intersect? Explain the method(s) you used to find your answer.

Enemy Lane 1: x y = 4

Enemy Lane 2: 3x y = 10

Enemy Lane 3: x 2y = 2 3. Bike Trip

You are planning your vacation and are going to rent a bike for the trip. The Green Bicycle Company charges $20 plus $10 per day to rent a bicycle. The Blue Bicycle Company charges $30 plus $8 per hour. You are going to rent your bike for 7 days. Your sister will only be renting hers for 4 days. Where should you and your sister rent to get the best prices on renting a bike? Explain. What will be your total charge to rent the bike? Your sisters charge? Justify your answer.

4. Skate Park Comparison

You and your friends want to go to a skate park on Saturday. There are two parks in your neighborhood, Sams Skate Park, and Brads Skate Park. The parks both charge for skating at their park. Each parks price is described below. Which park will you use and why? Consider different situations.

Sams Skate Park: $3 to get into the park and $1 for every hour. Brads Skate Park: $5 to get into the park and $0.50 for every hour.

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5. Car Rental Comparison Your family is planning a summer vacation to San Diego this summer. You are in charge of choosing the best car rental company to rent a car from. You are comparing the Rent-a-Wreck and Kurts. The price plans for each company is described below. Which company should you go with and why? Consider different situations.

Rent-a-Wreck: Charges a flat rate (meaning there is no charge for mileage) of $40 a day. Kurts: Charges $25 a day and $0.50 per mile. 6. Renting Skates

Young Rental rents inline skates for $5 plus $3.00 per hour. They charge to the nearest quarter hour. Riddle Rental competes with Young Rental by renting inline skates for $8 plus $2.50 per hour. Compare the rental programs. Consider when one is cheaper than the other. Can you draw any conclusions about their strategies?

7. Selling Lemonade Your younger sister wants to earn money by selling lemonade. The cost of starting the business is $1.20. The cost to make the lemonade is $.06 per cup. She sells the lemonade for $.25 per cup. How many cups of lemonade must she sell before making a profit?

8. Two Job Offers You have been offered a job marketing medical products. The company has two salary plans. Plan A pays a monthly salary of $700 plus a commission of $50 for each medical product you sell. Plan B pays a monthly salary of $500 plus a commission of $70 for each product sold. Analyze the income for the two plans.

9. Ed and Glendas Road Trip Mr. Ed and Ms. Glenda are both taking road trips to LA. Mr. Ed (in his green Toyota) started 50 miles from San Jose and drives at an average speed of 30 mph. Ms. Glenda (in her pink Chevy) started 10 miles from San Jose and drives at an average speed of 40 mph. Create a data table, make a graph, and write equations. Use the table, graph and equations to answer the following questions.

a. Who will get to LA first? How do you know? b. When will their cars be next to each other on the freeway? How can you tell? c. How far from San Jose will they be when they meet?

77


2.4Graph & Solve Inequality in Two Variables, Write & Solve System of Inequalities. Explain.


78


2.4a (build)Graphing Inequalities in Two Variables You have graphed many equations like y = 2x + 1. In the following investigation you will learn how to graph inequalities such as y < 2x + 1 and y > 2x + 1. Complete the following steps in your notebooks:

Make the Graphs (like the one at right) (note: your group will present your graph and analysis with the class) Step 1. Each group will be assigned one equation from below. (The

inequality signs are left out because you will use , or = to help you fill in the circlessee step 2.)

i. y 0.5x + 1 ii. y -2x 1 iii. y -0.5x + 1 iv. y -2x + 1

Step 2: Use the coordinates of each point shown with a circle to test the

statement you were given. Fill in each circle with the relational symbol, , or =, that makes the statement true. For example, to test the upper left point in statement i, substitute (-3, 3) for (x, y) as follows:

y 0.5x + 1 3 0.5(-3) + 1 3 > -0.5

Place a > in the upper left circle because this symbol makes statement i true.

Step 3: Repeat Step 2 until your 49 circles are filled in with one of the three symbols.

Analyze the Results Step 4: What do you notice about the circles filled with the equal sign? Tell any other patterns you see.

Step 5: Test a fractional/decimal coordinate-point that is not represented by a circle on the grid. Compare your result with the symbols on the same side of the line of equation signs as your point.

Step 6: Each person in the group will draw an xy-axes (example at right). Label

from 3 to 3 on each axis. Each group member will use the same equation given to the group above. However, each group member will choose a different sign (, )

Each person will write their equation under the graph, then shade the region of points (on the axes) that makes their statement true. If the points on the line make an inequality true, draw a solid line through them. If not, draw a dashed line.

Step 7: As a group, prepare to present and explain your graphs. You might use the document camera or put your graphs on a poster.

Conclusions: Step 7: Compare graphs with those of others in your class. What graphs require a solid line? A dashed line? Step 8: What graphs require shading? Shading above the line? Below the line? Step 9: Discuss how you would check the graph of an inequality with one point. Step 10: Write a log-entry for: graphing 2-variable inequalities (whats important, what to remember) .

Equation:

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2.4b (refine/apply) Graphing Inequalities

1. Suppose Mountain Sales Bicycle Shop makes $100 on each Model X bike sold and $50 on each Model Y bike sold. The bike shops overhead expenses are $1500 per month. At least how many of each model bike must be sold each month to avoid losing money?

a. Write the inequality. b. Rewrite the inequality in y = mx + b form. c. Graph the inequality. d. Color in the graph to show the numbers of x and y models which must be sold in order to make a

profit. Graph the inequality (use coordinate grids like the one at right). Identify the solutions. 2. 5x + 3y < 15 3. 3y 12 < -4x

Graph the solution to each linear inequality. Make sure to use the correct kind of border line.

4. 4

13

y x> +

5. 2 6 24x y+

Solve each inequality and graph the solution on a number line.

6. 56 5 96x

7. 2

30 223

x >

8. 5(2 1) 15 2(3 2)x x +

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2.4c (build)Ohio Jones, System of Inequalities Ohio Jones System of Inequality adventure (Use graph paper to find where he should stand.) Ohio Jones (Indianas lesser-known younger brother) is trying to gain entrance to the Lost Temple de los Dulces. When he arrives at the temple, there is a warning sign and three statues. The first one comes to life and says, Once you enter the main room, when the bell rings, you must be standing

on the floor where 1

52

y x< .

Ohio nods; just as Ohio is about to enter, the second statue comes to life and says, Beware! You will be turned to a statue if you do not stand on the floor where 6x 3y > 18 is true! Ohio stops hesitantly which statue should he listen to? Finally, the third statue comes to life and says Both of the other statues are telling the truth. You must listen to their words. And, you must only stand on squares where y > -4!

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2.4d (refine/apply)Write & Solve Systems of Inequalities

1. A Painting contractor estimates it will take 10 hours to paint a one-story house and 20 hours to paint a two-story house. The contractor submits a bid to paint 20 houses in less than 250 hours.

a. Write a system of inequalities to model the time to paint the houses and the number of houses to be painted.

b. Graph the system c. Give 5 whole number solutions to

the system d. Explain the solutions.

2. It costs 50 cents to make a bracelet and $1 to

make a necklace. To make a profit, the total cost for bracelets and necklaces must be less than $10. The jeweler can make no more than 14 pieces of jewelry each day.


b. Graph the system c. Explain the solutions.

3. The jeweler sells bracelets for $3 and

necklaces for $4. a. Write an inequality for profit needed

as $10 or more. b. Graph it. c. Test the three corner points on your

graph and determine how many bracelets and necklaces should be made to maximize profits.

Graph and then shade the solution area. 4.

3 2 -6

6

25

3

x y

y

y x

5.

-8

4 3

2 4 -28

y

y x

x y

>

6.

3

5

6

2

y

y

x

x

>

4. A clothing store manager wants to restock the mens department with two new types of shirt. A type x shirt

costs $20. A type y shirt costs $30. The store manager needs to stock at least $600 worth of shirts to be competitive with other stores, but the stores purchasing budget cannot exceed $1200 worth of shirts.

a. Write two inequalities demonstrating the minimum and maximum shirts to be stocked.

b. Rewrite in y mx b= + form.

c. Graph. Then shade in the graph to show the region that satisfies both inequalities. d. Name one combination of purchases that will satisfy both the minimum and maximum

requirements. 5. A lighting contractor estimates it will take 5 hours to wire a one-story house(x) and 10 hours to wire a two-

story house (y). The contractor submits a bid to wire 15 houses in less than 180 hours.


b. Graph and shade the system. c. Give 5 whole number solutions to the system.

6. It costs $.80 to make a bracelet and $2 to make a necklace. To make a profit, the total cost for bracelets and

necklaces must be less than $30. The jeweler can make no more than 20 pieces of jewelry each day. a. Write a system of inequalities to model the number of bracelets and necklaces to be made each

day. b. Graph and shade the system to show the solution c. Explain the solution

Extend the problem (this leads into cycle 2.5try it if you want). The jeweler sells bracelets for $5 and necklaces for $15.

d. Write an equation for profit as $30 or more. e. Graph and shade. f. Test the three corner points on your graph and determine how many bracelets and necklaces

should be made to maximize profits.

83


2.5Write constraint equations, apply linear programming, explain solutions.


84


2.5a (build)Systems of Inequalities: Linear ProgrammingReview the help section for 2.4a. Then graph and show the feasible regions and list the coordinates of the vertices of the polygon which represents the feasible region.

1. x 1

y x + 1 y 8

2. x 3

x 7 y 10 y -x + 8

3. y x + 6 x + y 6 x 4

4. y 0 y -3x + 6 y 3x x 7 y 6

5. x 0 y 0 x + y 4 2x + y 6

85


2.5b (refine) Linear Programming: Constraints in Problems

PROBLEM 1: A calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators must be shipped each day. If each scientific calculator sold results in a $2 loss, but each graphing calculator produces a $5 profit, how many of each type should be made daily to maximize net profits?

x: number of scientific calculators produced y: number of graphing calculators produced

1. What do the following constraints mean?

a. 100x b. 80y

c. 200x d. 170y

e. 200x y+

2. The above constraints are graphed below. One of the vertices is (120, 80). Name the rest of the vertices of the bounded region (4).

3. Each scientific calculator sold results in a $2 loss, but each graphing

calculator produces a $5 profit. The equation 2 5P x y= + represents this

situation. Explain each part of the equation. a. P represents ______ b. -2x represents______ c. 5y represents ______

4. Using the profit equation and vertices, find how many of each type of calculator should be made daily to maximize net profits?

a. Scientific Calc _________ b. Graphing Calc _________ c. Maximum Profit _________

PROBLEM 2: The area of a parking lot is 600 square meters. A car requires 6 square meters and a bus requires 30 square meters of space. The lot can handle a maximum of 60 vehicles.

5. Explain each of the following inequalities given x = cars and y = buses.

a. 0x b. 0y

c. 6 30 600x y+

d. 60x y+

6. Given the graph, name the vertices of the feasible region. (four points).

7. If a car costs $4 and a bus costs $7 to park in the lot, the function for the total

profit is: ( , ) 4 8f x y x y= + .

Determine the profit for each vertex above.

8. What is the maximum profit the parking lot can make? How many cars and buses will there be for that amount of profit?

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PROBLEM 3: A backpack manufacturer produces an internal frame pack and an external frame pack. Let x represent the number of internal frame packs produced in one hour and let y represent the number of external frame packs produced in one hour. These inequalities describe the constraints for manufacturing both packs.

3 18

2 16

0

0

x y

x y

x

y

+

+

9. Graph the constraints and name the vertices of the feasible region. (you will need graph paper)

10. Use the profit function to determine the maximum profit for manufacturing both backpacks for the given

constraints. Max Profit_____, Internal Frames _______, External Frames _______

PROBLEM 4: The Cupcake Boutique plans to purchase ads in a local newspaper to promote the grand opening of their new store in Herriman. Their advertising budget will allow them to spend a maximum of $2,300 for this purpose. They plan to run at most 20 ads. The cost of an advertisement in the local paper is $50 for a weekday paper and $200 in a weekend edition. Let x = the number of weekend ads and y = the number of weekday ads. 11. Explain each of the following constraints:

a. 20x y+

b. 200 50 2200x y+

c. 0 and 0x y

12. Using the graph, identify the vertices of the bounded region (4).

13. What does the point (0, 20) mean?



16. Which point gives the optimum combination if both types of ads are to be purchased?

PROBLEM 5: Abdul is the owner of a clothing store. He is getting ready to order his fall inventory. He wants to order up to 500 items of clothing (shirts and pants only). He knows that he should have at least as many shirts as pants. He also wants to have at least 100 pants. 17. Write a system of three inequalities that describes how many pants and shirts Abdul should order.

18. Graph the system.

19. Explain what the graph shows.

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PROBLEM 6: The Drama club is selling tickets to its play. An adult ticket costs $15 and a student ticket costs $11. The auditorium will seat 300 ticket-holders. The drama club wants to collect at least $3630 from ticket sales. 20. Write a system of four inequalities that describe how many of each type of ticket the club must well to meet

its goal.


22. Explain what the graph shows.

PROBLEM 7: A carpentry shop makes dinner tables and coffee tables. Each week the shop must complete at least 8 dinner tables and 12 coffee tables to be shipped to the furniture stores. The shop can produce at most 24 dinner tables and coffee tables combined each week. 23. Let c represent the number of coffee tables and d represent the number of dinner tables. Write a system of

inequalities to represent the number of coffee and dinner tables that can be produced in a week.


25. Find the coordinates of the feasible region

26. Suppose the shop sells coffee tables for $130 and dinner tables for $150. Write a function for the total

income for the week. ( , )f c d = ____________________

27. How many of each item should be produced for a maximum weekly income? What is the maximum weekly

income? Coffee tables __________, Dinner Tables__________, Maximum profit ______________

88


2.5c (apply)Linear Programming

Solve each linear programming problem given.

1. The area of a parking lot is 600 square meters. A car requires 6 square meters and a bus requires 30 square

meters of space. The lot can handle a maximum of 60 vehicles. Let c represent the number of cars and b represent the number of buses. a. Write a system of inequalities to represent the number of vehicles that can be parked. b. Graph the inequalities then find the coordinates of the feasible region. c. If a car costs $4 and a bus costs $7 to park in the lot, write a function for the total profit, then determine

the number of each vehicle to maximize the amount collected. Find maximum profits, buses, cars.

2. A painter has exactly 32 units of yellow dye and 54 units of green dye. He plans to mix as many gallons as possible of color A and color B. Each gallon of color A requires 4 units of yellow dye and 1 unit of green dye. Each gallon of color B requires 1 unit of yellow dye and 6 units of green dye. Find the maximum number of gallons he can mix.

89


2.6 (Tasks)Create a System, Writing Constraint Equations & Linear Programming Project

2.6 (task)Create a System of Equations Create a system of equations that has exactly one solution. Prove that your system of equations has

exactly one solution using two different methods.

Create a system of equations that has no solution. Prove that your system of equations has no solution

using two different methods.

Create a system of equations that has infinite solutions. Prove that your system of equations has infinite

solutions using two different methods.

2.6 (task)Writing Constraint Equations (from illustrativemathematics.org)

For each situation below, (i) write a constraint equation, (ii) determine two solutions, and (iii) graph the equation and mark your solutions.

a. The relation between quantity of chicken and quantity of steak if chicken costs $1.29/lb and steak costs $3.49/lb, and you have $100 to spend on a barbecue.

b. The relation between the time spent walking and driving if you walk at 3 mph then hitch a ride in a car traveling at 75 mph, covering a total distance of 60 miles.

c. The relation between the volume of titanium and iron in a bicycle weighing 5 kg, if titanium has a density of 4.5g/cm3 and iron has a density of 7.87 g/cm3 (ignore other materials).

d. The relation between the time spent walking and the time spent canoeing on a 30 mile trip if you walk at 4 mph and canoe at 7 mph.

___________________________

1. In your notebook, record your solutions. Explain your thinking using mathematical evidence to support your conclusions.


90


2.6 (project)Linear Programming Project

OBJECTIVE: Students will demonstrate an understanding of linear programming by solving a linear programming problem of their own design within the following guidelines.

I. Select a partner.

II. Choose two products that can be made with the same two materials. You need to be realistic because you need a prototype. (Example: toy car and toy train. Both are made of wood and paint.)

III. Make your prototypes. Keep a record of time and cost for each item.

IV. Set up the constraints of your problem. Keep in mind that according to Utah state law, full time

students under 18 may not work more than 20 hours per week. The amount of money you are able to invest in materials depends on your source of capital.

V. Make the neatest and most accurate graph of your problem. 8 x 11 graph paper should be

used.

VI. Investigate a possible selling price for your prototypes. You need to be realistic and competitive. Determine the profit per item.

VII. Use the linear programming theorem to maximize profit.

VIII. Summarize your results. Include such things as follows:

a plan of action

how competitive are your prices

what other variables could affect your profit

when will you break even

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Chapter 2 Help

2.0 2.3 Help: Systems of Equations SYSTEM OF TWO EQUATIONS Students have solved systems of linear equations in the previous course. The focus in Secondary 1 should be on the meaning of each equation, values that do or do not satisfy each of the equations, the meaning of a solution to the system, and understanding, justifying and explaining the methods used to solve the system. Students should explore systems where both equations describe the same line and result in infinitely many solutions. Systems with no solution, i.e. the equations describe parallel lines, should be tied into Unit 6.1.2 where the students look at how to show that parallel lines have the same slope. Examples:

3 4

3 2

y x

y x

=

= +

3 4

6 2 8

x y

x y

+ =

=

This system has no point of intersection and thus no solution. It

is an inconsistent system.

This has lines that coincide and an infinite number of solutions. The

system is consistent and dependent.

92


A SYSTEM WITH THE SAME SOLUTION There are three possibilities regarding a system of two linear functions:

1. the lines intersect at exactly one point, 2. there is no intersection, or 3. the two lines intersect at all points.

Suppose (2, -3) is the point where two lines intersect. For this point, we know that x = 2 and y = -3. We can think of each of the equations as a point on a number line, or as a line on the coordinate plane. (See the graph of x = 2 (red) and y = -3 (blue) on the same plane.)

Using the properties of equality (see pages 1 and 2 in this document), we can operate on these equations. If we add the equations together: x = 2 The sum is: x + y = -1

y = -3 This equation is based on the original equality, and can function as our first equation in the system. If we multiply x = 2 by three and add it to the second equation we get:

-3(x = 2) -3x = -6 y = -3 y = -3 -3x + y = -9 This equation is also based on the original equality and it will function as the second equation in the system. (Graph both x + y = -1 (blue) and -3x + y = -9 (red) on the same planealso have x = 2 (lighter red) and y = -3 (lighter blue) as lighter lines on the same plane.) The new lines also go through the point (2, -3). They are rotations of the original x = 2 and y = -3 around the point (2, -3).

x = 2

(2, -3)

y = -3

y = -3

x = 2

x + y = -1

-3x + y = -9

(2, -3)

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A SYSTEM WITH THE SAME SOLUTION For the line x + y = -1 the slope is -1, discuss how this is related to the lines x = 2 and y = -3. Further, relate it to how this line was created. For the line -3x + y = -9, the slope is 3, discuss how this line is related to each of x=2 and y=-3 and how it was generated. The discussion should lead to the conclusion that the new lines are rotations of x = 2 and y = -3 about the point (2, -3) and that the rotation is related to the slope. If we add the two equations together:

x + y = -1 -3x + y = -9 -2x + 2y = -10

We get: -2x + 2y = -10, which can be simplified to -x + y = -5. (Graph this line as a dark line (green) with all the previous lines as lighter lines). Again, this is a rotation of the original lines and lines generated for the original lines all rotated on the point (2, -3). It is important that students understand that all combinations of equations render lines that go through the same point of intersection. This point is the solution of the system (in this case the point (2, -3)). Any new line formed by combining two equations is just a rotation of the lines from which it was generated.

Every NON-vertical or NON-horizontal line that passes through the point (2, -3) has both an x and a y where x and y depend on each other. In other words, each line that passes through (2, -3) is some combination (with or without scalar multiplication) of the sum of x=2 and y=-3. Solving the system in essence means that you are pulling apart (through rotations) the lines x=2 and y=-3. Another way to think about it is that the combinations are systematically separating and rotating the system back to x=2 and y=-3. In solving this system of equations, the goal then is to rotate the line back to the original x = 2 and y = -3. Multiply the first equation by 3 and then add to the second:

3(x + y = -1) 3x + 3y = -3 -3x + y = -9 -3x + y = -9

4y = -12 (or 4 times y = -3)

y = -3 is just a rotation of x + y = -1 and/or -3x + y = -9. It just happens to be the original line. We can add y = -3 to either x + y = -1 or -3x + y = -9 and get another rotation, but the new equation will just be a non-vertical or non-horizontal rotation on the point (2, -3) of either equation where x and y are dependent on each other. Instead, because we know that y = -3 is a true equation, we can substitute -3 in for y in either of the equations and get x = 2.

x + y = -1 -3x + y = -9 x + -3 = -1 -3x + -3 = -9 x = 2 -3x = -6

x = 2

y = -3

x = 2

x + y = -1

-3x + y = -9

(2, -3)

-x + y = -5

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REPRESENTING RELATIONSHIPS WITH A SYSTEM OF EQUATIONS

Example: The admission fee at a county fair is $1.50 for children and $4.00 for adults. On the last day of the fair, 2200 people enter the fair and $5,050 is collected. How many children and how many adults attended?

In the past this problem would have been set up by picking a variable for one of the groups (say, "c" for "children") and then use "(total) less (what I've already accounted for)" (in this case, "2200 c") for the other group. Using a system of equations, however, allows us to use two different variables for the two different unknowns.

number of adults: a number of children: c

total number: a + c = 2200 total income: 4a + 1.5c = 5050

Now we can solve the system for the number of adults and the number of children, by using two different methods.

Substitution Solving Method: Solve the first equation for one of the variables, and then substitute the result into the other equation.

Elimination Solving Method: By adding the two equations together, we will eliminate one variable, solve for that variable, then substitute back in to solve for the other variable


total number: a + c = 2200, so a = 2200 c

4(2200 c) + 1.5c = 5050 8800 4c + 1.5c = 5050 8800 2.5c = 5050 2.5c = 3750 c = 1500

a = 2200 (1500) = 700

Solution: There were 1500 children and 700 adults.


To eliminate the variable a, multiply the top equation by -4, then add the equations. -4a 4c = -8800 4a + 1.5c = 5050

- 2.5c = -3750 c = 1500

a + 1500 = 2200 a = 2200 1500 Solution: There were 1500 children and 700 adults.

95


2.3 2.4 Help: Systems of Linear Inequalities SYSTEM OF LINEAR INEQUALITIES Students have not graphed linear inequalities in a previous course. However, this is just an extension of graphing linear equations with which they have had a great deal of experience in previous courses. They should be able to explain that any point in the shaded region is a solution that is, they make a true statement when the x and y-values are substituted back into the inequality. When finding the solution of a system of inequalities, we are interested in the area where the shaded regions overlap. Discussions should include similarities and differences with systems of linear equations.

Graphing an inequality in two-variables Graph the inequalities:

y < 2x + 1 y -x 1

1. Create a coordinate grid. 2. Graph the inequality y < 2x + 1, making sure to use the correct

notation: a. A dashed line to indicate that the boundary line is not

included if the inequality contains < or >. These are sometimes referred to as a strict inequality.

b. A solid line to indicate that the boundary line IS

included in the solution if the inequality contains or

. 3. Choose a test point to determine which side of the line, or half-

plane needs to be shaded. Always pick a point that is easy to work with, such as (0, 0).

a. Substitute the coordinates of the test point into the inequality.

b. If TRUE, shade the side of the boundary line where the test point is located.

c. If FALSE, shade the other half-plane.

y -x 1

y < 2x + 1

Solving a system of inequalities in two variables To solve a system of inequalities we are going to be asked to graph two or more two-variable inequalities and find areas where the inequalities overlap. This area of overlap will be the solution to the system. Look at the two inequalities we solved above. To find the solution of this system of inequalities we simply solve each of them by graphing them on the same coordinate grid and then find the area where they overlap. In this case, the solution area is the area shaded purple, not simply red only or blue only. A point located in an area shaded by only one of the equations will only be a solution for that equation. For example, the point (-4, 6) is only a solution for the equation y 2x + 1 and the point (-4, -10) is

only a solution for the equation y < 2x + 1.

y < 2x + 1 AND y -x 1

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SYSTEM OF INEQUALITIES TO DESCRIBE A CONTEXTUAL PROBLEM

Example: The Cupcake Boutique plans to purchase ads in a local newspaper to promote the grand opening of their new store in Herriman. Their advertising budget will allow them to spend a maximum of $2,200 for this purpose. They plan to run at most 20 ads. The cost of an advertisement in the local paper is $50 for a weekday paper and $200 to appear in a weekend edition. Write a system of linear inequalities to describe this situation. Create a graph that will represent the solutions. Solution: Let x = the number of weekend ads Let y = the number of weekday ad Since there will be at most 20 ads, x + y 20. The most that the ads can cost is $2,200, so: 200x + 50y 2200. Or the system of inequalities that represents the situation is:

x + y 20 200x + 50y 2200 For this problem, x and y cannot be negative numbers (cant have a negative number of ads) so we can restrict our graph to the first quadrant. This is an important idea to discuss with students when looking at the system of inequalities from a strictly mathematical standpoint, there are not restrictions on x or y. However, in context of the problem we are trying to solve, x and y should be positive whole numbers, since it does not make sense to talk about a negative number of ads or to have part of an ad. In the graph, all points in the region that is shaded twice and on the boundary lines of that region represent solutions to the system. Students should recognize that points on the boundaries are included in the solution set because of the way that the inequalities are written. Another point to highlight for students is that within the context of the problem we are only interested in whole number solutions since it is not feasible to purchase part of an ad. The vertices of the quadrilateral formed by the solution region are points of interest. The point (0, 20) means that if 20 weekend ads are purchased, then no weekend ads can be purchased. The point (11, 0) means that under the requirements given the maximum number of weekend ads that can be purchased is 11. Although any point in the solution region gives a solution to the inequality, the point (8, 12) gives us the optimum combination if both types of ads are to be purchased while remaining within the given constraints.

Number of Weekend Ads

Nu

mb

er o

f W

eek

da

y A

ds

Ads for Cupcake Boutique

(0, 0) (11, 0)

(8, 12)

(0, 20)

97


2.4 Help: Linear Programming (from Regents Exam Prep Center, created by Donna Roberts)

Linear Programming ("Planning") is an application of mathematics to such fields as business, industry, social science, economics, and engineering. The process is used to establish feasible regions and locate maximum and minimum values which can take place under certain given conditions.

Linear programming was developed as a discipline in the 1940's by George Dantzig, John von Neumann, and Leonid Kantorovich.

Let's start our investigation into Linear Programming by establishing feasible regions. These feasible regions are simply the solutions to systems of inequalities, such as those we have been studying. Feasible regions are all locations that represent "feasible" (viable, possible, correct) solutions to the set of inequalities.

Example 1:

Establish a feasible region for the following set of inequalities:

0

1

8

x

y

y x

+

Determine the coordinates of the vertices of the polygon formed by the feasible region.

The feasible region is shaded in yellow. The coordinates of the polygon are (0, 8), (0, 1), (7, 1).

Once the feasible region has been established, linear programming then examines the function which is to be maximized or minimized within this feasible region.

y = -x + 8

(0, 8)

(7, 1)

(0, 1) y = 1

x = 0

98


Linear Programming continued

For our study of Linear Programming, we will be limiting our investigation to the locations of feasible regions and the vertices of the polygons formed.

Example 2: Find a feasible region to represent this situation. Student Council is making colored armbands for the football team for an upcoming game. The school's colors are orange and black. After meeting with students and teachers, the following conditions were established:

1. The Council must make at least one black armband but not more than 4 black armbands since the black armbands might be seen as representing defeat.

2. The Council must make no more than 8 orange armbands.

3. Also, the number of black armbands should not exceed the number of orange armbands.

Let x = black armbands y = orange armbands

1. x > 1 and x < 4 2. y < 8 3. x < y

It will be assumed that these numbers are not negative

at any time.

The feasible region is shaded in yellow. The coordinates of the polygon are (1,8), (1,1),

y = x

x = 4 x = 1

y = 8

99


Chapter 2 Help Links Systems of equations:

Substitution: http://cstl.syr.edu/fipse/algebra/unit5/subst.htm

Elimination: http://www.purplemath.com/modules/systlin5.htm

Explain Elimination: http://math4teaching.com/2009/11/25/solving-systems-of-equation-by-

elimination-why-it-works-and-how-to-teach-it-with-conceptual-understanding/

One solution, no solution, infinite solutions:

http://www.mathwarehouse.com/algebra/linear_equation/systems-of-equation/index.php

Video: http://www.youtube.com/watch?v=xB-oXaCoJoc

Systems of Inequalities:

Help: http://www.purplemath.com/modules/syslneq.htm

Video: http://www.youtube.com/watch?v=6oehycq06vo

Linear Programming:

http://www.purplemath.com/modules/linprog.htm

http://www.purplemath.com/modules/linprog3.htm

Video: http://www.youtube.com/watch?v=M4K6HYLHREQ

chapter 2: reasoning with systems of equations 2: reasoning with systems of ... by equations or...

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