naMe: Unit 1 • Relationships Between Quantities …windrivermath.wikispaces.com/file/view/M1+Unit+1+Lesson...Unit 1 • Relationships Between Quantities Lesson 4: Representing Constraints

Unit 1 • Relationships Between QuantitiesLesson 4: Representing Constraints

naMe:

Assessment

CCSS IP Math I Teacher Resourceu1-166

© Walch Education

Pre-AssessmentCircle the letter of the best answer.

1. Given the equation y = 5x – 7, which point is a solution?

a. (1, 2)

b. (0, 7)

c. (–1, 2)

d. (–2, –17)

2. Given the inequality y ≤ –3x + 6, which point is NOT a solution?

a. (1, –3)

b. (0, –2)

c. (–1, –9)

d. (2, 3)

3. Julia has $6.50 to spend on peaches and apples at the farmer’s market. She bought 4 peaches at $0.75 each. How much money can she spend on apples? Determine which system of inequalities represents this situation.

a. a

a

4(0.75) 6.50

0

+ ≤≥

b. a

a

4(0.75) 6.50

0

+ ≤≤

c. a

a

4(0.75) 6.50

0

+ ≥≤

d. a

a

4(0.75) 6.50

0

+ ≥≥

4. Your cell phone company charges $29.99 a month plus $0.25 for each text message sent. You have budgeted no more than $35.00 for cell phone service each month. Given this situation, determine the minimum and maximum number of texts you can send without going over budget. Let x represent the number of texts.

a. x < 20.04

b. x ≥ 0 and x ≤ 20.04

c. x > 0 and x < 20

d. x ≥ 0 and x ≤ 20

5. Your doctor recommends that you eat at least 46 grams of protein each day. One serving of peanuts contains 9 grams of protein, while one egg contains 6 grams of protein. Determine which system of inequalities represents the number of servings of eggs and peanuts you must eat in order to reach the minimum recommendation.

a. x y

x

y

9 6 46

0

0

+ ≤≤≤

b. x y

x

y

9 6 46

0

0

+ ≤≥≥

c. x y

x

y

9 6 46

0

0

+ ≥≥≥

d. x y

x

y

9 6 46

0

0

+ ≥≤≤

CCSS IP Math I Teacher Resource© Walch Educationu1-167

Lesson 4: Representing ConstraintsUnit 1 • Relationships Between Quantities

InstructionCommon Core State Standard

A–CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.★

Essential Questions

1. How can you model real-world applications using equations?

2. How can you solve real-world applications by graphing systems of equations?

3. How can you model real-world applications using inequalities?

4. Why are there constraints when solving and graphing real-world applications?

WORDS TO KNOW

algebraic inequality an inequality that has one or more variables and contains at least one of the following symbols: <, >, ≤, ≥, or ≠

constraint a restriction or limitation on either the input or output values

inequality a mathematical sentence that shows the relationship between quantities that are not equivalent

solution set the value or values that make a sentence or statement true

system of equations a set of equations with the same unknowns

system of inequalities a set of inequalities with the same unknowns

Recommended Resources• NCTM Illuminations. “Dirt Bike Dilemma.”

http://walch.com/rr/CAU1L4SysEquations

Students use a system of equations to maximize profits for a dirt bike manufacturer.

• Purplemath.com. “Linear Programming: Word Problems.”

http://walch.com/rr/CAU1L4SysInequalities

This site offers a review of systems of inequalities and constraints associated with real-world situations. It also includes graphs of feasible regions.

http://walch.com/rr/CAU1L4SysEquations

http://walch.com/rr/CAU1L4SysInequalities


naMe:


© Walch Education

Lesson 1.4.1: Representing Constraints

Warm-Up 1.4.1Read the scenario and answer the questions that follow.

Roshanda pays $5 in tolls and uses 3 gallons of gasoline each day she drives to work. In one day, Roshanda spent a total of $15.23 on tolls and gasoline.

1. How much did each gallon of gas cost? Explain how you found your answer.

2. Roshanda needs to work 5 days next week and has set aside $75 for tolls and gas. Will Roshanda have enough money for her workweek? Explain how you found your answer.


Instruction


Lesson 1.4.1: Representing ConstraintsCommon Core State Standard

A–CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.★

Warm-Up DebriefRoshanda pays $5 in tolls and uses 3 gallons of gasoline each day she drives to work. In one day, Roshanda spent a total of $15.23 on tolls and gasoline.

1. How much did each gallon of gas cost? Explain how you found your answer.

Write an equation to represent this situation. Let x represent the cost of a gallon of gas.

5 + 3x = 15.23

Solve for x.

5 + 3x = 15.23 Subtract 5 from both sides of the equation.

3x = 10.23 Divide both sides by 3.

x = 3.41

Interpret the solution.

The cost of 1 gallon of gas is $3.41.

2. Roshanda needs to work 5 days next week and has set aside $75 for tolls and gas. Will Roshanda have enough money for her workweek? Explain how you found your answer.

Multiply the amount of money Roshanda spends on her travels for 1 day of work by the number of days she will be working.

15.23(5) = 76.15

Compare this amount to the amount that Roshanda has saved.

Roshanda has saved $75 and will need $76.15. She will not have enough money.

Connection to the Lesson

• The upcoming lesson will have students examining limitations and constraints on equations and inequalities. Students will need to think about their answers and determine if they are appropriate given the constraints.


Instruction


© Walch Education

Prerequisite Skills

This lesson requires the use of the following skills:

• reading and writing inequalities

• creating and evaluating inputs and outputs of equations and inequalities

IntroductionSituations in the real world often determine the types of values we would expect as answers to equations and inequalities. When an inequality has one or more variables and contains at least one inequality symbol (<, >, ≤, ≥, or ≠), it is called an algebraic inequality.

Sometimes there are limits or restrictions on the values that can be substituted into an equation or inequality; other times, limits or restrictions are placed on answers to problems involving equations or inequalities. These limits or restrictions are called constraints.

Key Concepts

• Many real-world situations can be modeled using an equation, an inequality, or a system of equations or inequalities. A system is a set of equations or inequalities with the same unknowns.

• When creating a system of equations or inequalities, it is important to understand that the solution set is the value or values that make each sentence in the system a true statement.

• Being able to translate real-world situations into algebraic sentences will help with the understanding of constraints.

Common Errors/Misconceptions

• incorrectly translating verbal descriptions to algebraic sentences

• not including appropriate constraints as related to the situation


Instruction


Guided Practice 1.4.1Example 1

Determine whether the coordinate (–2, 9) is a solution to the inequality y ≤ 5x + 6.

1. Substitute the values for x and y into the original inequality.

y ≤ 5x + 6

9 ≤ 5(–2) + 6

2. Simplify the sentence.

9 ≤ 5(–2) + 6 Multiply 5 and –2.9 ≤ –10 + 6 Add –10 and 6.9 ≤ –4

3. Interpret the results.

9 is NOT less than or equal to –4; therefore, (–2, 9) is not a solution to the inequality y ≤ 5x + 6.

Example 2

A taxi company charges $2.50 plus $1.10 for each mile driven. Write an equation to represent this situation. Use this equation to determine how far you can travel if you have $10.00. What is the minimum amount of money you will spend?

1. Translate the verbal description into an algebraic equation. Let m represent the number of miles driven and let C represent the total cost of the trip.

2.50 + 1.10m = C

2. The total cost of the trip can’t be more than $10.00 because that is all you have to spend. Substitute this amount in for C.

2.50 + 1.10m = 10.00


Instruction


© Walch Education

3. Although you have $10.00 to spend, you could also spend less than that. Change the equal sign to a less than or equal to sign (≤).

2.50 + 1.10m ≤ 10.00

4. Solve the inequality by isolating the variable.

2.50 + 1.10m ≤ 10.00 Subtract 2.50 from both sides.

1.10m ≤ 7.50 Divide both sides by 1.10.

m ≤ 6.82

You can travel up to 6.82 miles and not pay more than $10.00. Because the company charges by the mile, you can travel no more than 6 miles.

5. The minimum amount the taxi driver charges is $2.50, but it is unlikely that he or she will charge you if you get in the cab and get right back out without going anywhere. You will pay $1.10 if you travel 1 mile or less; add this to the minimum charge of $2.50 to arrive at $3.60.

6. You will spend a minimum of $3.60, but no more than $10.00.

Example 3

A school supply company produces wooden rulers and plastic rulers. The rulers must first be made, and then painted.

• It takes 20 minutes to make a wooden ruler. It takes 15 minutes to make a plastic ruler. There is a maximum amount of 480 minutes per day set aside for making rulers.

• It takes 5 minutes to paint a wooden ruler. It takes 2 minutes to paint a plastic ruler. There is a maximum amount of 180 minutes per day set aside for painting rulers.


Instruction


Write a system of inequalities that models the making and then painting of wooden and plastic rulers.

1. Identify the information you know.

There is a maximum of 480 minutes for making rulers.

• It takes 20 minutes to make a wooden ruler.

• It takes 15 minutes to make a plastic ruler.

There is a maximum of 180 minutes for painting rulers.

• It takes 5 minutes to paint a wooden ruler.

• It takes 2 minutes to paint a plastic ruler.

2. Write an inequality to represent the amount of time needed to make the rulers. Let w represent the wooden rulers and p represent the plastic rulers.

20w + 15p ≤ 480

3. Write an inequality to represent the amount of time needed to paint the rulers. Use the same variables to represent wooden and plastic rulers.

5w + 2p ≤ 180

4. Now consider the constraints on this situation. It is not possible to produce a negative amount of either wooden rulers or plastic rulers; therefore, you need to limit the values of w and p to values that are greater than or equal to 0.

w ≥ 0

p ≥ 0

5. Combine all the inequalities related to the situation and list them in a brace, {. These are the constraints of your scenario.

w p

w p

w

p

20 15 480

5 2 180

0

0

+ ≤+ ≤

≥≥


Instruction


© Walch Education

Example 4

Use the system of inequalities created in Example 3 to give a possible solution to the system.

1. We know from this situation that you cannot produce a negative amount of rulers, so none of our solutions can be negative.

2. In future lessons, we discuss more precise ways of determining the solution set to a system. For now, we can use our knowledge of numbers and ability to solve algebraic sentences to find possible solutions.

3. Choose a value for w.

Let w = 0. Substitute 0 for each occurrence of w in the system and solve for p.

20w + 15p ≤ 480

20(0) + 15p ≤ 480 Substitute 0 for w.

15p ≤ 480 Divide both sides by 15.

For the first inequality, p ≤ 32.

5w + 2p ≤ 180

5(0) + 2p ≤ 180 Substitute 0 for w.

2p ≤ 180 Divide both sides by 2.

For the second inequality, p ≤ 90.

4. Interpret the results.

In 480 minutes, the company can make no more than 32 plastic rulers if 0 wooden rulers are produced.

In 180 minutes, the company can paint no more than 90 plastic rulers if there are no wooden rulers to paint.


naMe:


Problem-Based Task 1.4.1: Skate ConstraintsA sporting goods company produces figure skates and hockey skates. One group of workers makes the blades for both types of skates. Another group makes the boots for both types of skates.

• It takes 2 hours to make the blade of a figure skate. It takes 3 hours to make the blade of a hockey skate. There is a maximum of 40 hours per week in which the blades can be made for both types of skates.

• It takes 3 hours to make the boot of a figure skate. It takes 1 hour to make the boot for a hockey skate. There is a maximum of 20 hours per week in which boots can be made for both types of skates.

What are possible combinations of the number of figure skates and hockey skates that can be produced given the constraints of this situation?


naMe:


© Walch Education

Problem-Based Task 1.4.1: Skate Constraints

Coaching a. What information do you know about the amount of time needed to make the blade of a

figure skate?

b. What information do you know about the amount of time needed to make the blade of a hockey skate?

c. How many hours each week can be spent making skate blades?

d. What inequality can be used to represent the amount of time it takes to make blades for both figure skates and hockey skates?

e. What information do you know about the amount of time needed to make the boot of a figure skate?

f. What information do you know about the amount of time needed to make the boot of a hockey skate?

continued


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g. How many hours each week can be spent making skate boots?

h. What inequality can be used to represent the amount of time it takes to make the boots for both figure skates and hockey skates?

i. What other constraints are needed in this situation?

j. What is the system of inequalities that represents this situation?

k. Is it possible to construct 3 figure skates and 4 hockey skates given the constraints of this situation?

l. Is it possible to construct 8 figure skates and 5 hockey skates given the constraints of this situation?

m. What is another possible combination of the number of figure skates and hockey skates that can be produced given the constraints of this situation?


Instruction


© Walch Education

Problem-Based Task 1.4.1: Skate Constraints

Coaching Sample Responsesa. What information do you know about the amount of time needed to make the blade of a

figure skate?

It takes 2 hours to make the blade of a figure skate.

b. What information do you know about the amount of time needed to make the blade of a hockey skate?

It takes 3 hours to make the blade of a hockey skate.

c. How many hours each week can be spent making skate blades?

No more than 40 hours can be spent making skate blades.

d. What inequality can be used to represent the amount of time it takes to make blades for both figure skates and hockey skates?

The total amount of time needed to make the blades can be represented by the sum of the time needed for each type of blade. Let f represent figure skates and h represent hockey skates.

The total number of hours needed for making the blades can’t be more than 40 hours.

2f + 3h ≤ 40

e. What information do you know about the amount of time needed to make the boot of a figure skate?

It takes 3 hours to make the boot of a figure skate.

f. What information do you know about the amount of time needed to make the boot of a hockey skate?

It takes 1 hour to make the boot of a hockey skate.

g. How many hours each week can be spent making skate boots?

No more than 20 hours can be spent making skate boots.

h. What inequality can be used to represent the amount of time it takes to make the boots for both figure skates and hockey skates?

The total amount of time needed to make the boots can be represented by the sum of the time needed for each type of boot.

The total number of hours needed for making the boots can’t be more than 20 hours.

3f + h ≤ 20


Instruction


i. What other constraints are needed in this situation?

The number of figure skates can’t be less than 0.

f ≥ 0

The number of hockey skates can’t be less than 0.

h ≥ 0

j. What is the system of inequalities that represents this situation?

To create the system of inequalities, group all of the related inequalities.

f h

f h

f

h

2 3 40

3 20

0

0

+ ≤+ ≤

≥≥

k. Is it possible to make 3 figure skates and 4 hockey skates given the constraints of this situation?

Substitute 3 for f and 4 for h in each inequality of the system.

2f + 3h ≤ 40 First inequality in the system

2(3) + 3(4) ≤ 40 Substitute values for f and h, then multiply.

6 + 12 ≤ 40 Simplify.

18 ≤ 40 This is a true statement.

3f + h ≤ 20 Second inequality in the system

3(3) + (4) ≤ 20 Substitute values for f and h, then multiply.

9 + 4 ≤ 20 Simplify.

13 ≤ 20 This is also a true statement.

f ≥ 0 Third inequality; substitute 3 for f.

3 ≥ 0 This is a true statement.

h ≥ 0 Last inequality; substitute 4 for h.


The substitution of 3 for f and 4 for h results in true statements for each inequality of the system. Therefore, it is possible to make 3 figure skates and 4 hockey skates given the constraints of this situation.


Instruction


© Walch Education

l. Is it possible to make 8 figure skates and 5 hockey skates given the constraints of this situation?

Substitute 8 for f and 5 for h in each inequality of the system.

2f + 3h ≤ 40 First inequality in the system

2(8) + 3(5) ≤ 40 Substitute values for f and h, then multiply.

16 + 15 ≤ 40 Simplify.

31 ≤ 40 This is a true statement.

3f + h ≤ 20 Second inequality in the system

3(8) + (5) ≤ 20 Substitute values for f and h, then multiply.

24 + 5 ≤ 20 Simplify.

29 ≤ 20 This is NOT a true statement.

f ≥ 0 Third inequality; substitute 8 for f.


h ≥ 0 Last inequality; substitute 5 for h.


The substitution of 8 for f and 5 for h results in true statements for 3 of the 4 inequalities of the system. Because one of the inequalities is not true, it is not possible to make 8 figure skates and 5 hockey skates given the constraints of this situation.

m. What is another possible combination of the number of figure skates and hockey skates that can be made given the constraints of this situation?

Students can use trial and error to find other possible combinations that will satisfy each of the inequalities in the system.

Possible solutions: 4 figure skates and 5 hockey skates; 2 figure skates and 10 hockey skates

Recommended Closure Activity

Select one or more of the essential questions for a class discussion or as a journal entry prompt.


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Practice 1.4.1: Representing ConstraintsDetermine whether each coordinate listed below is a solution to the given algebraic sentence.

1. Is the coordinate (–2, –4) a solution to the equation y = 3x – 2?

2. Is the coordinate (1, –3) a solution to the inequality y ≤ –4x + 6?

Read each scenario and use it to complete the parts that follow.

3. Given the inequalities y > 5x – 8 and y ≥ 3x + 4, find a point that

a. satisfies both inequalities.

b. satisfies neither inequality.

c. satisfies one inequality, but not the other.

4. You pay $12 to get into the fair, plus $3 per ticket for x ride tickets.

a. Write an equation to find the total cost of attending the fair.

b. Now write an inequality and solve it to determine the maximum number of tickets you can buy if you have $24 to spend.

c. What is the minimum amount of money you will spend?

5. Charlie borrowed $500 from his aunt. He has already paid back $75. His aunt doesn’t charge any interest and he is planning on making $15 payments each Friday.

a. Write an equation that represents the number of weeks it will take Charlie to repay his aunt if he pays $15 each Friday.

b. Is the solution to the equation the actual number of weeks it will take Charlie to repay his aunt? Explain your answer.

continued


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© Walch Education

Use the information in each scenario to complete problems 6–10.

6. The concession stand at the football game sells cans of soda for $0.75 and bottles of water for $1.25. You have $10.00. Write an inequality to represent this situation. What can you buy?

7. A stained glass artist has a fixed cost of $150. It costs the artist $15 to produce each piece, but each piece sells for $35. The equation C = 150 + 15n represents the total cost, C, for producing n pieces. The total revenue for n pieces is determined by the equation R = 35n. What constraint is necessary to include when modeling this situation?

8. Your dad needs to rent a chain saw to cut down trees in your yard. The rental company charges $20 plus $6.50 per hour to rent the chain saw. Your dad wants to spend no more than $50. What constraints apply to this situation? What is the maximum number of hours your dad can rent the chain saw?

9. Jermaine has $10.00 to spend on ice cream. Three scoops cost $5.99, plus $0.75 for each topping. He always leaves a 20% tip for the cashier. Write an inequality and use it to determine if Jermaine can afford to buy a three-scoop ice cream with three toppings plus tip the cashier.

10. The local florist never has more than a combined total of 40 daisy and carnation bouquets and never more than 12 carnation bouquets. Write a system of inequalities that represents this situation. Be sure to include all constraints.

Unit 1 • Relationships Between QuantitiesLesson 5: Rearranging Formulas

naMe:

Assessment


© Walch Education

Pre-AssessmentCircle the letter of the best answer.

1. Solve the equation 8x + 4y = 12 for y.

a. y = 2x – 3

b. y = –2x + 3

c. y = –3x + 2

d. y = 3x – 2

2. Solve the equation y x1

53 7− + = for y.

a. y = 15x – 35

b. y = –15x + 35

c. y = 35 – 15x

d. y = 35 + 15x

3. The formula P = 2l + 2w is used to calculate the perimeter of a rectangle. Solve this formula for l.

a. lw P

2

2=

−

b. lP w

2

2=

−

c. lw P2

2=

−

d. lP w2

2=

−

4. The formula V = lwh is used to calculate the volume of a prism. Solve this formula for l.

a. lwh

V=

b. l = V – wh

c. lV

wh=

d. l = wh – V

5. The speed, v, of a point on the edge of a spinning disk is found using the formula vr

T

2π= .

Solve this formula for T.

a. Tv

r2π=

b. Tr

v

2π=

c. T = 2πr – v

d. T = v – 2πr


Lesson 5: Rearranging FormulasUnit 1 • Relationships Between Quantities

InstructionCommon Core State Standard

A–CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.★

Essential Questions

1. How is solving a literal equation or formula for a specific variable similar to solving an equation with one variable?

2. How could solving a literal equation or formula for a specific variable be helpful?

3. How do you determine for which variable a literal equation or formula should be solved?

WORDS TO KNOW

formula a literal equation that states a specific rule or relationship among quantities

inverse a number that when multiplied by the original number has a product of 1

literal equation an equation that involves two or more variables

reciprocal a number that when multiplied by the original number has a product of 1

Recommended Resources• CRCTLessons.com. “Solving Equations Game.”

http://walch.com/rr/CAU1L5SolvingEquations

Practice solving equations for a given variable with this online basketball game.

• Purplemath.com. “Solving Literal Equations.”

http://walch.com/rr/CAU1L5LitEquations

This site has an overview of literal equations, with worked examples on how to solve equations and formulas for a given variable.

http://walch.com/rr/CAU1L5SolvingEquations

http://walch.com/rr/CAU1L5LitEquations


naMe:


© Walch Education

Lesson 1.5.1: Rearranging Formulas

Warm-Up 1.5.1Read the scenario below. Write an equation and use it to answer the questions that follow.

In January 2011, the national average for 5 gallons of gasoline was $17.20.

1. What was the national average for 1 gallon of gas?

2. What was the price for 18 gallons of gas?


Instruction


Lesson 1.5.1: Rearranging FormulasCommon Core State Standard

A–CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.★

Warm-Up 1.5.1 Debrief1. What was the national average for 1 gallon of gas?

Set up the equation.

Let x represent the cost per gallon of gasoline.

5x = 17.20

Using inverse operations, solve this equation for x.

x

x

x

5 17.20 Original equation

5

5

17.20

5Divide each side by 5.

3.44

=

=

=Interpret the solution.

The average cost of 1 gallon of gasoline was $3.44.

2. What was the price for 18 gallons of gas?

To find the cost of 18 gallons of gasoline, multiply the number of gallons by the cost per gallon.

18 • 3.44 = $61.92

Interpret the solution.

The cost of 18 gallons of gasoline at $3.44 a gallon is $61.92.

Connection to the Lesson

• Students are asked to solve equations, much like they will be asked to solve a literal equation or formula for a specified variable.


Instruction


© Walch Education

Prerequisite Skills

This lesson requires the use of the following skills:

• order of operations

• solving multi-step equations

IntroductionLiteral equations are equations that involve two or more variables. Sometimes it is useful to rearrange or solve literal equations for a specific variable in order to find a solution to a given problem. In this lesson, literal equations and formulas, or literal equations that state specific rules or relationships among quantities, will be examined.

Key Concepts

• It is important to remember that both literal equations and formulas contain an equal sign indicating that both sides of the equation must remain equal.

• Literal equations and formulas can be solved for a specific variable by isolating that variable.

• To isolate the specified variable, use inverse operations. When coefficients are fractions,

multiply both sides of the equation by the reciprocal. The reciprocal of a number, also

known as the inverse of a number, can be found by flipping a number. Think of an integer

as a fraction with a denominator of 1. To find the reciprocal of the number, flip the fraction.

The number 2 can be thought of as the fraction 2

1. To find the reciprocal, flip the fraction:

2

1

becomes 1

2. You can check if you have the correct reciprocal because the product of a number

and its reciprocal is always 1.

Common Errors/Misconceptions

• solving for the wrong variable

• improperly isolating the specified variable by using the opposite inverse operation

• incorrectly simplifying terms


Instruction


Guided Practice 1.5.1Example 1

Solve 6y – 12x = 18 for y.

1. Begin isolating y by adding 12x to both sides.

6y – 12x = 18

+ 12x + 12x

6y = 18 + 12x

2. Divide each term by 6.

y x6

6

18

6

12

6= +

y = 3 + 2x

Example 2

Solve 15x – 5y = 25 for y.

1. Begin isolating y by subtracting 15x from both sides of the equation.

x y

x x

y x

15 5 25

15 15

5 25 15

− =− −

− = −

2. To further isolate y, divide both sides of the equation by the coefficient of y. The coefficient of y is –5. Be sure that each term of the equation is divided by –5.

y x

y x

y x

5

5

25 15

55

5

25

5

15

55 3

−−

=−−

−−

=−

−−

= − +


Instruction


© Walch Education

Example 3

Solve 4y + 3x = 16 for y.

1. Begin isolating y by subtracting 3x from both sides of the equation.

y x

x x

y x

4 3 16

3 3

4 16 3

+ =− −

= −

2. To further isolate y, divide both sides of the equation by the coefficient of y. The coefficient of y is 4. Be sure that each term of the equation is divided by 4.

y x

yx

y x

4

4

16 3

416

4

3

4

43

4

=−

= −

= −


Instruction


Example 4

The formula for finding the area of a triangle is A bh1

2= , where b is the length of the base and h is

the height of the triangle. Suppose you know the area and height of the triangle, but need to find the

length of the base. In this case, solving the formula for b would be helpful.

1. Begin isolating b by multiplying both sides of the equation by the

reciprocal of 1

2, or 2.

A bh

A bh

A bh

1

2

2 21

2

2

=

• = •

=

Multiplying both sides of the equation by the reciprocal is the same as

dividing both sides of the equation by 1

2. The result will be the same.

2. To further isolate b, divide both sides of the equation by h.A

h

bh

hA

hb

2

2

=

=

or bA

h

2=

3. The formula for finding the length of the base of a triangle can be found by doubling the area and dividing the result by the height of the triangle.


Instruction


© Walch Education

Example 5

The distance, d, that a train can travel is found by multiplying the rate of speed, r, by the amount of time that it is travelling, t, or d = rt. Solve this formula for t to find the amount of time the train will travel given a specific distance and rate of speed.

1. Isolate t by dividing both sides of the equation by r.d

r

rt

r

td

r

=

=

2. The formula for finding the amount of time it will take a train

to travel a given distance at a given speed is td

r= .


naMe:


Problem-Based Task 1.5.1: BricklayersThe formula N = 7LH is used to determine N, the number of bricks needed to build a wall that is L feet in length and H feet high. A customer would like a wall constructed that is 4 feet high. If the bricklayer wants to use all of the 1,820 bricks that he has readily available, how long will the wall be?


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© Walch Education

Problem-Based Task 1.5.1: Bricklayers

Coachinga. What does each of the variables represent?

b. What variable is the given formula solved for?

c. Which variable does the formula need to be solved for?

d. Solve the given formula for the unknown variable.

e. What values are given in the problem statement?

f. How long can the wall be if the bricklayer has 1,820 bricks and the wall must be 4 feet high?


Instruction


Problem-Based Task 1.5.1: Bricklayers

Coaching Sample Responsesa. What does each of the variables represent?

N represents the number of bricks.

L represents the length of the wall in feet.

H represents the height of the wall.

b. What variable is the given formula solved for?

The formula is solved for N, the number of bricks.

c. Which variable does the formula need to be solved for?

The formula needs to be solved for L, the length of the wall.

d. Solve the given formula for the unknown variable.

N LH

N

H

LH

H

LN

H

7

7

7

7

7

=

=

=

e. What values are given in the problem statement?

The number of bricks, N, is 1,820.

The height of the wall, H, is 4 feet.

Original equation

Divide both sides by 7H.


Instruction


© Walch Education

f. How long can the wall be if the bricklayer has 1,820 bricks and the wall must be 4 feet high?

Substitute the known values into the formula.

LN

H71820

7(4)1820

2865

=

=

=

=

The length of the wall, L, can be 65 feet.

Recommended Closure Activity

Select one or more of the essential questions for a class discussion or as a journal entry prompt.

Original equation

Substitute 4 for H.

Multiply 7 and 4.

Simplify.


naMe:


Practice 1.5.1: Rearranging Equations and FormulasFor problems 1–4, solve each equation for y.

1. 9y + 18 = 27x

2. 6y + 24x = 66

3. 10x – 77 = 7y

4. 44 – 4y = 20x

Read each scenario and solve for the given variable.

5. To convert degrees Celsius to Kelvin, the formula K = C + 273.15 is used. Solve this formula for C.

6. The formula C = 2πr is used to calculate the circumference of a circle. Solve this formula for r.

7. The formula V = lwh is used to calculate the volume of a prism. Solve this formula for w.

8. The formula S = 2πr2 + 2πrh is used to find the surface area of a cylinder. Solve this formula for h.

9. The formula for converting degrees Celsius to degrees Fahrenheit is F C9

532= + . Solve this

formula for C.

10. The formula for calculating the volume of a cone is V r h1

32π= . Solve this formula for h.

Documents

naMe: Unit 1 • Relationships Between Quantities …windrivermath.wikispaces.com/file/view/M1+Unit+1+Lesson...Unit 1 • Relationships Between Quantities Lesson 4: Representing Constraints