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Topic 18: Other methods for solving systems 191

Copyright © 2015 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, Agile Mind, Inc.

OTHER METHODS FOR SOLVING SYSTEMS Lesson 18.1 The substitution method

18.1 OPENER 1. Evaluate +2ab c when 2= −a , 3=b , and 5=c .

2. Following is a set of three shape equations. The value for each shape is constant in the three equations. Find the values for the shapes. Then explain your reasoning.

+ + = 15

3 i = 12

− = 2

a. Answer:

= ____ = ____ = ____

b. Here is how I figured out the answer:

c. Here is how I know that my answer is correct:

18.1 CORE ACTIVITY 1. In the Opener, you figured out the values of the circle, square, and triangle in a shape equation.

a. Use the same type of reasoning you used to solve the shape equation to figure out the values of x and y in the following system.

26

=⎧⎨ + =⎩

y xx y

Answer: =x ______; =y ______

b. Here is how I figured out the answer:

2. The swamp has a perimeter of 124 feet. The length of the swamp is 10 feet less than 5 times its width. If l = length of the swamp in feet and w = width of the swamp in feet, these equations represent the swamp situation:

a. Using the answer choices provided, fill in the blanks in the following statement to describe these equations.

system Together, the two equations that model this situation form a of

in .

set four variables equations

two variables

b. Just as you did in the Opener and in question 1, use substitution to solve the system representing the swamp situation.

192 Unit 6 – Systems of linear equations and inequalities


3. Your class will review the solution to the Swamp Problem using the substitution method.

a. Record the solution steps on the notepad image. This example will be an important reference for you as you solve linear system problems using the substitution method.

b. What substitution did you make in the second equation, 2 2 124+ =l w ?

c. After you made this substitution into the second equation, how many variables were left in that equation?

d. A common mistake when solving systems of equations is to only partially solve the system. In the example on the notepad image, you found a value for the variable w. What do you need to do now to get a full solution?

e. List two ways that you can report the answer to the system of equations for the Swamp Problem.

4. In the animation you saw that you could find l by substituting the value you found for w, 12, into the equation 5 10= −l w . Could you have instead substituted 12 for w in the other equation, 2 2 124+ =l w , to find l ?

Try it to see if you get the same result.

5. Check to make sure that the solution you found makes both equations true.



6. Your class will discuss the following questions. After the class reaches agreement, record the answers.

a. Here is the basic idea involved with the substitution method for solving a system of equations:

b. Here are the steps to take when using the substitution method for solving a system of equations:

7. Work with your partner to solve the following systems. Use the substitution method. Check your answers.

2 356 3 15= −⎧

⎨ + =⎩

y xx y

2 3 1020 6+ = −⎧

⎨ + =⎩

x yx y



18.1 CONSOLIDATION ACTIVITY 1. Simplify each expression by applying the Distributive Property.

a. 2(3 4)−x

b. ( 3)− −x

c. 5(2 7 )− − x d. 14(6 10)−x

2. Simplify each equation by combining like terms.

a. + − + =3 2 3 6 24x y x y

b. 2 3 2 3 96+ − + + − =xy x y xy y x

c. 2 3 3 2− = −n m m n

d. 2.5 4.2 16 4.2 2.5 3= + − + −a b b a c

3. Solve each equation for the variable indicated. a. 7+ =x y for x

b. 4 5 12 0+ − =x y for y

c. 4 3 2+ =y x for x

d. 3( 3 2 ) 16 2− + = +x y x for y

e. =PV nRT for T



HOMEWORK 18.1 Notes or additional instructions based on whole-‐class discussion of homework assignment:

1. Why might you want to use methods other than graphing or tables to solve systems of equations?

2. Consider the following system of equations:

2 42 3 28− =⎧

⎨ + =⎩

y xx y

a. Solve the first equation for y.

b. Now solve the system of equations using the substitution method.

3. Solve each of the following systems by substitution.

a. 2 15

+ =⎧⎨ = −⎩

x yx y

b. 43

− =⎧⎨ =⎩

y xy x

c.

2=⎧

⎨ = − +⎩

y xy x

d. 5 43 4

= +⎧⎨ + =⎩

y xx y

e. 123

− =⎧⎨ + =⎩

x yx y

f.

2 35

+ =⎧⎨ + =⎩

x yx y

4. Use the six-‐step process for solving systems of equations problems to solve the following problem. Use the substitution

method.

The sum of two numbers is 10 and the difference of those two numbers is 18. (Use f to represent the first number and s to represent the second number.)



STAYING SHARP 18.1 Re

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ing pre-‐algebra ideas

1. Lisa set a weekly goal of jogging 18 miles. On each of

the first 3 days, she jogged 328miles. How many

more miles must she jog to reach her goal?

Answer with supporting work:

2. The population of Anytown is 8000. It is predicted to increase by 5% in one year. What is the predicted population after one year?


Practic

ing algebra skills & con

cepts

3. Simplify each of the following expressions by applying the Distributive Property:

− − + =1( 2 )x y

14 2

2⎛ ⎞− =⎜ ⎟⎝ ⎠

c b

3(12 5 )n− − =

4. Write a situation that matches the following equation. (There are many correct answers.)

7.50 2.50= +y x

Situation:

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5. Matthew and Karen are shopping for vegetables to make soup. Karen buys 1 onion and 2 carrots. Her total cost (before tax) is $1.30. Matt buys 3 onions and 6 carrots. His total cost before tax is $3.90. Create two equations that you could use to find the cost of a single onion and a single carrot. Use the variable n to represent 1 onion and the variable c to represent 1 carrot.

Equations:

6. For the situation in question 5, find the cost of a single carrot and the cost of a single onion.




Lesson 18.2 More on the substitution method

18.2 OPENER

Consider the following system of linear equations: 3 4 10

1+ =⎧

⎨ = −⎩

x yy x

1. Solve the system using the substitution method.

2. Following is a graph of the two equations in the system. Explain how the graph shows the solution to the system. Note that the first equation, shown here as Y1, has been converted to slope-‐intercept form.

3. Following is a table that shows the two equations in the system. Explain how the table shows the solution to the system. Note that the first equation, shown here as Y1, has been converted to slope-‐intercept form.

18.2 CORE ACTIVITY 1. Solve the following systems of linear equations. Use the substitution method. Check your answer.

a.

316

=⎧⎨ + =⎩

y xx y

b. 2 5

= −⎧⎨ + =⎩

x yx y

c. 2 60

30+ =⎧

⎨ = +⎩

x yx y



2. You were introduced to the following problem earlier in the unit. Set up and solve a system of linear equations to find a solution to the problem. Use the substitution method.

A farmer raises chickens and cows. There are 34 animals in all. The farmer counts 110 legs on these animals. How many of each kind of animal does the farmer have?

Step 1. Read the problem carefully and understand the situation.

Step 2. Identify what you are looking for and assign variables.

Step 3. Write equations to model the conditions in the problem. Report the two equations as a system:

Step 4. Solve the system of equations. Use the substitution method.

Step 5. Check the solution in both equations.

Step 6. Write your answer in a sentence.

18.2 CONSOLIDATION ACTIVITY

1. Line ℓ has the equation 2 4= −y x . Line m has the equation 5= −y x .

a. Graph each line on the coordinate plane provided.

b. Complete the following:

Name a point that is on line ℓ but not on line m.

Name a point that is on line m but not on line ℓ:

Name a point that is on both line ℓ and line m:



2. Line ℓ has the equation 3 4= −y x . Line m has the equation 3 2= +y x .



Name a point that is on line ℓ but not on line m:

Name a point that is on line m but not on line ℓ:


3. Graph the following system of equations on the coordinate plane. Then state the solution to the system and check the solution.

2 41

22

+ =⎧⎪⎨ = +⎪⎩

x y

x y

Solution:

Check:

4. Graph the following system of equations on the coordinate plane. Then state the solution to the system and check the solution.

2 22 3

+ =⎧⎨ − =⎩

y xy x

Solution:

Check:



5. Line ℓ has a slope of 23and passes through

the point (1,1) . Line m has a slope of 2 and passes through the point( , )0 3 .


b. List the intersection point of the two lines.

6. Line ℓ has a slope of 32 and passes through the point( , )4 2

.

Line m has a slope of 32 and passes through the origin.


b. List the intersection point of the two lines.

7. Based on questions 1-‐6, when do two lines meet at one point?

8. Based on questions 1-‐6, when do two lines have no intersection point?



HOMEWORK 18.2

Notes or additional instructions based on whole-‐class discussion of homework assignment:

1. Solve the following systems of linear equations. Use the substitution method. Check your answers. Use notebook paper.

a. 3 2 20

=⎧⎨ + =⎩

y xx y

b.

3 52

= − +⎧⎨ =⎩

y xy x

c. 2 2

2= +⎧

⎨ = +⎩

y xx y

d. 4

4 12=⎧

⎨ + =⎩

x yx

e. 62 18= −⎧

⎨ + =⎩

y xx y

f. 2 7

3 2 3+ =⎧

⎨ − = −⎩

x yx y

2. Line ℓ has the equation 3 3= +y x .

Line m has the equation 1

42

= − −y x .



Name a point that is on line ℓ but not on line m:

Name a point that is on line m but not on line ℓ :


3. You were introduced to the following problem earlier in the unit. Set up and solve a system of linear equations to find a solution to the problem. Use the substitution method.

The school auditorium seats 310 people. For a particular performance, all of the seats in the auditorium are reserved. The number of seats reserved for students is 25 more than twice the amount reserved for adults (faculty, staff, and parents). How many seats are reserved for students? How many seats are reserved for adults?



4. Create a linear system of equations in which the two lines do not intersect. Then graph your system on the grid provided.

⎧⎨⎩

5. Create a linear system of equations in which the two lines intersect at one point. Graph your system of equations on the

grid provided. State the intersection point and then check the solution.

⎧⎨⎩

Solution: ( , )

Check:




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1. A rectangle in the coordinate plane has vertices at(0,0), (4,0), and (0, 2)− . What are the coordinates of the fourth vertex?


2. Find the sum:

3 45 9− +


Practic


cepts

3. What is the value of 2 23 ( 3)x x− + − when 1x = − ?


4. In the following set of shape equations, the value of the symbols is constant. Find the values of the shapes.

+ + = 3

4 · = 8−

+ = 5

Answer: = ____; = ____; = ____

Evidence for Answer:

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5. Is the following equation true? ( )( )2 2 3 3 4 4 2 3 4 2 3 4⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅


6. Is the following equation true?

( )3(6 5)(6 5)(6 5) 6 5⋅ ⋅ ⋅ = ⋅ ?




Lesson 18.3 Applying the substitution method

18.3 OPENER Consider each of the following situations and then state what conclusion can be drawn.

1. Fran is the same height as Jan. Fran is the same height as Nan.

Conclusion:

2. The cost of tuition at East Central State College is the same as the cost of tuition at West Central State College. The cost of tuition at East Central State College is the same as the cost of tuition at North Central State College.

Conclusion:

18.3 CORE ACTIVITY Here is a summary of key facts from the Hot Chocolate Problem:

• 15 hot chocolates (some large and some small) were ordered. • A small hot chocolate costs $2. • A large hot chocolate costs $3. • The total amount of the order was $42.

Use the six-‐step process and the substitution method to solve the problem. Step 2 is done for you.


Step 2. Choose variables to represent the unknowns in the problem

Let S = the number of small hot chocolates ordered

L = the number of large hot chocolates ordered

Step 3. Write two equations to model the conditions in the problem.

Report the two equations as a system:

Step 4. Solve the system of equations you wrote. Use the

substitution method. Step 5. Check your answer.

Step 6. Report your answer in the context of the problem situation.

EXTENSION: Solve the problem using another system of equations solution method (tables or graphing).



18.3 CONSOLIDATION ACTIVITY 1. The substitution method is easier to use with some systems than others. Here are three different systems of equations.

Solve each of these systems using substitution.

a. 2 5 182 10− =⎧

⎨ = −⎩

x yy x

b. 5 4

3 10 10+ =⎧

⎨ + =⎩

x yx y

c. 2 3 113 2 4

+ =⎧⎨ + =⎩

x yx y

2. Were some of them more difficult than others to solve using substitution? Why?

System Rate the difficulty level of solving using the substitution method.

(Circle your choice.) Explanation of your rating

2 5 18

2 10− =⎧

⎨ = −⎩

x yy x

Easy

Medium Difficult

5 4

3 10 10+ =⎧

⎨ + =⎩

x yx y

Easy

Medium Difficult

2 3 113 2 4

+ =⎧⎨ + =⎩

x yx y

Easy

Medium Difficult

3. The following problem involves a system with three variables and three equations. Use the substitution method to

solve this system.

3 126

1

=⎧⎪ + + =⎨⎪ = −⎩

aa b cc a

4. How can you apply reasoning similar to that which you used in the Opener to solve the following problems? Think, for example, about the Jan, Fran, and Nan height problem. (Do not solve the problems yet. Just provide an explanation to answer the question.)

a. 2 32

= −⎧⎨ = +⎩

y xy x

b. 2 39

= − +⎧⎨ = +⎩

y xy x

c. 0.5 42 13x − =⎧

⎨ = −⎩

yy x

Explanation:

5. Apply the method you described in question 4 to solve the following problems. Show your work. (Note: We will call this method the y-‐equals method. You should know that this is not “official” mathematics terminology. Actually, the “y-‐equals method” is a special case of the substitution method. Can you see why?)

a. 2 32

= −⎧⎨ = +⎩

y xy x

b. 2 39

= − +⎧⎨ = +⎩

y xy x

c. 0.5 42 13x − =⎧

⎨ = −⎩

yy x



HOMEWORK 18.3


1. Identify the error that was made in the following problem when applying the substitution method. Then correctly solve the

problem using the substitution method.

Incorrect solution: 7

2 8= +⎧

⎨ + =⎩

x yx y

2( 7) 82 7 8

3 7 83 1

13

y yy y

yy

y

+ + =+ + =

+ ==

=

Description of error:

71

731 22

73 3

x y

x

x

= +

= +

= =

Solution: 22 1

,3 3

⎛ ⎞⎜ ⎟⎝ ⎠

Corrected solution: 7

2 8= +⎧

⎨ + =⎩

x yx y

2. Solve the following problems using the substitution method. Use notebook paper. Here are some reminders:

• Consider which variable will be easier to substitute for.

• Consider instances where the y-‐equals method might be useful.

a. 3 84

= −⎧⎨ = −⎩

y xy x

b.

3 8=⎧

⎨ + = −⎩

x yx y

c. 4

3 2 28=⎧

⎨ + =⎩

x yx y

d. 6 5 113 13

− + =⎧⎨ − =⎩

x yy x

e. 84

= − +⎧⎨ = +⎩

x yx y

f.

2 53 2 4

+ =⎧⎨ − =⎩

a ba b

3. Set up and solve the following problem using the substitution method. Use notebook paper.

You and your friends decide to rent some studio time to make a CD. Big Notes Studio rents for $100 plus $60 per hour. Great Sounds Studio rents for $25 plus $80 per hour. Determine the number of hours for which the cost of renting the studios is the same.




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1. In ΔABC , the measure of ∠A is 108° and the measure of ∠B is 38° . What is the measure of ∠C ?


2. Find the least common multiple (LCM) of each pair of numbers:

LCM of 3, 9: LCM of 7, 3: LCM of 6, 9:

Practic


cepts

3. What is the equation for the graph below?

1 2 3 4 5

1

2

3

4

5

6

7

8

9

10

x

y

Answer:

4. Solve the following equation for x:

2 46 8x x +=


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5. A joke is being spread by a group of students. Each day, any student who has heard the joke tells it to 1 student who has not already heard it. If 2 students knew the joke at the end of the first day and 4 students knew it at the end of the second day, how many students knew the joke at the end of the third day? How many knew it at the end of the fourth day?


6. For the situation in question 5, how many students knew the joke at the end of the tenth day?




Lesson 18.4 The linear combination method

18.4 OPENER Use the information from Scale 1 and Scale 2 to explain why Scale 3 must balance.

Scale 1 Scale 2 Scale 3

18.4 CORE ACTIVITY 1. You watched an animation that showed another method, the linear combination method, for solving linear systems

problems. The system that was solved in the animation is shown here. Now solve the system yourself, using the linear combination method.

2 123 2 4+ =⎧

⎨ − = −⎩

x yx y

2. Why did the system 2 12

3 2 4+ =⎧

⎨ − = −⎩

x yx y

solve so easily using the linear combination method?

3. Summarize the linear combination method for solving systems of equations and then write down the steps for using this method to solve a system.

a. Here is the basic idea of the linear combination method for solving a system of equations:

b. Here are the steps to take when using the linear combination method for solving a system of equations:

4. Why does the linear combination method work? You can use ideas from the balance scale animation in your explanation.



5. Use the linear combination method to solve these systems, then answer the questions that follow.

System 1

2 3 762 5 20x yx y+ =⎧

⎨− + =⎩

System 2

5 6 195 3 8a ba b− =⎧

⎨− + =⎩

a. What makes linear combination a good method for solving these systems of equations?

b. Can you think of a system for which linear combination might not be a good method? Write your system below.

c. Why might it not be a good method for the system of equations that you created?

6. Consider the following system of equations. 4 5 224 3 18x yx y+ =⎧

⎨ + =⎩

a. Why doesn’t adding the two equations work?

b. What can you do to the equations so you can use the linear combination method to solve this system?

c. Carry out your plan to find a solution for this system using the linear combination method. (Be sure to get an

answer for both x and y.)

7. Consider this system of equations: 2 9 124 3 6

+ =⎧⎨ + = −⎩

x yx y

a. What can you do to the equations so you can use the linear combination method to solve this system? b. Carry out your plan to find a solution for this system using the linear combination method. (Be sure to get an

answer for both x and y.)




1. Consider this system, then answer the following questions. 3 10 1618 2 34x yx y− =⎧

⎨ + =⎩

a. What first step could be applied to use the linear combination method to solve this system using the coefficients of the x-‐terms?

b. What first step could be applied to use the linear combination method to solve this system using the coefficients of the y-‐terms?

2. Now solve the system in question 1 using the two different methods. In the left-‐hand column, apply the method by

manipulating the coefficients of the x-‐terms. In the right column, apply the method by manipulating the coefficients of the y-‐terms.

Linear combination method solution by manipulating the coefficients of the x-‐terms

Linear combination method solution by manipulating the coefficients of the y-‐terms

3. What conclusion/generalization can you make based on your work in question 2?



4. Think back to today’s Opener. In what ways is solving the Fruit Balancing Problem similar to solving a system of linear equations problem using the linear combination method?

Scale 1 Scale 2 Scale 3




Use notebook paper to record your work for these problems.

1. Solve each of the following systems using the linear combination method. Check your answers.

a. 2 92 5

+ =⎧⎨ − =⎩

x yx y

b. 9

2 0+ =⎧

⎨ − =⎩

x yx y

c. 3 2 7

5 9+ = −⎧

⎨ − =⎩

x yx y

d. 2 7 52 3 9

+ =⎧⎨ + =⎩

x yx y

e. 8 3 212 9 24

− = −⎧⎨ + =⎩

a ba b

f. 8 9 194 7− =⎧

⎨ + = −⎩

x yx y

2. Identify the error that was made in the following problem when applying the linear combination method. Then correctly solve the problem using linear combination.

Incorrect solution: 2 3 242 8x yx y+ =⎧

⎨ − =⎩

(2 3 ) (2 ) (24 8)

2 168

2 3(8) 240

x y x yyy

xx

+ − − = −==

+ ==

Solution: ( )0,8

Description of error:

Corrected solution: 2 3 242 8x yx y+ =⎧

⎨ − =⎩

3. You were introduced to the following problem earlier in the unit. Set up and solve the problem using the linear

combination method.

Maggie and Mia go shopping together. At the Fashion Bee, shirts cost one price and sweaters cost one price. Maggie buys 2 shirts and 2 sweaters for $86. Mia buys 3 shirts and 1 sweater for $81. What is the cost of a shirt? What is the cost of a sweater?




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1. How long is the rope that can be used to totally enclose the rectangular space at the beach, as shown in this diagram?


2. In this right triangle, what is the length of BC?

Answer with supporting explanation:

Practic


cepts

3. Complete the table, then make a graph, for the rule

21

2y x= − .

x Y

-‐4

-‐2

0

2

4

4. What is the story of the graph?

Answer:

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5. Jefferson was solving the system of equations shown here. He substituted 2y x= into the other equation for y, but forgot to finish solving the problem. Find the solution for Jefferson by continuing where he left off.

5 6 14x y− = and 2y x=

Jefferson’s work: 5 6(2 ) 14x x− =


6. Is the following equation true? 32 2 4 4 5 5 (2 4 5)⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅




Lesson 18.5 The linear combination method continued

18.5 OPENER

Candi, Sandy, and Andy are working as a group to solve the linear system shown. They plan to use the linear combination method. They discuss how to do this.

2 3 185 2 7

+ =⎧⎨ − =⎩

a ba b

Candi: I don’t think this system can be solved because if we want to eliminate the a terms in both equations, we can’t multiply the 2 in the first equation by an integer to get a −5, which is what we would want so that when we add the equations the a terms drop out. And if we want to eliminate the b terms in both equations, we can’t multiply the −2 in the second equation by an integer to get a −3, which is what we would want so that when we add the equations the b terms drop out.

Sandy: But what if we multiply the first equation by −5 and the second equation by 2. Won’t that cause the a terms to drop out when we add the equations?

Andy: I think we can multiply the first equation by 2 and the second equation by 3. Then, when we add the equations, we’ll get rid of the b terms.

Which student is (or students are) correct? Explain your answer.

18.5 CORE ACTIVITY 1. Solve the following systems of linear equations. Use the linear combination method. Be sure to check your answers.

a.

4 2 27 3 23x yx y− =⎧

⎨ + =⎩

b.

5 2 14 9 14x yx y− =⎧

⎨− + =⎩

c. 7 2 35 3 10x yx y+ =⎧

⎨ + =⎩



2. You were introduced to the following problem earlier in the unit. Set up

and solve a system of linear equations to find a solution to the problem. Use the linear combination method. Investment A starts with $1,000 and adds $60 per week. Investment B starts with $1,500 and loses $40 per week. After how many weeks will the two investments have the same balance?


Step 2. Identify what you are looking for and assign variables.

Step 3. Write equations to model the conditions in the problem. Report the equations as a system:

Step 4. Solve the system of equations you wrote. Use the

linear combination method. Step 5. Check the solution in both equations.

Step 6. Write your answer in a sentence.


1. Ciarra and Ashley sold tickets for the high school basketball tournament. For the opening game of the tournament, the girls sold 20 student tickets and 30 non-‐student tickets and made $190. For the championship game, they sold 50 student tickets and 40 non-‐student tickets and made $300. What was the price of a student ticket? What was the price of a non-‐student ticket?

a. Assign variables to the unknown quantities.

b. Write a system of equations to model the problem.



c. Solve the system using the linear combination method.

d. Check the answer in both equations.

e. Report your answer in a complete sentence.

2. Solve the following systems using the linear combination method. Check your answers.

a.

4x − 3y = −7−8x + 2y = −6

⎧⎨⎩⎪

b.

5x − 4y = −73x − 3y = −21

⎧⎨⎩⎪



HOMEWORK 18.5


Use notebook paper to record your work for the problems in this homework assignment.

1. Solve each of the following systems using the linear combination method. Check your answers.

a. 4 2 124 18x yx y+ =⎧

⎨− + = −⎩

b. 3 17

2 14x yx y+ =⎧

⎨ + =⎩

c. 2 5 10

10 3 6x yx y

− + =⎧⎨ + =⎩

d. 4 2 147 3 8x yx y+ =⎧

⎨ − = −⎩

e. 7 20 483 10 22x yx y+ =⎧

⎨ + =⎩

f. 3 5 115 8 18x yx y+ =⎧

⎨ + =⎩

2. Earlier in the unit you solved a problem similar to the following problem. Set up and solve this problem using the linear combination method.

Joseph and Patrick purchase school supplies in the school bookstore. Joseph purchases 4 notebooks and 3 pens for $11. Patrick purchases 3 notebooks and 5 pens for $11. What is the price of a notebook? What is the price of a pen?



STAYING SHARP 18.5

Review


1. Find the least common multiple (LCM) of each pair of numbers:

LCM of 2, 8:

LCM of 3, 4:

LCM of 6, 8:

2. Jonathan has 7 hours to finish three jobs. If the

first job takes him 324hours and the second job

takes him 516hours, how much time does he have

to complete the third job?


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3. If you follow the Magic Number Puzzle steps below, how is the ending number related to the starting number?

Directions How your number changes

Write down a number.

Add 6.

Multiply by 2.

Divide by 2.

Answer:

4. Solve for y:

2(2 5) 3 8y y+ = −


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5. Use the following graph to find the coordinates of the point where the lines 2 6y x= − + and 3 1y x= +intersect.

-0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5

1234

5678

9

x

y

6. Solve the system of equations graphed in question 5 algebraically.




Lesson 18.6 Connecting the algebra and geometry of systems of equations

18.6 OPENER Eugene and Jennifer are saving money for a summer art class. Eugene starts with $60 and adds $15 each week. Jennifer starts with $45 and adds $15 each week.

1. Write an equation for each student that represents the money saved, m, as a function of the number of weeks that have passed, w.

2. Use your graphing calculator to graph each equation. Pick an appropriate viewing window in quadrant I. Sketch your graph below and report the viewing rectangle you used.

3. Based on the equations and graphs, will Jennifer be able to “catch up” to Eugene in terms of money saved? If so, report when this will take place. If not, explain why.

18.6 CORE ACTIVITY 1. Solve the following system of equations using either the substitution method or the linear combination method.

2 34 2 14= +⎧

⎨ + =⎩

y xx y

Which statement (or statements) best describes what happened when you solved the system? Check all that apply.

The answer includes one value for x and one value for y.

Both the x and y simplified out of the equations, leaving only numbers.

The numbers made a false equation, like 0 12= .

The numbers made a true equation, like 0 0= or 18 18= .

Convert the equations in the system to slope-‐intercept form (write your equations in Y1 and Y2) and then graph the equations in the standard viewing rectangle. Sketch the graph in the space provided and, if it applies, state the intersection point.

Intersection point: ________

2. Solve the following system of equations using either the substitution method or the linear combination method.



4 2 62 2− = −⎧

⎨− + = −⎩

x yx y


The answer included one value for x and one value for y.






3. Solve the following system of equations using either the substitution method or the linear combination method.

2 6

3 6 18+ =⎧

⎨ + =⎩

x yx y


The answer included one value for x and one value for y.








4. Based on these three examples, what conjectures can you make about the relationship between algebraic and geometric solutions to systems of linear equations? Complete the following table to describe your conjectures.

Algebraic result What does the graph look like? What might this tell you about the number of solutions for the system?

Answer includes one value for x and one value for y.

Equations simplify to a false equation containing only numbers (for example, 0 = 12).

Equations simplify to a true equation containing only numbers (for example, 18 = 18).

5. Create a graph of a system that is a single line. For this case, it is easier to write the system first. To see how to create such a system, look at the system in question 3. Look for a pattern in the values of the coefficients of the x-‐terms, the y-‐terms, and the constants in the two equations.

Graph System of equations Algebraic solution

⎧⎪⎪⎨⎪⎪⎩

18.6 ONLINE ASSESSMENT Today you will take an online assessment.




1. Solve each system by substitution OR linear combination. Try to make good choices about which method to use to solve each problem efficiently. Report your solution and check your answer. Write no solution or infinitely many solutions where appropriate. Use notebook paper to record your work.

a.

5 115 1

= −⎧⎨ = −⎩

y xy x

b.

6 3 62 5

− =⎧⎨ − = −⎩

x yx y

c.

3 63 6= −⎧

⎨− + = −⎩

y xx y

d.

4 2 82 2+ =⎧

⎨− − =⎩

x yx y

e.

3 6 62 3 4

+ =⎧⎨ − =⎩

x yx y

f.

3 103 4

+ =⎧⎨ = − +⎩

x yy x

2. Complete the triple-‐entry journal below following the directions.

Directions: In the FIRST column is a list of types of solutions to system of equations problems. In the MIDDLE column, describe in your own words what happens when you solve each type of system algebraically. In the LAST column, describe in your own words what happens when you solve each type of system graphically.

Type of solution to systems of equations problem

What happens when you solve algebraically?

What happens when you solve graphically?

a. One-‐solution case

b. No-‐solution case

c. Infinitely many solutions case



3. Antonio is starring in the high school play and has lots of family members and friends who want to see him perform. He has purchased 17 tickets for the play. Some of these are adult tickets and some are child tickets. The cost of an adult ticket is $4 and the cost of a child ticket is $2. If the total cost of the 17 tickets was $60, find the number of each type of ticket purchased. a. Set up a system of equations to represent the problem. b. Solve the problem using the substitution method.

c. Solve the problem using the linear combination method.



4. In today’s lesson, you tested a conjecture about a system of equations in which the graph was a single line. Now, you will test your conjectures for the other two cases. Create a graph of a system that fits the description given. Then write equations for the system and solve the system algebraically. Do your results support your conjectures?

a. Graph of the system is two lines that intersect at a single point.

Graph System of equations Algebraic solution

⎧⎪⎪⎨⎪⎪⎩

b. Graph of the system is two parallel lines.

System of equations Graph Algebraic solution

⎧⎪⎪⎨⎪⎪⎩



STAYING SHARP 18.6

Review


1. Using the chart shown, what is the total distance, in miles, of biking for the five different legs of the trip?


2. A high school play performance had two performances, one on Friday and one on Saturday. Of the tickets sold for both performances, 60% were sold for the Friday performance. If 120 tickets were sold for the Friday performance, how many were sold for the Saturday performance?


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3. What is the rule for the Input-‐Output table shown?

Input Output -‐2 4 -‐1 1 0 0 1 1

2 4 3 9

Answer:

4. Change the following equation to slope-‐intercept form: 3 2 5+ =x y


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5. Write and solve a system of equations for the following Square Box Problem:

ab=⎧

⎨ =⎩

6. What are the coordinates of the point where the lines 2 1y x= + and 4 37x y+ = intersect?


a

32

-‐8

b



Lesson 18.7 Choosing a linear system solution method

18.7 OPENER For each of the following problems, decide what method you would use to solve the system of equations. Think about which method would be most efficient for each problem. Circle your answer choice, then write a brief explanation of why you would choose that method. You do not need to solve the system of equations.

1. 3 102 5

= −⎧⎨ = −⎩

y xy x

Tables Graphing Substitution Linear combination

Reason for choosing method:

2. 2 3 185 3 3

− =⎧⎨ + =⎩

x yx y



3. 4

2 4 10= −⎧

⎨ − =⎩

x yx y



4. 2 2 96

3+ =⎧

⎨ =⎩

l wl w



18.7 CORE ACTIVITY For each of the following problems, choose a method to solve the system of equations. Solve the system using that method, then explain why you chose that method. Use each method exactly once. You will need graph paper for this activity.

1. 2 7

4 3 11= − +⎧

⎨− + =⎩

y xx y

Method chosen (circle): • Tables

• Graphing

• Substitution method

• Linear combination method

State why this was a good method to use for this particular problem:



2. 3 12 11

= +⎧⎨ = +⎩

y xy x


• Graphing




3. 7 2.5 13 2.5 4

− =⎧⎨ + =⎩

x yx y


• Graphing




4. 7

2 4 30+ =⎧

⎨ + =⎩

a ba b


• Graphing






18.7 REVIEW ONLINE ASSESSMENT You will work with your class to review the online assessment questions. Problems we did well on: Skills and/or concepts that are addressed in these problems:

Problems we did not do well on: Skills and/or concepts that are addressed in these problems:

Addressing areas of incomplete understanding

Use this page and notebook paper to take notes and re-‐work particular online assessment problems that your class identifies. Problem #_____ Work for problem:

Problem #_____ Work for problem:

Problem #_____ Work for problem:




Next class period, you will take the end-‐of-‐unit assessment. One good study strategy to prepare for tests is to review what you have learned. Here is a list of some of the important skills and ideas that you have worked on in this unit. Use this list to help you review these skills and concepts, especially by looking at your course materials. Another good study strategy to prepare for tests is to re-‐work problems that you did in class. Some specific activities to study/re-‐work are listed after each concept or skill.

Important skills and concepts from the unit:

• Identify the variables and conditions in a situation and write a system of equations;

• Understand the meaning of a solution of a system of equations and verify a solution;

• Solve a system of equations using tables;

• Solve a system of equations using graphs by hand;

• Understand the importance of your mindset as a factor that can impact your motivation and learning;

• Solve systems of linear equations using the substitution method;

• Solve systems of linear equations using the linear combination method;

• Recognize and write the solution set for “special cases” of linear systems (systems in which the two lines are parallel and systems in which the two lines are collinear); connect the algebraic solution of a system of linear equations to the geometry of the case (one solution, no solution, infinite solution cases).

Homework Assignment

Part I: Study for the end-‐of-‐unit assessment by reviewing the key ideas listed above.

Part II: Complete the online More practice in the topic Other methods for solving systems. Note the skills and ideas for which you need more review, and refer back to related activities and animations from this topic to help you study.

Part III: Complete Staying Sharp 18.7

As you complete the More practice, record below any questions you may have or challenges you encountered with the items.



STAYING SHARP 18.7

Review


1. A stack of 500 sheets of paper is 3 inches in height. What is the thickness of one sheet of paper, in inches? Answer with supporting work:

2. Convert 40 feet per minute to inches per second. Answer with supporting work:

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3. Use the rule 23 2y x= − + to complete the missing entries in the following table.

x y

-‐10

-‐5

-‐1

0

1

5

10

4. In the following sequence, 2 is the first term , −4 is the second term, −14 is the third term, and so on.

2, -‐4, -‐14, -‐28, -‐46, -‐68, …

If n represents the term number, show that the following rule will generate the sequence:

22 4n− + .

Show:

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5. Dhara graphs the functions 10 7 18x y− = and 5 3 2x y− = − to see where they intersect. She finds that

22y = − . What value does she find for x? -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

-24

-22

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

xy


6. After Dhara graphs the functions in question 5, she decides to use the linear combination method to verify her answer. Show how she could solve the system. Answer with supporting work:



Lesson 18.8 Assessing understanding

18.8 OPENER

A Fun Park refreshment stand sells only hotdogs and nachos. Hotdogs cost $2 and nachos cost $3. Marcus and his friends spent $34 at the refreshment stand. All together, they ordered a total of 14 items. How many hotdogs and how many nachos did they buy? Write a system of two linear equations in two variables that can be used to model the situation. Then choose a method to solve the system. Show your work and check your answer.

18.8 END-OF-UNIT ASSESSMENT

Today you will take the end-‐of-‐unit assessment.


1. What is the solution to the system of linear equations 2 5 10+ =x y and 2 3 2+ =x y ?

2. In the coordinate plane, what are the coordinates of the point where the lines 1+ =x y and 2 1+ =x y intersect?

3. If 2 6− =x y and 4 12+ =x y , what is the value of y?

4. Given 3 2 10+ =x y , what does 12 8+x y equal?

5. Christine went to a sale at a media store. She bought 8 videos and CDs for $92. If each video cost $16 and each CD cost $10, how many of each did she buy?

a. Using v to represent the number of videos that Christine bought and c to represent the number of CDs that she bought, set up a system of equations that models the information in the problem.

b. Solve the system of equations and report how many of each product Christine bought.

6. For what value of a would the following system of equations have an infinite number of solutions?

2 68 4 3

− =− =

x yx y a



7. Answer the following questions to reflect on your performance and effort this unit.

a. Summarize your thoughts on your performance and effort in math class over the course of this unit of study. Which areas were strong? Which areas need improvement? What are the reasons that you did well or did not do as well as you would have liked?

b. Set a new goal for the next unit of instruction. Make your goal SMART.

Description of goal:

Description of enabling goals that will help you achieve your goal:




In questions 1-‐3, you will apply your knowledge about systems of equations to a setting involving an amusement park. Use notebook paper to record your work for questions 1-‐3. 1. The admission fee at Fun Park is $13 for adults and $9 for children.

On a certain day, 940 people entered the park and $10,148 was collected. How many children attended Fun Park on that day? Use a system of equations approach to find an answer. Show all work.

2. The popcorn stand sells only soft drinks and popcorn, and only one

size of each. In fact, the same cup is used for both products—this is part of the stand’s “sales pitch.” A soft drink costs $2.00 and a popcorn costs $4.00. On a certain day, 120 cups were used and $330.00 was collected. How many soft drinks and how many popcorns were sold on that day? Use a system of equations approach to find an answer. Show all work.

3. Write your own Fun Park Problem. The problem should involve a

system of linear equations. Then, find an answer to the problem, showing all of your work. For your Fun Park Problem, you can use the information about admissions fees and the popcorn stand given in problems 1 and 2, you can use information about Snack Shack prices given here, or you can make up your own information.




view


1. Express the rate 50 yards per minute in feet per second.


2. Henry and Adam each mow lawns during the

summer months. Henry takes 122hours to mow a

particular yard. Adam can mow the same yard in 2

13hours. How much faster can Adam mow the

lawn than Henry? Express your answer in minutes.


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3. Use the rule 3xy = to complete the missing entries in the table.

x y 0 1 1 3 2 9 3 4 5 6

4. Graph 2y x=

-5 -4 -3 -2 -1 1 2 3 4 5-2

2

4

6

8

10

12

14

16

18

20

x

y

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5. What are the coordinates of the point where the lines 2 4x y+ = and 6 10x y+ = intersect?


6. Explain how you can tell that the lines 4 10 3x y− = − and 2 5 1.5x y− = − are collinear by looking at the equations.

Answer with reasoning:

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