Algebra - White Plains Middle School...9 Chapter 6-2 Solving Systems by Substitution SWBAT: solve...

Algebra

Chapter 6: Systems of Equations and

Inequalities

Name:______________________________

Teacher:____________________________

Pd: _______

Table of Contents

Chapter 6-1: SWBAT: Identify solutions of systems of linear equations in two variables; Solve systems of linear equations in two variables by graphing Pgs: 1 – 5 HW: Pgs: 6 – 8 Chapter 6-2: SWBAT: solve systems of linear equations in two variables by substitution Pgs: 9 - 14 HW: Pgs: 15 – 17

Lesson 6-3: SWBAT: solve systems of linear equations in two variables by elimination Pgs: 18 - 22 HW: Pgs: 23 – 24

Word Problems: SWBAT: Write and solve word problems whose solution requires solving systems of linear equations in two variables Pgs: 25 - 30 HW: Pgs: 31 – 33

Review Lesson 6-1 to 6-3: SWBAT: Demonstrate their knowledge of solving systems of linear equations in two variables Pgs: 34 - 38

Lesson 6-5: SWBAT: graph and solve linear inequalities in two variables Pgs: 39 – 43 HW: Pgs: 44 – 45

Lesson 6-6: SWBAT: graph and solve systems of linear inequalities in two variables Pgs: 46 - 49 HW: Pgs: 50 – 52

Review

CHAPTER 6 EXAM

1

Chapter 6 – 1 Solving Systems by Graphing SWBAT: Identify solutions of systems of linear equations in two variables; Solve systems of linear

equations in two variables by graphing

Warm Up

Midterm Review #1 – See attached Sheet

If two or more equation are given, we call this a system of equations. The solution to a system of

equations consists of the set of all ordered pairs, x, y, that satisfy (make true) all of the equations in the

system. In today’s lesson, we will investigate ways of finding this solution set for two linear equations.

Practice: Use the graph below to estimate a solution to the system. Then check your solution algebraically.

Solution: ( ___ , ___ )

Check

2

Practice: Identifying Solutions of Systems

Tell whether the ordered pair is a solution of the given system.

A) (4, 1); 2 6

3

x y

x y

2 6x y

3x y

B) (–1, 2); 2 5 8

3 2 5

x y

x y

2 5 8x y

3 2 5x y

Example 2: Solving Systems of Linear Equations by Graphing

Directions: Solve each system by graphing. Check your answer.

C)

Check:

D)

Check:

All solutions of a linear equation are on its graph. To find a solution of a system of linear equations, you

need a point that each line has in common. In other words, you need their point of intersection.

3

Practice: Solving Systems of Linear Equations by Graphing

Directions: Solve each system by graphing. Check your answer.

5

Challenge / Regents Problem:

Summary:

Exit Ticket:

y = 3x + 1

y = -x + 5

6

Chapter 6- 1 Solving Systems by Graphing HW

Tell whether the ordered pair is a solution of the given system.

1) (3, 1);

754

63

yx

yx 2) (6, –2);

325

1423

yx

yx

Solve each system by graphing. Check your answers.

3)

92

6

xy

xy Solution: __________ 4)

63

6

xy

xy Solution: _________

Check: Check:

63 yx

325 yx

754 yx

1423 yx

7

5)

yx

yx

2

2 Solution: __________ 6)

83

62

xy


Check: Check:

7)

73

43

yx

yx Solution: _________

8) Solution:_________

Check: Check:

8

9.

10.

9

Chapter 6-2 Solving Systems by Substitution

SWBAT: solve systems of linear equations in two variables by substitution

Warm Up


Solving Systems of Equations by Substitution

Step 1 Solve for one variable in at least one equation, if necessary.

Step 2 Substitute the resulting expression into the other equation.

Step 3 Solve that equation to get the value of the first variable.

Step 4 Substitute that value into one of the original equations and solve.

Step 5 Write the values from Steps 3 and 4 as an ordered pair, (x, y).

Step 6 Check!

10

Practice:

Solve the system by substitution.

1)

11

Practice:


2)

12

Practice:


3)

13

Sometimes you substitute an expression for a variable that has a coefficient. When solving for the

second variable in this situation, you can use the Distributive Property.

Word Problems

14

Challenge : Solve for x and y, given ABCD is a rectangle.

Summary:

Exit Ticket:

15

Chapter 6-2 Solving Systems by Substitution – HW Solve each system by substitution. Check your answers.

1)

14

2

xy

xy Solution: __________

Check:

2)

2

4

xy


Check:

3)

35

13

xy


Check:

4)

3

62

yx


Check:

16

5)

7

82

xy


Check:

7) Solution: _________

Check:

6)

12

032

yx


Check:

8) Solution: _________

Check:

17

11.

9. 10.

18

Chapter 6-3 Solving Systems by Elimination

SWBAT: solve systems of linear equations in two variables by elimination

Warm Up


Practice: Solve using Elimination by Addition.

19

In some cases, you will first need to multiply one or both of the equations by a number so that one

variable has opposite coefficients. This will be the new Step 1.

Practice: Solve using Elimination by Multiplication

2.

20

Word Problems At a sale on winter clothing, Cody bought two pairs of gloves and four hats for $43.00. Tori bought two pairs of

gloves and two hats for $30.00. What were the prices for the gloves and hats?

21

Closure

1. Write addition or Multiplication to tell which operation it would be easiest to use to eliminate a variable

of the system. Explain your choice.

2. Tell how you can decide whether to use addition or multiplication to eliminate a variable in a system of

equations.

22

Challenge Problem: Write the equation of a line that contains the point of intersection of the graphs

8x – 3y = 7 and 10x + 4y = -1 and is perpendicular to the line .73

1 xy

Summary:

Exit Ticket:

23

Chapter 6-3 Solving Systems by Elimination HW

24

10. 11.

12.

25

Systems of Equations – Word Problems

SWBAT: Analyze and solve verbal problems whose solution requires solving systems of linear equations in two

variables

Warm – Up


Example 1:

Two health clubs offer different membership plans. The graph below represents the total cost of belonging to Club A and

Club B for one year.

Part A

Write an equation to show the cost of each membership plan at Club A and Club B.

Club A’s Equation: ___________________ Club B’s Equation: __________________

Part B

What is the number of the month when the total cost is the same for both clubs? __________

Part C

What is the total cost when both plans are the same? __________

26

2. Angela is planning to go to an amusement park for the fourth of July. She can either go to Playworld or

Fantastic Adventures.

Part A Write an equation to show how the cost at each amusement park relates to the number of rides.

Playworld: ___________________ Fantastic Adventures: __________________

Part B For what number of rides would the cost be the same? _____________ rides

KEY Playworld

Fantastic Adventures

Amusement Park Costs

Number of rides

C

ost

(doll

ars)

2

4

6

8

10

12

0 1 2 4 5

14

16

18

3

20

27

Example 3:

At Connie’s Couches, a person can rent a couch for $10 a month plus a one-time “wear-and-tear” fee of $50. At

Harry’s Homes, the charge is $20 a month and an additional charge of $20 for delivery with no “wear-and-tear” fee.

Part A - If c equals the cost, write an equation representing the cost of the rental for m months at Connie’s

Couches.

Equation:________________________________

Part B - If c equals the cost, write an equation representing the cost of the rental for m months at Harry’s Homes.

Equation:__________________________

Part C - On the accompanying grid, graph and label each equation.

Part D - From your graph, determine in which month Harry’s cost will equal Connie’s cost.

___________________

28

4. Angela wants to go to an amusement park for the 4th

of July. She can go to Playworld for $12 a day

plus $2 per ride or Great Action Park for $2 a day plus $ 4 per ride.

PART A Write an equation that represents the cost (c) of going to each amusement park in relationship to the

number of rides (r).

Playworld:________________ Great Action Park: __________________

PART B: On the accompanying grid, graph each equation.

PART C For what number of rides would the cost be the same? _______________

PART D For what range of rides would you visit Playworld? __________

For what range of months would you visit Great Action Park? _________

Amusement Park Costs

Number of rides

C

ost

(doll

ars)

2

4

6

8

10

12

0 1 2 4 5

14

16

18

3

20

22

24

26

28

29

Word Problems 5. The total attendance at a school play was 850. The tickets for senior citizens were $1.50 each, and

the regular tickets were $2.00 each. If the total receipts were

$1, 650.00, how many tickets of each kind were sold?

6. Troy has 25 coins in dimes and nickels. The value of his coins is $1.60. How many dimes and

nickels does he have?

30

Challenge Problem:

Summary:

Exit Ticket:

31

Homework – Word Problems

1. The accompanying diagram represents the monthly cost of exercising at two local sports clubs.

Part A

Write an equation to show the cost of each membership plan at NY Sports World and Platinum Gym.

NY Sports World’s Equation: ___________________ Platinum Gym’s Equation: __________________

Part B

What is the number of the month when the total cost is the same for both gyms? __________

Part C

In what month will the total cost for both gyms be the same? __________

KEY NY Sports World

Platinum Gym

Number of months

C

ost

(doll

ars)

40

80

120

160

200

4 8 12 0

240

280

320

360

(x +

10)

Monthly Sports Club Costs 400

32

2. Taylor’s Department Store sells CDs for $15.00 each.

Buyer’s Warehouse sells each CD for $10 each and has a $25 membership fee.

Part A: Write an equation that represents the cost (c) of going to each store in relationship to the number of

CDs bought.

Taylor’s:___________________ Buyer’s: __________________

Part B: Make a graph that shows the cost of purchasing several different quantities of CDs at each store.

Part C: How many CDs would Dee have to purchase so that the cost is the same for Buyer’s Warehouse and

Taylor’s Department Store?

Answer: ____________ CDs

30

CD Costs

Number of CDs

C

ost

(doll

ars)

5

10

15

20

25

0 1 2 4 5

35

45

3

50

55

60

65

70

6 7

75

80

85

90

95

100

105

40

30

33

3. The Town Recreation Department ordered a total of 100 balls and bats for the summer baseball

camp. Balls cost $4.50 each and bats cost $20.00 each. The total purchase was $822.00. How many

of each item were ordered?

4. In a store a total of 70 hammers and wrenches were sold. Hammers sold for $10.00 and wrenches sold for

$5.00. A total of $600.00 were sold. How many hammers and wrenches were sold?

5. Juan has 11 coins in dimes and quarters. The value of his coins is $2.15. How many dimes and quarters does

he have?

34

Chapter 6-1 to 6-3 Review

SWBAT: Demonstrate their knowledge of solving systems of linear equations in two variables

37

Word Problems 21. A baseball manager bought four bats and nine balls for $76.50. On another day, he bought

three bats and twelve balls at the same prices and paid $81.00. How much did he pay for each

bat and each ball? 22. Sharu has $2.35 in nickels and dimes. If he has a total of thirty-two coins, how many of

each coin does he have?

38

23. At Ron’s Rental, a person can rent a big-screen television for $10 a month plus a one-time “wear-and-tear”

fee of $100. At Josie’s Rental, the charge is $20 a month and an additional charge of $20 for delivery with no

“wear-and-tear” fee.

a) If c equals the cost, write one equation representing the cost of the rental for m months at Ron’s Rental

and one equation representing the cost of the rental for m months at Josie’s Rental.

b) On the accompanying grid, graph and label each equation.

c) From your graph, determine in which month Josie’s cost will equal Ron’s cost.

39

Chapter 6-5 - Graphing Linear Inequalities

SWBAT: graph and solve linear inequalities in two variables

Warm Up

Directions: Graph each inequality.

Solve for y: –6x + 2y = –4

What is a Linear Inequality?

Which of the following points is a solution to the inequality above?

(2, 1) (-1, -5) (-3, 2) (0, -2)

Example 1: Identify Solutions of Inequalities

Determine if the ordered pair is a solution of the inequality.

A) (7, 3); 1 xy B) (4,5) 23 xy

Practice: Identify Solutions of Inequalities

1 xy

40

A linear inequality describes a region of a coordinate plane called a half-plane. All points in the region are

solutions of the linear inequality. The boundary line of the region is the graph of the related equation.

Example 2: Graphing Linear Inequalities in Two Variables

C) Graph the solutions of each linear inequality.

43 xy

Step 1: Solve for y.

Step 2: Graph the boundary line.

(Solid or dashed)

Step 3: Shade the half-plane.

Step 4: Check by plugging in

a point in the shaded region.

Graphing Linear Inequalities

Step 1: Solve the inequality for y (slope-intercept form).

Step 2: Graph the boundary line. Use a solid line for ≤ or ≥ . Use a dashed line for < or >.

Step 3:

Pick a point and plug it into the inequality to determine what area needs to be shaded.

Shade the region above the line for y > or ≥.

Shade the region below the line for y < or ≤ .

Step 4: Check your answer.

41

Practice: Graphing Linear Inequalities in Two Variables

Graph the solutions of each linear inequality.

4) 12 xy

5) 25

3 xy

6) 3y

42

Example 3: Writing Linear Inequalities from a Graph

D) E)

_______________________ _______________________

Practice: Writing Linear Inequalities from a Graph

Challenge / Regents Problem: Graph the inequality below.

Challenge Problem

y

x

y

x

43

Summary:

Exit Ticket:

44

Chapter 6-5 - Graphing Linear Inequalities – Homework Tell whether the ordered pair is a solution of the given inequality.

1) (1, 6); 6 xy 2) (–3, –12); 52 xy 3) (5, –3); 2 xy

________________ ________________ ________________

Graph the solution of each linear inequality. Check your answer.

4) 4 xy 5) 22 yx

Check: Check:

45

6) 01 yx 7) 2y 63 x

Check: Check:

Write an inequality to represent each graph.

8) 9) 10)

________________________ ________________________ ________________________

46

Chapter 6-6 Solving Systems of Linear Inequalities

SWBAT: graph and solve systems of linear inequalities in two variables

Warm Up

Graph the solution of 1234 yx

What is a System of Inequalities??

How do we know if a point is a solution to the inequality??

Which of the following points is a solution to the system above?

(2, 7) (-1, -5) (2, 1) (-5, -2)

47

Practice: Identify Solutions of Systems of Linear Inequalities

To show all the solutions of a system of linear inequalities, graph the solutions of each inequality. The solutions

of the system are represented by the overlapping shaded regions.

Below are graphs of Examples 1 and 2.

Example #1 Example #2

Ex ample 2: Solving a System of Linear Inequalities by Graphing

Graph the systems of inequalities. Give two ordered pairs that (a) are solutions (b) are not solutions.

C) D)

1

42

xy

xy

48

Practice: Solving a System of Linear Inequalities by Graphing


2a) 2b)

Example 3: Solving a System of Linear Inequalities by Graphing


E)

1248

22

1

yx

xy F)

34

223

xy

yx

G) H) 7

3 6 12

y x

x y

xy

xy

2

1

49

Challenge / Regents Problem:

Summary:

Exit Ticket:

50

Graphing Linear Inequalities Systems – Homework

Tell whether the ordered pair is a solution of the given inequality.

1) (2, –2); 3

1

y x

y x

2) (2, 5);

2

2

xy

xy 3) (1, 3);

14

2

xy

xy

________________ ________________ ________________

Graph the system of linear inequalities.

a) Give two ordered pairs that are solutions.

b) Give two ordered pairs that are NOT solutions.

4)

xy

xy

2

4 5)

3

12

1

yx

xy

a) __________________________ a) __________________________

b) __________________________ b) __________________________

51

6. 7.

52

8. 9.

Algebra - White Plains Middle School...9 Chapter 6-2 Solving Systems by Substitution SWBAT: solve...

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