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 – 7 Chapter 6-2: SWBAT: solve systems of linear equations in two variables by substitution Pgs: 8 - 13 HW: Pgs: 14 – 16

Lesson 6-3: SWBAT: solve systems of linear equations in two variables by elimination Pgs: 17 - 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 - 29 HW: Pgs: 30 – 31

Review Lesson 6-1 to 6-3: SWBAT: Demonstrate their knowledge of solving systems of linear equations in two variables Pgs: 32 - 35

Lesson 6-5: SWBAT: graph and solve linear inequalities in two variables Pgs: 36 – 40 HW: Pgs: 41 – 42

Lesson 6-6: SWBAT: graph and solve systems of linear inequalities in two variables Pgs: 43 - 46 HW: Pgs: 47 – 50

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

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.

4

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

xy Solution: _________

Check: Check:

7)

73

43

yx

yx Solution: _________

8) Solution:_________

Check: Check:

8

Chapter 6-2 Solving Systems by Substitution

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

Warm Up

Solve the system below by graphing.

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!

9

Solution: _________ Solution: _________

Solution: _________

10

Practice:

Solve the system by substitution.

2)

Solution: _________

Solution: _________

x = 2y – 4

x + 8y = 16

11

Practice:

Solve the system by substitution.

Solution: _________

Word Problems

12

Practice

One cable television provider has a $60 setup fee and $80 per month and the second has a $160

equipment fee and $70 per month.

In how many months will the cost be the same? What will that cost be?

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

13

Summary:

Exit Ticket:

14

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

1)

14

2

xy

xy Solution: __________

2)

2

4

xy

xy Solution: _________

3)

35

13

xy

xy Solution: _________

4)

3

62

yx

yx Solution: _________

15

5)

7

82

xy

yx Solution: _________

7) Solution: _________

6)

12

032

yx

yx Solution: _________

8) Solution: _________

16

11.

9. 10.

17

Chapter 6-3 Solving Systems by Elimination

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

Warm Up

18

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.

19

Practice: Solve the system of equations by using elimination.

a)

b)

20

Word Problems

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.

13.

25

Day 4 - Systems of Equations – Word Problems

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

variables

Warm – Up

Word Problems that involve “Substitution Method” 1.

You Try It!

26

Word Problems that involve Elimination: Version 1 2.

You Try it!

27

Word Problems that involve Elimination: Version 2 3.

You Try It!

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?

28

Coin Problems Find the number of dimes and quarters that Alexis has.

29

Challenge Problem:

Summary:

Exit Ticket:

30

Homework – Word Problems

1.

2. 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?

3. In a store ties sell for $10 and shirts sell for $25. Together 80 items were sold. A total of $1250 was

collected. How many shirts and how many ties were sold?

31

4. Three ties and four belts cost $125.00. Five ties and two belts cost $115.00. Find the cost of 1 tie and 1 belt.

5.

6.

7. Vanessa has 52 coins in dimes and quarters which are worth $6.25. How many of each coin does she have?

32

Chapter 6-1 to 6-3 Review

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

33

34

35

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. Max has $2.35 in nickels and dimes. If he has a total of thirty-two coins, how many of each

coin does he have?

36

Chapter 6-5 - Graphing Linear Inequalities

SWBAT: graph and solve linear inequalities in two variables

Warm Up

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

1 xy

37

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.

38

Practice: Graphing Linear Inequalities in Two Variables

Graph the solutions of each linear inequality.

4) 12 xy

5) 25

3 xy

6) 3y

39

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

40

Summary:

Exit Ticket:

41

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.

4) 4 xy 5) 22 yx

6) 01 yx 7) 2y 63 x

42

Write an inequality to represent each graph.

8) 9) 10)

________________________ ________________________ ________________________

11) Solve the system below.

3x + 3y = -12

5x – y = 2

12) You buy 3 DVD’s and 4 CD’s and the cost is $120. On another day you buy 2 DVD’s and 5 CD’s and the

cost is $115. Find the cost of 1 DVD and 1 CD.

13) The high school basketball team sold 500 tickets for a tournament. If you bought tickets in advance you

paid $3. If you bought tickets at the door paid $5. The basketball team ended up earning $1700. How many of

both kinds of tickets were sold?

43

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)

44

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

Example 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

45

Practice: 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.

2a) 2b)

Example 3: 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.

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

46

Challenge / Regents Problem:

Summary:

Exit Ticket:

47

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) __________________________

48

6. 7.

49

8. 9.

50

10) Solve the system below.

3x + 8y = -46

-2x – 4y = 24

11) John has a total of 9 stamps, which consists of 25 cent and 2 cent stamps. His stamps have a value of $1.10. How

many of each stamp does he have?

12) One day you bought 3 Almond Joys and 4 cans of Pepsi for $8.50. The next weekend at the same price you

bought 5 Almond Joys and 8 cans of Pepsi for $15.50. How much is each can of Pepsi and each Almond Joy?