Unit 3

1
Unit 3 Chapter 7
2
Unit 3

• Section 7.1 Solve Linear Equations by graphing
• Section 7.2 Solve Linear Equations by
  substitution.
• Section 7.3 Solve Linear Equations by Adding or
  Subtracting
• Section 7.4 Solve Linear Systems by multiplying
  first
• Section 7.5 Solve Special Types of Linear
  Systems
• Section 7.6 Solve systems of linear inequalities

3
Warm-Up 7.1
4
Lesson 7.1, For use with pages 426-434
1. Graph the equation 2x y 1.
2. It takes 3 hours to mow a lawn and 2 hours to
trim hedges. You spend 16 hours doing yard
work. What are 2 possible numbers of lawns you
mowed and hedges you trimmed?
5
Vocabulary 7.1

• System of Linear Equations
• 2 or more linear equations with the same
  variables
• Solution of a Linear Equation
• The solution set that MAKES ALL THE EQUATIONS
  TRUE AT THE SAME TIME!!
• Usually a single point!
• Consistent Independent System
• A linear system that has EXACTLY one solution.

6
Notes 7.1 Solving Linear Systems by Graphing

• Remember the steps to convert English to
  Mathlish
• Read and highlight key words
• DEFINE THE VARIABLES (MOST CRITICAL STEP!!)
• Write Mathlish sentence left to right (careful
  with subtraction and division!)
• 3 Step Process To Solve Linear Eqns by graphing
• Write equations in slope-intercept form
• Graph them on calculator
• Find intersection of lines
• CHECK ANSWERS!

7
Examples 7.1

8

Use the graph-and-check method
EXAMPLE 2
Solve the linear system
x y 7
Equation 1
Equation 2
x 4y 8
SOLUTION
STEP 1
Graph both equations.
9
Use the graph-and-check method
EXAMPLE 2
STEP 2
Estimate the point of intersection. The two lines
appear to intersect at (4, 3).
STEP 3
Check whether (4, 3) is a solution by
substituting 4 for x and 3 for y in each of the
original equations.
Equation 1
Equation 2
x y 7
x 4y 8
10
Use the graph-and-check method
EXAMPLE 2
11
Use the graph-and-check method
EXAMPLE 2
for Examples 1 and 2
GUIDED PRACTICE
Solve the linear system by graphing. Check your
solution.
Put eqns in slope- Intercept form
y 5x
5x y -5x 10
The Intersection point is at (1,5).
12
EXAMPLE 3
Standardized Test Practice
13
EXAMPLE 3
Standardized Test Practice
SOLUTION
Write a system of equations where y is the total
cost (in dollars) for x sessions.
EQUATION 1
14
EXAMPLE 3
Standardized Test Practice
EQUATION 2
ANSWER
The correct answer is y 13x and y 4x 90.
15
for Example 3
GUIDED PRACTICE
4. Solve the linear system in Example 3 to find
the number of sessions after which the total cost
with a season pass, including the cost of the
pass, is the same as the total cost without a
season pass.
SOLUTION
Let the number of sessions be x
13x 90 4x
9x 90
x 10
16
for Example 3
GUIDED PRACTICE
5. WHAT IF? In Example 3, suppose a season pass
costs 135. After how many sessions is the total
cost with a season pass, including the cost of
the pass, the same as the total cost without a
season pass?
SOLUTION
Let the number of sessions be x
13x 135 4x
9x 135
x 15
17

Solve a multi-step problem
EXAMPLE 4
RENTAL BUSINESS
A business rents in-line skates and bicycles.
During one day, the business has a total of 25
rentals and collects 450 for the rentals. Find
the number of pairs of skates rented and the
number of bicycles rented.
18
Solve a multi-step problem
EXAMPLE 4
SOLUTION
STEP 1
Write a linear system. Let x be the number of
pairs of skates rented, and let y be the number
of bicycles rented.
x y 25
Equation for number of rentals
15x 30y 450
Equation for money collected from rentals
STEP 2
Graph both equations.
19
Solve a multi-step problem
EXAMPLE 4
STEP 3
Estimate the point of intersection. The two lines
appear to intersect at (20, 5).
STEP 4
Check whether (20, 5) is a solution.
20
Solve a multi-step problem
EXAMPLE 4
for Example 4
GUIDED PRACTICE
SOLUTION
STEP 1
Write a linear system. Let x be the number of
pairs of skates rented, and let y be the number
of bicycles rented.
x y 20
Equation for number of rentals.
15x 30y 420
Equation for money collected from rentals.
21
Solve a multi-step problem
EXAMPLE 4
for Example 4
GUIDED PRACTICE
STEP 2
Graph both equations.
STEP 3
Estimate the point of intersection. The two lines
appear to intersect at (12, 8).
STEP 4
Check whether (12, 8) is a solution.
12 8 20
20 20
420 420
22
Solve a multi-step problem
EXAMPLE 4
EXAMPLE 4
for Example 4
GUIDED PRACTICE
23
Warm-Up 7.2
24
Lesson 7.2, For use with pages 435-441
Solve the equation.
1. 6a 3 2a 13
2. 4(n 2) n 11
25
Lesson 7.2, For use with pages 435-441
Solve the equation.
3. You burned 8 calories per minute on a
treadmilland 10 calories per minute on an
elliptical trainer for a total of 560 calories
in 60 minutes. How many minutes did you spend on
each machine?
26
Lesson 7.2, For use with pages 435-441
Solve the system of equations BUT YOU CANT USE
THE CALCULATOR OR GRAPH IT!!! WORK WITH YOUR
GROUP!!.

1. 2x 3y 40
2. y x 5

27
Vocabulary 7.2

• None!!

28
Notes 7.2 Solving Systems w/Substitution

• If its EASY to get one of the variables in an
  equation by itself, substitution may be the
  easiest way to solve the system of equations.
• To Solve Linear Systems with Substitution
• Get ONE of the variables in ONE of the equations
  by itself.
• Substitute that variable into the OTHER equation.
• Solve the equation from 2.
• Plug the answer back into the original equation.

29
Examples 7.2

30

EXAMPLE 2
Use the substitution method
Solve the linear system
x 2y 6
Equation 1
4x 6y 4
Equation 2
SOLUTION
STEP 1
Solve Equation 1 for x.
x 2y 6
Write original Equation 1.
x 2y 6
Revised Equation 1
31
EXAMPLE 2
Use the substitution method
STEP 2
Substitute 2y 6 for x in Equation 2 and solve
for y.
4x 6y 4
Write Equation 2.
Substitute 2y 6 for x.
4(2y 6) 6y 4
8y 24 6y 4
Distributive property
14y 24 4
Simplify.
14y 28
Add 24 to each side.
y 2
Divide each side by 14.
32
EXAMPLE 2
Use the substitution method
STEP 3
Substitute 2 for y in the revised Equation 1 to
find the value of x.
x 2y 6
Revised Equation 1
x 2(2) 6
Substitute 2 for y.
x 2
Simplify.
33
EXAMPLE 2
Use the substitution method
GUIDED PRACTICE
CHECK
Substitute 2 for x and 2 for y in each of the
original equations.
Equation 1
Equation 2
4x 6y 4
x 2y 6
34
EXAMPLE 1
for Examples 1 and 2
Use the substitution method
GUIDED PRACTICE
Solve the linear system using the substitution
method.
Equation 1
Equation 2
3x y 10
SOLUTION
STEP 1
Solve for y. Equation 1 is already solved for y.
35
EXAMPLE 2
for Examples 1 and 2
Use the substitution method
GUIDED PRACTICE
STEP 2
Substitute 2x 5 for y in Equation 2 and solve
for x.
3x y 10
Write Equation 2.
Substitute 2x5 for x.
3x (2x 5) 10
5x 5 10
Simplify.
5x 5
x 2
36
EXAMPLE 2
for Examples 1 and 2
GUIDED PRACTICE
STEP 3
Substitute 1 for x in the revised Equation 1 to
find the value of y.
y 2x 5 2(1) 5 7
37
Use the substitution method
for Example 1
EXAMPLE 1
for Examples 1 and 2
GUIDED PRACTICE
GUIDED PRACTICE
CHECK
Substitute 1 for x and 7 for y in each of the
original equations.
Equation 1
Equation 2
3x y 10
y 2x 5
38
EXAMPLE 2
for Examples 1 and 2
Use the substitution method
GUIDED PRACTICE
Equation 1
x 2y 6
Equation 2
SOLUTION
STEP 1
Solve Equation 1 for x.
x y 3
Write original Equation 1.
x y 3
Revised Equation 1
39
EXAMPLE 2
for Examples 1 and 2
Use the substitution method
GUIDED PRACTICE
STEP 2
Substitute y 3 for x in Equation 2 and solve
for y.
x 2y 6
Write Equation 2.
Substitute y 3 for x.
( y 3) 2y 6
3y 3 6
Simplify.
3y 9
Add 3 to each side.
y 3
Divide each side by 3.
40
EXAMPLE 2
for Examples 1and 2
Use the substitution method
GUIDED PRACTICE
STEP 3
Substitute 3 for y in the revised Equation 1 to
find the value of x.
x y 3
Revised Equation 1
Substitute 3 for y.
x 3 3
x 0
Simplify.
41
Use the substitution method
EXAMPLE 2
GUIDED PRACTICE
for Examples 1 and 2
GUIDED PRACTICE
CHECK
Substitute 0 for x and 3 for y in each of the
original equations.
Equation 1
Equation 2
x 2y 6
x y 3
42

EXAMPLE 3
Solve a multi-step problem
WEBSITES
Many businesses pay website hosting companies to
store and maintain the computer files that make
up their websites. Internet service providers
also offer website hosting. The costs for website
hosting offered by a website hosting company and
an Internet service provider are shown in the
table. Find the number of months after which the
total cost for website hosting will be the same
for both companies.
43
EXAMPLE 3
Solve a multi-step problem

SOLUTION
STEP 1
Write a system of equations. Let y be the
total cost after x months.
Equation 1 Internet service provider
44
EXAMPLE 3
Solve a multi-step problem

Equation 2 Website hosting company
The system of equations is
y 10 21.95x
Equation 1
y 22.45x
Equation 2
45
EXAMPLE 3
Solve a multi-step problem

STEP 2
Substitute 22.45x for y in Equation 1 and
solve for x.
y 10 21.95x
Write Equation 1.
22.45x 10 21.95x
Substitute 22.45x for y.
0.5x 10
Subtract 21.95x from each side.
x 20
Divide each side by 0.5.
46
for Example 3
GUIDED PRACTICE
SOLUTION
Let y be the total cost.
449
47
for Example 3
GUIDED PRACTICE
48
EXAMPLE 4
Solve a mixture problem
ANTIFREEZE
For extremely cold temperatures, an automobile
manufacturer recommends that a 70 antifreeze and
30 water mix be used in the cooling system of a
car. How many quarts of pure (100) antifreeze
and a 50 antifreeze and 50 water mix should be
combined to make 11 quarts of a 70 antifreeze
and 30 water mix?
SOLUTION
STEP 1
Write an equation for the total number of quarts
and an equation for the number of quarts of
antifreeze. Let x be the number of quarts of 100
antifreeze, and let y be the number of quarts of
a 50 antifreeze and 50 water mix.
49
EXAMPLE 4
Solve a mixture problem
Equation 1 Total number of quarts
x y 11
Equation 2 Number of quarts of antifreeze
x 0.5y 7.7
50
EXAMPLE 4
Solve a mixture problem
x y 11
The system of equations is
Equation 1
x 0.5y 7.7
Equation 2
STEP 2
Solve Equation 1 for x.
Write Equation 1.
x y 11
x 11 y
Revised Equation 1
STEP 3
Substitute 11 y for x in Equation 2 and
solve for y.
x 0.5y 7.7
Write Equation 2.
51
EXAMPLE 4
Solve a mixture problem

Substitute 11 y for x.
(11 y) 0.5y 7.7
Solve for y.
y 6.6
STEP 4
Substitute 6.6 for y in the revised Equation 1
to find the value of x.
x 11 y 11 6.6 4.4
52
for Example 4
GUIDED PRACTICE
SOLUTION
STEP 1
Write an equation for the total number of quarts
and an equation for the number of quarts of
antifreeze. Let x be the number of quarts of 100
antifreeze, and let y be the number of quarts of
a 50 antifreeze and 50 water mix.
53
for Example 4
GUIDED PRACTICE
Equation 1 Total number of quarts
x y 16
Equation 2 Number of quarts of antifreeze
x 0.5y 11.2
54
for Example 4
GUIDED PRACTICE
x y 16
The system of equations is
Equation 1
x 0.5y 11.2
Equation 2.
STEP 2
Solve Equation 1 for x.
Write Equation 1.
x y 16
x 16 y
Revised Equation 1
55
for Example 4
GUIDED PRACTICE
STEP 3
Substitute 16 y for x in Equation 1 and
solve for x.
x 0.5y 7.7
Write Equation 2.
Substitute 16 y for x.
(16 y) 0.5y 7.7
Solve for y.
y 9.6
56
for Example 4
GUIDED PRACTICE
STEP 4
Substitute 9.6 for y in the revised Equation 1
to find the value of x.
x 16 y 16 9.6 6.4
57
Warm-Up 7.3
58
Lesson 7.2, For use with pages 435-441
Solve the linear systems by GRAPHING!!!

1. x y -2
2. -x y 6

1. x y 0
2. 5x 2y -7

59
Lesson 7.2, For use with pages 435-441
Solve the linear systems by SUBSTITUTION!!!!!!

1. y x 4
2. -2x y 18

1. 5x 4y 27
2. -2x y 3

60
Lesson 7.3, For use with pages 443-450
1. Solve the linear system using
substitution. 2x y 12 3x 2y 11

• One auto repair shop charges 30 for a
  diagnosis and 25 per hour for labor. Another
  auto repair shop charges 35 per hour for labor.
  For how many hours are the total charges for
  both of the shops the same?
• HINT FIND EQUATIONS FOR TOTAL AND SUBSTITUTE!

61
Lesson 7.3, For use with pages 443-450
1. Add the two equations together (combine like
terms) and solve for x and y. 2x 3y 11 -2x
5y 13
62
Vocabulary 7.3 - REVIEW

• System of Linear Equations
• 2 or more linear equations with the same
  variables
• Solution of a Linear Equation
• The solution set that MAKES ALL THE EQUATIONS
  TRUE AT THE SAME TIME!!
• Consistent Independent System
• A linear system that has EXACTLY one solution.

63
Notes 7.3 Solving systems with elimination

• If we have two equations and two variables, how
  many solutions should we USUALLY have??
• Whats the goal of solving every algebra eqn. you
  will ever see?
• You can eliminate variables from some systems by
  adding or subtracting equations to eliminate
  variables.
• RULES/HINTS TO MAKE PROCESS EASIER!
• Remember the goal You are trying to eliminate
  one variable!
• Line up like terms under each other.
• NEVER subtract. Add the negative instead!

64
Examples 7.3

65

EXAMPLE 1
Use addition to eliminate a variable
Solve the linear system
2x 3y 11
Equation 1
2x 5y 13
Equation 2
SOLUTION
STEP 1
Solve for y.
8y 24
STEP 2
y 3
STEP 3
Substitute 3 for y in either equation and Solve
for x.
66

EXAMPLE 1
Use addition to eliminate a variable
2x 3y 11
Write Equation 1
2x 3(3) 11
Substitute 3 for y.
x 1
Solve for x.
67

EXAMPLE 2
Use subtraction to eliminate a variable
Solve the linear system
4x 3y 2
Equation 1
5x 3y 2
Equation 2
SOLUTION
STEP 1
Solve for x.
x 4
STEP 2
x ?4
Substitute ?4 for x in either equation and
solve for y.
STEP 3
68

EXAMPLE 2
Use subtraction to eliminate a variable
4x 3y 2
Write Equation 1.
4( 4) 3y 2
Substitute 4 for x.
y 2
Solve for y.
69

EXAMPLE 3
Arrange like terms
Solve the linear system
8x 4y 4
Equation 1
4y 3x 14
Equation 2
SOLUTION
STEP 1
Rewrite Equation 2 so that the like terms are
arranged in columns.
8x 4y 4
8x 4y 4
4y 3x 14
STEP 2
STEP 3
STEP 4
Substitute 2 for x in either equation and solve
for y.
70

EXAMPLE 3
Arrange like terms
4y 3x 14
Write Equation 2.
4y 3(2) 14
Substitute 2 for x.
y 5
Solve for y.
71
for Example 1,2 and 3
GUIDED PRACTICE
Solve the linear system
1.
4x 3y 5

Equation 1
2x 3y 7
Equation 2
SOLUTION
STEP 1
Solve for x.
2x 2
STEP 2
x 1
STEP 3
Substitute 1 for y in either equation and
Solve for x.
72
for Example 1,2 and 3
GUIDED PRACTICE
4x 3y 5
Write Equation 1.
2( 1) 3y 5
Substitute 1 for x.
y 3
Solve for x.
73
for Example 1,2 and 3
GUIDED PRACTICE
4x 3y 5
2x 3y 7
74
for Example 1,2 and 3
GUIDED PRACTICE
Solve the linear system
4.
7x 2y 5
Equation 1
7x 3y 4
Equation 2
SOLUTION
STEP 1
y 1
STEP 2
Solve for y.
STEP 3
Substitute 1 for y in either and solve for x.
75
for Example 1,2 and 3
GUIDED PRACTICE
Write Equation 1.
7x 2y 5
Substitute 1 for y.
7x 2(1) 5
x 1
Solve for x.
76
for Example 1,2 and 3
GUIDED PRACTICE
Solve the linear system
5.
3x 4y 6
Equation 1
Equation 2
SOLUTION
STEP 1
Rewrite Equation 2 so that the like terms are
arranged in columns.
3x 4y 6
STEP 2
Subtract the equations.
6y 0
STEP 3
Solve for y.
y 0
STEP 4
Substitute 0 for y in either equation and solve
for x.
77
for Example 1,2 and 3
GUIDED PRACTICE
Write Equation 2.
Substitute 0 for y.
x 2
Solve for x .
78
for Example 1,2 and 3
GUIDED PRACTICE
Solve the linear system
6.
2x 5y 12
Equation 1
Equation 2
SOLUTION
STEP 1
Rewrite Equation 2 so that the like terms are
arranged in columns.
2x 5y 12
STEP 2
Subtract the equations.
6x 6
STEP 3
Solve for x.
x 1
STEP 4
Substitute 1 for x in either equation and solve
for y.
79
for Example 1,2 and 3
GUIDED PRACTICE
2x 5y 12
Write Equation 2.
Substitute 1 for x.
x 2
Solve for y .
80
EXAMPLE 4
Write and solve a linear system
KAYAKING
81
EXAMPLE 4
Write and solve a linear system
STEP 1
Write a system of equations. First find the speed
of the kayak going upstream and the speed of the
kayak going downstream.
4 r
6 r
82
EXAMPLE 4
Write and solve a linear system
Use the speeds to write a linear system. Let x be
the average speed of the kayak in still water,
and let y be the speed of the current.
83
EXAMPLE 4
Write and solve a linear system
84
EXAMPLE 4
Write and solve a linear system
STEP 2
Solve the system of equations.
x y 4
Write Equation 1.
Write Equation 2.
2x 10
Add equations.
x 5
Solve for x.
Substitute 5 for x in Equation 2 and solve for y.
85
EXAMPLE 4
Write and solve a linear system
5 y 6
Substitute 5 for x in Equation 2.
y 1
Subtract 5 from each side.
86
for Example 4
GUIDED PRACTICE
7. WHAT IF? In Example 4, suppose it takes the
kayaker 5 hours to travel 10 miles upstream and
2 hours to travel 10 miles downstream. The speed
of the current remains constant during the trip.
Find the average speed of the kayak in still
water and the speed of the current.
SOLUTION
STEP 1
Write a system of equations. First find the speed
of the kayak going downstream.
87
for Example 4
GUIDED PRACTICE
2 r
5 r
Use the speeds to write a linear system. Let x be
the average speed of the kayakar in still water,
and let y be the speed of the current.
88
for Example 4
GUIDED PRACTICE
89
for Example 4
GUIDED PRACTICE
STEP 2
Solve the system of equations.
x y 2
Equation 1.
Equation 2.
2x 7
Add equations.
x 3.5
Solve for x.
Substitute 3.5 for x in Equation 2 and solve for
y.
90
for Example 4
GUIDED PRACTICE
3.5 y 6
Substitute 3.5 for x in Equation 2.
y 1.5
Subtract 3.5 from each side.
91
Warm-Up 7.4
92
Lesson 7.4, For use with pages 451-457
Solve the linear system.
1. 4x 3y 15 2x 3y 9
2. 2x y 8 2x 2y 8
93
Lesson 7.4, For use with pages 451-457
Solve the linear system.
3. You can row a canoe 10 miles upstream in 2.5
hours and 10 miles downstream in 2 hours. What
is the average speed of the canoe in still water?
Multiply the second equation by -2 and rewrite
it. Then use it to solve the system.

1. 8x 6y 30
2. 2x 3y 9

94
Vocabulary 7.4

• Least Common Multiple
• Smallest POSITIVE number that is a multiple of
  two or more factors

95
Notes 7.4 Solving systems by multiplying
first.

• In order to add equations and eliminate a
  variable, two of the coefficients must be
  opposite signs.
• Learned 3 ways to solve systems of linear eqns
• Graphing
• Easiest when I can get y by itself and have a
  calculator!
• Substitution
• Easiest when I can get one variable by itself.
• Elimination
• Easiest when I can get opposite coefficients.
• There is a 4th way - multiply and then
  eliminate.
• To get coefficients with opposite signs, you can
  multiply one or more equations by constants.
• May need to identify LCM of two coefficients.

96
Examples 7.4

97
EXAMPLE 1
Multiply one equation, then add
Solve the linear system
6x 5y 19
Equation 1
2x 3y 5
Equation 2
SOLUTION
STEP 1
Multiply Equation 2 by 3 so that the
coefficients of x are opposites.
6x 5y 19
6x 5y 19
2x 3y 5
STEP 2
Add the equations.
4y 4
98
EXAMPLE 1
Multiply one equation, then add
STEP 3
Solve for y.
y 1
STEP 4
Substitute 1 for y in either of the original
equations and solve for x.
99
EXAMPLE 2
Multiply both equations, then subtract
Solve the linear system
4x 5y 35
Equation 1
2y 3x 9
Equation 2
SOLUTION
STEP 1
Arrange the equations so that like terms are in
columns.
4x 5y 35
Write Equation 1.
3x 2y 9
Rewrite Equation 2.
100
EXAMPLE 2
Multiply both equations, then subtract
STEP 2
Multiply Equation 1 by 2 and Equation 2 by 5 so
that the coefficient of y in each equation is the
least common multiple of 5 and 2, or 10.
4x 5y 35
3x 2y 9
STEP 3
Subtract the equations.
23x 115
Solve for x.
STEP 4
101
for Examples 1 and 2
GUIDED PRACTICE
Solve the linear system using elimination
Equation 1
2x 3y 5
Equation 2
SOLUTION
y 2
102
for Examples 1 and 2
GUIDED PRACTICE
Equation 1
3x 10y 3
Equation 2
SOLUTION
103
for Examples 1 and 2
GUIDED PRACTICE
Equation 1
9y 5x 5
Equation 2
SOLUTION
104
EXAMPLE 3
Standardized Test Practice
Darlene is making a quilt that has alternating
stripes of regular quilting fabric and sateen
fabric. She spends 76 on a total of 16 yards of
the two fabrics at a fabric store. Write a system
of equations can be used to find the amount x (in
yards) of regular quilting fabric and the amount
y (in yards) of sateen fabric she purchased?
105
EXAMPLE 3
Standardized Test Practice
SOLUTION
Write a system of equations where x is the number
of yards of regular quilting fabric purchased and
y is the number of yards of sateen fabric
purchased.
Equation 1 Amount of fabric
106
EXAMPLE 3
Standardized Test Practice
Equation 2 Cost of fabric
The system of equations is
x y 16
Equation 1
4x 6y 76
Equation 2
ANSWER
The correct answer is x 10 and y 6.

107
for Example 3
GUIDED PRACTICE
SOLUTION
Write a system of equations where x is the cost
of soccer ball and y is the cost of soccer ball
bag.
108
for Example 3
GUIDED PRACTICE
Equation 1
109
for Example 3
GUIDED PRACTICE
Equation 2
The system of equations is
10x 2y 155
Equation 1
12x 3y 189
Equation 2
110
for Example 3
GUIDED PRACTICE
STEP 1
Multiply equation 1 by 3 and equation 2 by 2
so that the coefficient of y in each equation is
the least common multiple of 3 and 2 .
STEP 2
30x 6y 465
10x 2y 155
12x 3y 189
Add the equation
?6x ?87
STEP 3
Solve for x
x 14.50
STEP 4
111
for Example 3
GUIDED PRACTICE
STEP 5
Substitute 14.50 for x in either of the original
equations and solve for y.
10x 2y 155
Write Equation 1.
Substitute 14.50 for x.
10(14.50) 2y 155
solve for y
y 5
112
Warm-Up 7.5
113
Lesson 7.5, For use with pages 459-465
1. Solve the linear system. 2x 3y 9 x
2y 6
2. You buy 8 pencils for 8 at the bookstore.
Standard pencils cost .85 and specialty pencils
cost 1.25. How many specialty pencils did you
buy?
114
Lesson 7.5, For use with pages 459-465
1. Solve the linear system by GRAPHING. Describe
the lines on your whiteboard. x y -2 y
-x5
1. Solve the linear system by SUBSTITUTION. What
do you get? x y -2 y -x5
1. Solve the linear system by ELIMINATION. What
do you get? x y -2 y -x5
115
Vocabulary 7.5

• Consistent Independent System
• System of equations with ONE solution
• Inconsistent System
• System of equations with NO solution.
• Consistent Dependent System
• System of equations with INFINITE solutions.

116
Notes 7.5Special Types of Systems

• Systems can have one soln, no soln, or infinite.
• Easiest ways to check for solutions
• Graph them (put them in slope-intercept form)
• Intersect 1 solution
• Parallel No solution
• Same line Infinite solutions
• Check if equations are multiples of each other.
• Yes infinite solutions
• No Check some more!
• Eliminate the variables (using Add. or Mult.)
• Always False statement No solutions
• Always True statement Infinite solutions

117
Notes 7.5Special Types of Systems
118
Examples 7.4

119
EXAMPLE 1
A linear system with no solution
Solve the linear system by graphing and by
elimination!
3x 2y 10
Equation 1
3x 2y 2
Equation 2
SOLUTION
METHOD 1 Graphing
Graph the linear system.
Answer NO SOLUTION.
120
EXAMPLE 1
A linear system with no solution
METHOD 2 Elimination
Subtract the equation.
121
EXAMPLE 2
A linear system with infinitely many solutions
Solve the system by graphing and substitution.
x 2y 4
Equation 1
Equation 2
SOLUTION
Graph the linear system.
122
EXAMPLE 2
A linear system with infinitely many solutions
METHOD 2 Substitution
Write Equation 1
Simplify.
123
for Examples 1 and 2
GUIDED PRACTICE
Tell whether the linear system has no solution or
infinitely many solutions. Explain.
1. 5x 3y 6
Equation 1
5x 3y 3
Equation 2
METHOD 2 Elimination
Subtract the equations.
5x 3y 6
0 9
124
for Examples 1 and 2
GUIDED PRACTICE
125
for Examples 1 and 2
GUIDED PRACTICE
2. y 2x 4
Equation 1
6x 3y 12
Equation 2
METHOD 2 Elimination
Substitute 2x 4 for y in Equation 2 and solve
for x.
6x 3y 12
Write Equation 2
Substitute (2x 4) for y.
6x 3(2x 4) 12
12 12
Simplify.
126
for Examples 1 and 2
GUIDED PRACTICE
127
Warm-Up 7.6
128
Lesson 7.6, For use with pages 466-472
1. Solve -3x gt 12
ANSWER x lt -4
129
Lesson 7.6, For use with pages 466-472
2. You are running one ad that costs 6 per day
and another that costs 8 per day. You can spend
no more than 120. Graph this inequality.
HINT WRITE THE EQUATION FIRST!

• Graph the following on a number line
• x lt 5 and x gt 0

ANSWER
130
Vocabulary 7.6

• System of Linear Inequalities
• Two or more linear inequalities in the same
  variables.
• Solution of a system of linear inequalities
• An ordered pair that makes ALL the inequalities
  true at the same time.
• Graph of a system of linear inequalitites
• Graph of all the solutions of the system.

131
Notes7.6Solve Systems of linear inequalities.

• REVIEW
• Graphing inequalities - similar to graphing
  lin.eqns
• Play the pretend game and let equation be .
• Dotted line is lt or gt
• Solid line is lt or gt
• Pick a point NOT ON THE LINE, check the answer,
  and shade the correct side of the line.
• TO GRAPH SYSTEMS OF INEQUALITIES
• Graph each inequality
• Find the AREA where solutions intersect.
• Pick a point and check your solution.

132
Examples 7.6

133
EXAMPLE 1
Graph a system of two linear inequalities
SOLUTION
134
EXAMPLE 1
Graph a system of two linear inequalities
Choose a point in the dark blue region, such as
(0, 1). To check this solution, substitute 0 for
x and 1 for y into each inequality.
CHECK
135
EXAMPLE 2
Graph a system of three linear inequalities
SOLUTION
Graph all three inequalities in the same
coordinate plane. The graph of the system is the
triangular region shown.
136
for Examples 1 and 2
GUIDED PRACTICE
ANSWER
137
for Examples 1 and 2
GUIDED PRACTICE
ANSWER
138
for Examples 1 and 2
GUIDED PRACTICE
ANSWER
139
EXAMPLE 3
Write a system of linear inequalities
SOLUTION
INEQUALITY 2 Another boundary line for the
shaded region has a slope of 2 and a y-intercept
of 1. So, its equation is y 2x 1. Because the
shaded region is above the dashed line, the
inequality is y gt 2x 1.
140
EXAMPLE 3
Write a system of linear inequalities
141
EXAMPLE 4
Write and solve a system of linear inequalities
BASEBALL
The National Collegiate Athletic Association
(NCAA) regulates the lengths of aluminum baseball
bats used by college baseball teams. The NCAA
states that the length (in inches) of the bat
minus the weight (in ounces) of the bat cannot
exceed 3. Bats can be purchased at lengths from
26 to 34 inches.
a. Write and graph a system of linear
inequalities that describes the information
given above.
b. A sporting goods store sells an aluminum bat
that is 31 inches long and weighs 25 ounces. Use
the graph to determine if this bat can be used by
a player on an NCAA team.
142
EXAMPLE 4
Write and solve a system of linear inequalities
SOLUTION
a. Let x be the length (in inches) of the bat,
and let y be the weight (in ounces) of the bat.
From the given information, you can write the
following inequalities
The difference of the bats length and weight can
be at most 3.
x 26
The length of the bat must be at least 26 inches.
x 34
The length of the bat can be at most 34 inches.
y 0
The weight of the bat cannot be a negative number.
Graph each inequality in the system. Then
identify the region that is common to all of the
graphs of the inequalities. This region is shaded
in the graph shown.
143
EXAMPLE 4
Write and solve a system of linear inequalities
144
for Examples 3 and 4
GUIDED PRACTICE
Write a system of inequalities that defines the
shaded region.
145
for Examples 3 and 4
GUIDED PRACTICE
Write a system of inequalities that defines the
shaded region.
146
for Examples 3 and 4
GUIDED PRACTICE
6. WHAT IF? In Example 4, suppose a Senior League
(ages 1014) player wants to buy the bat
described in part (b). In Senior League, the
length (in inches) of the bat minus the weight
(in ounces) of the bat cannot exceed 8. Write and
graph a system of inequalities to determine
whether the described bat can be used by the
Senior League player.
ANSWER
147
Review Ch. 7 PUT HW QUIZZES HERE
148
Daily Homework Quiz
For use after Lesson 7.1
3x y 5
x 3y 5
149
Daily Homework Quiz
For use after Lesson 7.1
2. Solve the linear system by graphing.
2x y 3
6x 3y 3
150
Daily Homework Quiz
For use after Lesson 7.1
ANSWER
2 angel fish and 6 clown loaches
151
Daily Homework Quiz
For use after Lesson 7.2
Solve the linear system using substitution
152
Daily Homework Quiz
For use after Lesson 7.2
153

Daily Homework Quiz
For use after Lesson 7.3
Solve the linear system using elimination.
154

Daily Homework Quiz
For use after Lesson 7.3
155

Daily Homework Quiz
For use after Lesson 7.4
Solve the linear system using elimination.
156

Daily Homework Quiz
For use after Lesson 7.4
(-4,-2)
157

Daily Homework Quiz
For use after Lesson 7.5
Without solving the linear system, tell whether
the linear system has one solution, no solution,
or infinitely many solutions.
158

Daily Homework Quiz
For use after Lesson 7.3
159

Daily Homework Quiz
For use after Lesson 7.6
160

Daily Homework Quiz
For use after Lesson 7.6
161
Warm-Up X.X
162
Vocabulary X.X

• Holder
• Holder 2
• Holder 3
• Holder 4

163
Notes X.X LESSON TITLE.

• Holder
• Holder
• Holder
• Holder
• Holder

164
Examples X.X