Systems With Two Variables

1
Section 4.1
Systems With Two Variables
2
OBJECTIVES
Find solution of two linear equations using
3
OBJECTIVES
Find solution of two linear equations using
4
OBJECTIVES
Find solution of two linear equations using
5
OBJECTIVES
Find solution of two linear equations using
6
Solving Two Equations in Two Unknowns by
Elimination

• Clear any fractions or
• decimals.

7
Solving Two Equations in Two Unknowns by
Elimination

2. Multiply both sides of the equations (as
needed) by numbers that make the
coefficients of one of the variables
additive inverses.
8
Solving Two Equations in Two Unknowns by
Elimination

3. Add the two equations.
4. Solve for the remaining variable.
9
Solving Two Equations in Two Unknowns by
Elimination

5. Substitute this solution into one of the
equations and solve for second variable.
6. Check the solution.
10
Practice Test
Chapter 4Systems With Two VariablesSection 4.1A

• Exercise 2

11
Use the graphical method to solve the system.
12
Use the graphical method to solve the system.
13
Use the graphical method to solve the system.
There is no solution.
System is inconsistent.
Lines are parallel.
14
Use the graphical method to solve the system.
5
x
-5
5
y
-5
15
Practice Test
Chapter 4Systems With Two VariablesSection 4.1A

• Exercise 3

16
Use the graphical method to solve the system.
y
x
5
-5
17
Use the graphical method to solve the system.
Infinitely many solutions
y
x
5
-5
18
Practice Test
Chapter 4Systems With Two Variables Section 4.1B

• Exercise 5

19
Use the substitution method to solve the system.
20
Use the substitution method to solve the system.
NO solution
21
Practice Test
Chapter 4Systems With Two VariablesSection 4.1C

• Exercise 9

22
Solve the system.
Multiply by 6.
Multiply by 8.
Multiply by 2.
23
Solve the system.
24
Solve the system.
Substitute x 4 in
25
Solve the system.
26
Solve the system.
Solution is (4, 0).
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Section 4.2
Systems with Three Variables
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OBJECTIVES
29
OBJECTIVES
30
OBJECTIVES
31
PROCEDURE FOR SOLVING
Three Equations in Three Unknowns by Elimination

• Select a pair of equations
• and eliminate one variable
• from this pair.

32
PROCEDURE FOR SOLVING
Three Equations in Three Unknowns by Elimination

2. Select a different pair of equations and
eliminate the same variable as in step 1.
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PROCEDURE FOR SOLVING

Three Equations in Three Unknowns by Elimination
3. Solve the pair of equations resulting
from step 1 and 2.
34
PROCEDURE FOR SOLVING
Three Equations in Three Unknowns by Elimination

4. Substitute the values found in the
simplest of original equations. Solve for
third variable.
35
PROCEDURE FOR SOLVING
Three Equations in Three Unknowns by Elimination

5. Check by substituting the values in each
of the original equations.
36
Solving Three Equations in Three Unknowns by
Elimination

• The system is consistent
• independent it has one
• solution consisting of an
• ordered triple (x, y, z).

37
Solving Three Equations in Three Unknowns by
Elimination

• The system is inconsistent.
• It has no solution.

38
Solving Three Equations in Three Unknowns by
Elimination

• The system is consistent
• and dependent. It has
• infinitely many solutions.

39
Practice Test
Chapter 4Systems With Two VariablesSection 4.2A

• Exercise 11

40
Solve the system.
x 1
41
Solve the system.
x 1
42
Solve the system.
x 1
43
Section 4.3
Coin, Distance-Rate-Time, Investment and Geometry
Problems
44
OBJECTIVES
45
OBJECTIVES
46
OBJECTIVES
47
OBJECTIVES
48
OBJECTIVES
49
Practice Test
Chapter 4Systems With Two VariablesSection 4.3C

• Exercise 16

50
A motorboat can go 10 mi downstream on a river in
20 min. It takes 30 min for this boat to go back
upstream the same 10 mi. Find the speed of the
current.
51
A motorboat can go 10 mi downstream on a river in
20 min. It takes 30 min for this boat to go back
upstream the same 10 mi. Find the speed of the
current.
52
A motorboat can go 10 mi downstream on a river in
20 min. It takes 30 min for this boat to go back
upstream the same 10 mi. Find the speed of the
current.
53
A motorboat can go 10 mi downstream on a river in
20 min. It takes 30 min for this boat to go back
upstream the same 10 mi. Find the speed of the
current.
The rate of the current is 5 mi/hr.
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Section 4.4
Matrices
55
OBJECTIVES
56
OBJECTIVES
57
OBJECTIVES
58
DEFINITION
Matrix

A rectangular array of numbers enclosed in
brackets.
59
PROCEDURE
Elementary Operations on Systems of Equations

1. The order of equations may be changed. This
clearly cannot affect the solutions.
60
PROCEDURE
Elementary Operations on Systems of Equations

2. Any of the equations may be multiplied by
any nonzero real number.
61
PROCEDURE
Elementary Operations on Systems of Equations

3. Any equation of a system may be replaced
by the sum of itself and any other
equation of the system.
62
PROCEDURE
Elementary Row Operations on Matrices

• Change the order of the
• rows.

63
PROCEDURE
Elementary Row Operations on Matrices

2. Multiply all elements of a row by any
nonzero number.
64
PROCEDURE
Elementary Row Operations on Matrices

3. Replace any row by the element-by-element
sum of itself and any other row.
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Practice Test
Chapter 4Systems With Two VariablesSection 4.4A

• Exercise 18

66
Use matrices to solve the system.
67
Use matrices to solve the system.
68
Use matrices to solve the system.
69
Use matrices to solve the system.
70
Use matrices to solve the system.
71
Use matrices to solve the system.
72
Use matrices to solve the system.
73
Section 4.5
Determinants and Cramers Rule
74
OBJECTIVES
Evaluate a 2 by 2 determinant.
75
OBJECTIVES
76
OBJECTIVES
77
OBJECTIVES
78
Determinant

79
Cramers Rule - 2 Equations
80
Cramers Rule - 2 Equations
81
Cramers Rule - 2 Equations
82
Cramers Rule - 2 Equations
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Cramers Rule - 2 Equations
84
Cramers Rule - 2 Equations
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DEFINITION
Minor

The determinant that remains after deleting the
row and column in which the element appears.
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Minor

87
DEFINITION

Sign Array
88
Cramers Rule - 3 Equations
89
Cramers Rule - 3 Equations
90
Cramers Rule - 3 Equations
91
Cramers Rule - 3 Equations
92
Cramers Rule - 3 Equations
2.
93
Cramers Rule - 3 Equations
3.
94
Practice Test
Chapter 4Systems With Two VariablesSection 4.5A

• Exercise 19a

95
Evaluate.
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Practice Test
Chapter 4Systems With Two VariablesSection 4.5A

• Exercise 19b

97
Evaluate.
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Practice Test
Chapter 4Systems With Two VariablesSection 4.5B

• Exercise 20

99
Solve by Cramers rule.
100
Solve by Cramers rule.
101
Solve by Cramers rule.
102
Solve by Cramers rule.
103
Practice Test
Chapter 4Systems With Two VariablesSection 4.5C

• Exercise 21

104
Evaluate.
105
Evaluate.
106
Practice Test
Chapter 4Systems With Two VariablesSection 4.5D

• Exercise 23

107
Solve by Cramers rule.
108
Solve by Cramers rule.
109
Solve by Cramers rule.
110
Solve by Cramers rule.
111
Solve by Cramers rule.
112
Solve by Cramers rule.
113
Solve by Cramers rule.
114
Solve by Cramers rule.
115
Solve by Cramers rule.
116
Solve by Cramers rule.
117
Section 4.6
Systems of Linear Inequalities
118
OBJECTIVES
119
OBJECTIVES
120
PROCEDURE
Graphing Inequalities

121
PROCEDURE
Graphing Inequalities

• Use a test point to shade
• the half-plane that is the
• graph of each linear
• inequality.

122
PROCEDURE
Graphing Inequalities

• Graph is the intersection of
• the half-planes, that is, the
• region consisting of the
• points satisfying all inequalities.

123
Practice Test
Chapter 4Systems With Two VariablesSection 4.6B

• Exercise 25