1. Section 4.1
Systems With Two Variables

2. OBJECTIVES The graphical method.
Find solution of two linear equations using: A
The graphical method.

3. OBJECTIVES The substitution method.
Find solution of two linear equations using: B
The substitution method.

4. OBJECTIVES The elimination method.
Find solution of two linear equations using: C
The elimination method.

5. OBJECTIVES Solve applications involving systems of equations.
Find solution of two linear equations using: D
Solve applications involving systems of equations.

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. Exercise #2 Practice Test
Chapter 4 Systems With Two Variables Section 4.1A
Practice Test
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. Exercise #3 Practice Test
Chapter 4 Systems With Two Variables Section 4.1A
Practice Test
Exercise #3

16. Use the graphical method to solve the system.x 5 -5

17. Use the graphical method to solve the system.
Infinitely many solutions y x 5 -5

18. Exercise #5 Practice Test Chapter 4 Systems With Two Variables
Section 4.1B
Practice Test
Exercise #5

19. Use the substitution method to solve the system.

20. Use the substitution method to solve the system.
NO solution

21. Exercise #9 Practice Test
Chapter 4 Systems With Two Variables Section 4.1C
Practice Test
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).

27. Section 4.2
Systems with Three Variables

28. OBJECTIVES A
Solve a system of three equations and three unknowns by the elimination method.

29. OBJECTIVES B
Determine if a system of three equations in three unknowns is consistent, inconsistent, or dependent.

30. OBJECTIVES C
Solve applications involving systems of three equations.

31. Three Equations in Three Unknowns by Elimination
PROCEDURE FOR SOLVING
Three Equations in Three Unknowns by Elimination
Select a pair of equations
and eliminate one variable
from this pair.

32. Three Equations in Three Unknowns by Elimination
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.

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

34. Three Equations in Three Unknowns by Elimination
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. Three Equations in Three Unknowns by Elimination
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. Exercise #11 Practice Test
Chapter 4 Systems With Two Variables Section 4.2A
Practice Test
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 A
Solve coin problems with two or more unknowns.

45. OBJECTIVES B
Solve general problems with two or more unknowns.

46. OBJECTIVES C
Solve rate, time and distance problems with two or more unknowns.

47. OBJECTIVES D
Solve investment problems with two or more unknowns.

48. OBJECTIVES E
Solve geometry problems with two or more unknowns.

49. Exercise #16 Practice Test
Chapter 4 Systems With Two Variables Section 4.3C
Practice Test
Exercise #16

50. A motorboat can go 10 mi downstream on a river in 20 min
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
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
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.
(3)
(4)

53. A motorboat can go 10 mi downstream on a river in 20 min
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.

54. Section 4.4
Matrices

55. OBJECTIVES A
Perform elementary operations on systems of equations.

56. OBJECTIVES B
Solve systems of linear equations using matrices.

57. OBJECTIVES C
Solve applications using matrices.

58. A rectangular array of numbers enclosed in brackets.
DEFINITION
Matrix
A rectangular array of numbers enclosed in brackets.

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

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

61. Elementary Operations on Systems of Equations
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. Elementary Row Operations
PROCEDURE
Elementary Row Operations
on Matrices
Change the order of the
rows.

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

64. Elementary Row Operations
PROCEDURE
Elementary Row Operations
on Matrices
3. Replace any row by the
element-by-element sum of
itself and any other row.

65. Exercise #18 Practice Test
Chapter 4 Systems With Two Variables Section 4.4A
Practice Test
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 Cramer’s Rule

74. OBJECTIVES
Evaluate a 2 by 2 determinant. A

75. OBJECTIVES B
Use Cramer’s rule to solve a system of two equations in two unknowns.

76. OBJECTIVES C
Use minors to evaluate 3 by 3 determinants.

77. OBJECTIVES D
Use Cramer’s rule to solve a system of three equations.

78. Determinant

79. Cramer’s Rule - 2 Equations

80. Cramer’s Rule - 2 Equations

81. Cramer’s Rule - 2 Equations

82. Cramer’s Rule - 2 Equations

83. Cramer’s Rule - 2 Equations

84. Cramer’s Rule - 2 Equations

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

86. Minor

87. DEFINITION
Sign Array

88. Cramer’s Rule - 3 Equations

89. Cramer’s Rule - 3 Equations

90. Cramer’s Rule - 3 Equations

91. Cramer’s Rule - 3 Equations

92. Cramer’s Rule - 3 Equations2.

93. Cramer’s Rule - 3 Equations3.

94. Exercise #19a Practice Test
Chapter 4 Systems With Two Variables Section 4.5A
Practice Test
Exercise #19a

95. Evaluate.

96. Exercise #19b Practice Test
Chapter 4 Systems With Two Variables Section 4.5A
Practice Test
Exercise #19b

97. Evaluate.

98. Exercise #20 Practice Test
Chapter 4 Systems With Two Variables Section 4.5B
Practice Test
Exercise #20

99. Solve by Cramer’s rule.

100. Solve by Cramer’s rule.

101. Solve by Cramer’s rule.

102. Solve by Cramer’s rule.

103. Exercise #21 Practice Test
Chapter 4 Systems With Two Variables Section 4.5C
Practice Test
Exercise #21

104. Evaluate.

105. Evaluate.

106. Exercise #23 Practice Test
Chapter 4 Systems With Two Variables Section 4.5D
Practice Test
Exercise #23

107. Solve by Cramer’s rule.

108. Solve by Cramer’s rule.

109. Solve by Cramer’s rule.

110. Solve by Cramer’s rule.

111. Solve by Cramer’s rule.

112. Solve by Cramer’s rule.

113. Solve by Cramer’s rule.

114. Solve by Cramer’s rule.

115. Solve by Cramer’s rule.

116. Solve by Cramer’s rule.

117. Section 4.6
Systems of Linear Inequalities

118. OBJECTIVES A
Graphing systems of two linear inequalities.

119. OBJECTIVES B
Graphing systems of inequalities.

120. Graphing Inequalities
PROCEDURE
Graphing Inequalities

121. Graphing Inequalities
PROCEDURE
Graphing Inequalities
Use a test point to shade
the half-plane that is the
graph of each linear
inequality.

122. Graphing Inequalities
PROCEDURE
Graphing Inequalities
Graph is the intersection of
the half-planes, that is, the
region consisting of the
points satisfying all inequalities.

123. Exercise #25 Practice Test
Chapter 4 Systems With Two Variables Section 4.6B
Practice Test
Exercise #25