1. Linear Equations and Functions
CHAPTER 3
Linear Equations and Functions

2. Open Sentences in Two Variables
SECTION 3-1
Open Sentences in Two Variables

3. DEFINITIONS
Open sentences in two variables– equations and inequalities containing two variables

4. EXAMPLES
9x + 2y = 15
y = x2 – 4
2x – y ≥ 6

5. Solution– is a pair of numbers (x, y) called an ordered pair.
DEFINITIONS
Solution– is a pair of numbers (x, y) called an ordered pair.

6. DEFINITIONS
Solution set– is the set of all solutions satisfying the sentence. Finding the solution is called solving the open sentence.

7. Solve the equation: 9x + 2y = 15 if the domain of x is {-1,0,1,2}
EXAMPLE
Solve the equation:
9x + 2y = 15
if the domain of x is
{-1,0,1,2}

8. SOLUTION x (15-9x)/2 y Solution -1 [15-9(-1)]/2 12 (-1,12) [15-9(0)]/2
[15-9(0)]/2
15/2
(0,15/2) 1
[15-9(1)]/2 3
(1,3) 2
[15-9(2)]/2
-3/2
(2,-3/2)

9. SOLUTION
the solution set is
{(-1,12), (0, 15/2), (1,3), (2,-3/2)}

10. EXAMPLE
Roberto has $22. He buys some notebooks costing $2 each and some binders costing $5 each. If Roberto spends all $22 how many of each does he buy?

11. (n and b must be whole numbers) 2n + 5b =22 n = (22-5b)/2
SOLUTION
n = number of notebooks
b = number of binders
(n and b must be whole numbers)
2n + 5b =22
n = (22-5b)/2

12. If n is odd then 22-5b is odd and n is not a whole number
SOLUTION
If n is odd then 22-5b is odd and n is not a whole number

13. SOLUTION b (22-5b)/2 n Solution [22-5(0)]/2 11 (0,11) 2 [22-5(2)]/2 6
[22-5(0)]/2 11
(0,11) 2
[22-5(2)]/2 6
(2,6) 4
[22-5(4)]/2 1
(4,1)
[12-5(6)]/2 -4
Impossible

14. SOLUTION
the solution set is
{(0,11), (2, 6), (4,1)}

15. Graphs of Linear Equations in Two Variables
SECTION 3-2
Graphs of Linear Equations in Two Variables

16. COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants

17. COORDINATE PLANE

18. X-coordinate (abscissa)- the horizontal number line
Y-coordinate (ordinate) - the vertical number line
ORIGIN - the point where the x-coordinate and
y-coordinate cross

19. (-3, 5), (2,4), (6,0), (0,-3) DEFINITION
ORDERED PAIR - a unique assignment of real numbers to a point in the coordinate plane consisting of one x-coordinate and one y-coordinate
(-3, 5), (2,4), (6,0), (0,-3)

20. ONE-TO-ONE CORRESPONDENCE
There is exactly one point in the coordinate plane associated with each ordered pair of real numbers.

21. ONE-TO-ONE CORRESPONDENCE
There is exactly one ordered pair of real numbers associated with each point in the coordinate plane.

22. DEFINITION
GRAPH – is the set of all points in the coordinate plane whose coordinates satisfy the open sentence.

23. The graph of every equation of the form
THEOREM
The graph of every equation of the form
Ax + By = C (A and B not both zero) is a line. Conversely, every line in the coordinate plane is the graph of an equation of this form

24. is an equation whose graph is a straight line.
LINEAR EQUATION
is an equation whose graph is a straight line.

25. SECTION 3-3
The Slope of a Line

26. SLOPE
is the ratio of vertical change to the horizontal change. The variable m is used to represent slope.

27. FORMULA FOR SLOPE Or m = rise run m = change in y-coordinate
change in x-coordinate Or
m = rise
run

28. SLOPE OF A LINE
m = y2 – y1
x2 – x1

29. a horizontal line containing the point
(a, b) is described by the equation y = b and has slope of 0

30. VERTICAL LINE
a vertical line containing the point (c, d) is described by the equation x = c and has no slope

31. Find the slope of the line that contains the given points.
M(4, -6) and N(-2, 3)

32. THEOREM
The slope of the line
Ax + By = C (B ≠ 0) is
- A/B

33. THEOREM
Let P(x1,y1) be a point and m a real number. There is one and only one line L through P having slope m. An equation of L is
y – y1 = m (x – x1)

34. Write an equation of a line with the given slope and through a given point
m= -2
P(-1, 3)

35. Finding an Equation of a Line
SECTION 3-4
Finding an Equation of a Line

36. where m is the slope and (x1 ,y1) is a point on the line.
POINT-SLOPE FORM
y – y1 = m (x – x1)
where m is the slope and (x1 ,y1) is a point on the line.

37. Write an equation of a line with the given slope and passing through a given point in standard form

38. is the point where the line intersects the y -axis.
Y-Intercept
is the point where the line intersects the y -axis.

39. is the point where the line intersects the
X-Intercept
is the point where the line intersects the
x -axis.

40. where m is the slope and b is the y -intercept
SLOPE-INTERCEPT FORM
y = mx + b
where m is the slope and b is the y -intercept

41. Write an equation of a line with the given y-intercept and slope
m=3 b = 6

42. L1 and L2 are parallel if and only if m1=m2
THEOREM
Let L1 and L2 be two different lines, with slopes m1 and m2 respectively.
L1 and L2 are parallel if and only if m1=m2

43. L1 and L2 are perpendicular if and only if m1m2 = -1
THEOREM
and
L1 and L2 are perpendicular if and only if m1m2 = -1

44. Write an equation of a line passing through the given points
A(1, -3) B(3,2)

45. Find the slope of a line parallel to the line containing points M and N.

46. Find the slope of a line perpendicular to the line containing points M and N.

47. Write an equation of a line parallel to y=-1/3x+1 containing the point (1,1)
m=-1/3

48. Write an equation of a line perpendicular to y= -1/3x+1 containing the point (1,1)
m=-1/3
P(1, 1)

49. Systems of Linear Equations in Two Variables
SECTION 3-5
Systems of Linear Equations in Two Variables

50. SYSTEM OF EQUATIONS
Two linear equations with the same two variable form a system of equations.

51. The ordered pair that makes both equations true.
SOLUTION
The ordered pair that makes both equations true.

52. INTERSECTING LINES
The SOLUTION to the system of equations is a point (point of intersection of the two lines).

53. PARALLEL LINES
There is NO SOLUTION to the system of equations (no intersection of the two lines).

54. COINCIDING LINES
The graph of each equation is the same. The lines coincide and any point on the line is a solution.

55. Systems that have the same solution.
EQUIVALENT SYSTEMS
Systems that have the same solution.

56. METHODS FOR SOLVING SYSTEM OF EQUATIONS
Graphing
Substitution
Linear-Combination

57. SOLVE BY GRAPHING
4x + 2y = 8
3y = -6x + 12

58. SOLVE BY GRAPHING
y = 1/2x + 3
2y = x - 2

59. SUBSTITUTION
A method for solving a system of equations by solving for one variable in terms of the other variable.

60. 3x – y = 6 x + 2y = 2 Solve for y in terms of x. 3x = 6 + y
SOLVE BY SUBSTITUTION
3x – y = 6
x + 2y = 2
Solve for y in terms of x.
3x = 6 + y
3x – 6 = y then

61. Substitute the value of y into the second equation
SOLVE BY SUBSTITUTION
Substitute the value of y into the second equation
x + 2y = 2
x + 2(3x – 6) = 2
x + 6x – 12 = 2
7x = 14
x = 2 now

62. Substitute the value of x into the first equation
SOLVE BY SUBSTITUTION
Substitute the value of x into the first equation
3x – y = 6
y = 3x – 6
y = 3(2 – 6)
y = 3(-4)
y = -12

63. 2x + y = 0 x – 5y = -11 Solve for y in terms of x. y = -2x then
SOLVE BY SUBSTITUTION
2x + y = 0
x – 5y = -11
Solve for y in terms of x.
y = -2x
then

64. Substitute the value of y into the second equation
SOLVE BY SUBSTITUTION
Substitute the value of y into the second equation
x – 5y = -11
x – 5(-2x) = -11
x+ 10x = -11
11x = -11
x = -1

65. Substitute the value of x into the first equation
SOLVE BY SUBSTITUTION
Substitute the value of x into the first equation
2x + y = 0
y = -2x
y = -2(-1)
y = 2

66. LINEAR-COMBINATION
Another method for solving a system of equations where one of the variables is eliminated by adding or subtracting the two equations.

67. LINEAR COMBINATION
If the coefficients of one of the variables are opposites, add the equations to eliminate one of the variables. If the coefficients of one of the variables are the same, subtract the equations to eliminate one of the variables.

68. Solve the resulting equation for the remaining variable.
LINEAR COMBINATION
Solve the resulting equation for the remaining variable.

69. LINEAR COMBINATION
Substitute the value for the variable in one of the original equations and solve for the unknown variable.

70. Check the solution in both of the original equations.
LINEAR COMBINATION
Check the solution in both of the original equations.

71. LINEAR COMBINATION
This method combines the multiplication property of equations with the addition/subtraction method.

72. SOLVE BY LINEAR-COMBINATION
3x – 4y = 10
3y = 2x – 7

73. 3x – 4y = 10 -2x +3y = -7 Multiply equation 1 by 2
SOLUTION
3x – 4y = 10
-2x +3y = -7
Multiply equation 1 by 2
Multiply equation 2 by 3

74. SOLUTION
6x – 8y = 20
-6x +9y = -21
Add the two equations.
y = -1

75. Substitute the value of y into either equation and solve for
SOLUTION
Substitute the value of y into either equation and solve for
3x – 4y = 10
3x – 4(-1) = 10
3x + 4 = 10
3x = 6
x = 2

76. The system of equations has at least one solution.
CONSISTENT SYSTEM
The system of equations has at least one solution.

77. INCONSISTENT SYSTEM
The system of equations has no solution.

78. DEPENDENT SYSTEM
The graph of each equation is the same. The lines coincide and any point on the line is a solution.

79. Problem Solving: Using Systems
SECTION 3-6
Problem Solving: Using Systems

80. EXAMPLE
If 8 pens and 7 pencils cost $3.37 while 5 pens and 11 pencils cost $3.10, how much does each pen and each pencil cost?

81. Let x = cost of a pen y = cost of a pencil 8x + 7y = 3.37
SOLUTION
Let x = cost of a pen
y = cost of a pencil
8x + 7y = 3.37
5x + 11y = 3.10

82. Solve the system of equations.
SOLUTION
Solve the system of equations.
x = 29
y = 15

83. EXAMPLE
To use a certain computer data base, the charge is $30/hr during the day and $10.50/hr at night. If a research company paid $411 for 28 hr of use, find the number of hours charged at the daytime rate and at the nighttime rate.

84. Let x = number of hrs at daytime rate
SOLUTION
Let x = number of hrs at daytime rate
y = number of hrs at nighttime rate
30x y = 411
x + y = 28

85. Solve the system of equations.
SOLUTION
Solve the system of equations.
x = 6
y = 22

86. Linear Inequalities in Two Variables
SECTION 3-7
Linear Inequalities in Two Variables

87. SYSTEM OF INEQUALITIES
Linear equations that have the equal sign replaced by one of these symbols
<, ≤, ≥, >

88. SYSTEM OF LINEAR INEQUALITIES
The SOLUTION of an inequality in two variables is an ordered pair of numbers that satisfies the inequality.

89. SYSTEM OF LINEAR INEQUALITIES
A system of linear inequalities can be solved by graphing each associated equation and determining the region where the inequality is true.

90. Graphically its the region on either side of the line.
HALF-PLANE
Graphically its the region on either side of the line.

91. is the line separating the half-planes
BOUNDARY
is the line separating the half-planes

92. OPEN HALF-PLANE
is the region on either side of the boundary line (illustrated by a dashed-line).

93. is the solution that includes the boundary line.
CLOSED HALF-PLANE
is the solution that includes the boundary line.

94. GRAPH of an INEQUALITY
is a graph of all the solutions of the inequality and includes a boundary, either a solid line or dashed line and a shaded area.

95. TEST POINT
is a point that does not lie on the boundary, but rather above or below it.

96. GRAPHING INEQUALITIES
x + y ≥ 4
(0,4),(4,0)

97. Shade the region above a dashed line if y > mx + b.

98. Shade the region above a solid line if
y  mx + b.

99. Shade the region below a dashed line if y < mx + b.

100. Shade the region below a solid line if
y ≤ mx + b.

101. SOLVE BY GRAPHING THE INEQUALITIES
x + 2y < 5
2x – 3y ≤ 1

102. SOLVE BY GRAPHING THE INEQUALITIES
4x - y  5
8x + 5y ≤ 3

103. SECTION 3-8
Functions

104. A picture showing a correspondence between two sets
MAPPING DIAGRAM
A picture showing a correspondence between two sets

105. MAPPING – the relationship between the elements of the domain and range

106. DOMAIN – the set of all possible x-coordinates
RANGE – the set of all possible y-coordinates

107. FUNCTION
A correspondence between two sets, D and R, that assigns to each member of D exactly one member of R.

108. Introduction to Functions
Definition – A function f from a set D to a set R is a relation that assigns to each element x in the set D exactly one element y in the set R. The set D is the domain of the function f, and the set R contains the range

109. Characteristics of a Function
Each element in D must be matched with an element in R.
Some elements in R may not be matched with any element in D.
Two or more elements in D may be matched with the same element in R.
An element in D (domain) cannot be match with two different elements in R.

110. Example
A = {1,2,3,4,5,6} and B = {9,10,12,13,15} Is the set of ordered pairs a function? {(1,9), (2,13), (3,15), (4,15), (5,12), (6,10)}

111. Given f: x→4x – x2 with domain D= {1,2,3,4,5} Find the range of f.
EXAMPLE
Given f: x→4x – x2 with domain D= {1,2,3,4,5}
Find the range of f.
f, the function that assigns to x the number 4x – x2

112. SOLUTION x 4x-x2 (x,y) 1 4(1) – 1 = 3 (1,3) 2 4(2) – 4 = 4 (2,4) 3
4(3) – 9 = 3
(3,3) 4
4(4) – 16 = 0
(4,0) 5
4(5) – 25 = -5
(5,-5)

113. f(x) denotes the value of f at x
FUNCTIONAL NOTATION
f(x) denotes the value of f at x

114. The members of its range.
VALUES of a FUNCTION
The members of its range.

115. SECTION 3-9
Linear Functions

116. Linear Function Is a function f that can be defined by f(x) = mx + b
Where x, m and b are real numbers. The graph of f is the graph of y = mx +b, a line with slope m and y -intercept b.

117. Constant Function
If f(x) = mx + b and
m = 0, then f(x) = b for all x and its graph is a horizontal line y = b

118. Rate of Change m
Rate of Change m =
change in f(x)
Change in x

119. Find equations of the linear function f using the given information.
EXAMPLE
Find equations of the linear function f using the given information.
f(4) = 1 and f(8) = 7

120. SOLUTION
m = f(8) – f(4)
8 – 4
m = 3/2

121. SECTION 3-10
Relations

122. RELATION
Is any set of ordered pairs. The set of first coordinates in the ordered pairs is the domain of the relation, and

123. and the set of second coordinates is the range.
RELATION
and the set of second coordinates is the range.

124. FUNCTION
is a relation in which different ordered pairs have different first coordinates.

125. VERTICAL LINE TEST
a relation is a function if and only if no vertical line intersects its graph more than once.

126. Determine if Relation is a Function
{2,1),(1,-2), (1,2)}
{(x,y): x  + y = 3}

127. END