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CHAPTER 3
Linear Equations
and Functions
SECTION 3-1
Open Sentences
in Two Variables
DEFINITIONS
Open sentences in two
variables– equations and
inequalities containing
two variables
EXAMPLES
9x + 2y = 15
y = x2 – 4
2x – y ≥ 6
DEFINITIONS
Solution– is a pair of
numbers (x, y) called an
ordered pair.
DEFINITIONS
Solution set– is the set of
all solutions satisfying the
sentence. Finding the
solution is called solving
the open sentence.
EXAMPLE
Solve the equation:
9x + 2y = 15
if the domain of x is
{-1,0,1,2}
SOLUTION
x
-1
0
1
(15-9x)/2
[15-9(-1)]/2
[15-9(0)]/2
[15-9(1)]/2
y
12
15/2
3
Solution
(-1,12)
(0,15/2)
(1,3)
2
[15-9(2)]/2
-3/2
(2,-3/2)
SOLUTION
the solution set is
{(-1,12), (0, 15/2), (1,3), (2,3/2)}
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?
SOLUTION
n = number of notebooks
b = number of binders
(n and b must be whole
numbers)
2n + 5b =22
n = (22-5b)/2
SOLUTION
If n is odd then 22-5b is odd
and n is not a whole
number
SOLUTION
b
0
2
4
(22-5b)/2
[22-5(0)]/2
[22-5(2)]/2
[22-5(4)]/2
n
11
6
1
Solution
(0,11)
(2,6)
(4,1)
6
[12-5(6)]/2
-4
Impossible
SOLUTION
the solution set is
{(0,11), (2, 6), (4,1)}
SECTION 3-2
Graphs of Linear
Equations in Two
Variables
COORDINATE PLANE
consists of two
perpendicular number
lines, dividing the plane
into four regions called
quadrants
COORDINATE PLANE
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
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)
ONE-TO-ONE
CORRESPONDENCE
1. There is exactly one point
in the coordinate plane
associated with each
ordered pair of real
numbers.
ONE-TO-ONE
CORRESPONDENCE
2. There is exactly one
ordered pair of real
numbers associated with
each point in the
coordinate plane.
DEFINITION
GRAPH – is the set of all points
in the coordinate plane
whose coordinates satisfy
the open sentence.
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
LINEAR EQUATION
is an equation whose
graph is a straight
line.
SECTION 3-3
The Slope of a
Line
SLOPE
is the ratio of vertical
change to the horizontal
change. The variable m
is used to represent
slope.
FORMULA FOR SLOPE
m = change in y-coordinate
change in x-coordinate
m = rise
run
Or
SLOPE OF A LINE
m = y2 – y1
x2 – x1
HORIZONTAL LINE
a horizontal line containing
the point
(a, b) is described by the
equation y = b and has
slope of 0
VERTICAL LINE
a vertical line containing
the point (c, d) is described
by the equation x = c and
has no slope
Find the slope of the line that
contains the given points.
M(4, -6) and N(-2, 3)
m = 3 –(-6)
-2 – 4
m=3+6
-6
m = - 9/6 or -3/2
THEOREM
The slope of the line
Ax + By = C (B ≠ 0) is
- A/B
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)
Write an equation of a line with the given
slope and through a given point
m=-2
P(-1, 3)
y-3=-2[x-(-1)]
y-3= -2(x+1)
y-3= -2x-2
y=-2x+1
SECTION 3-4
Finding an
Equation of a
Line
POINT-SLOPE FORM
y – y1 = m (x – x1)
where m is the slope and
(x1 ,y1) is a point on the
line.
Write an equation of a line with the given
slope and passing through a given point
m=-2
P(-1, 3)
y-3=-2[x-(-1)]
y-3= -2(x+1)
y-3= -2x-2
y=-2x+1
Y-Intercept
is the point where the
line intersects the y axis.
X-Intercept
is the point where the
line intersects the
x -axis.
SLOPE-INTERCEPT FORM
y = mx + b
where m is the slope and b
is the y -intercept
Write an equation of a line with the
given y-intercept and slope
m=3 b = 6
y=3x + 6
THEOREM
Let L1 and L2 be two different
lines, with slopes m1 and m2
respectively.
1. L1 and L2 are parallel if and
only if m1=m2
THEOREM
and
2. L1 and L2 are
perpendicular if and only if
m1m2 = -1
Write an equation of a line passing
through the given points
A(1, -3) B(3,2)
y-(-3)=[2-(-3)](x-1)
3-1
y+3=5(x-1)
2
y+3=5x-5
2 2
y=5x-11
2
2
Find the slope of a line parallel to the
line containing points M and N.
M(-2, 5) and N(0, -1)
m = -1 -5
0 – (-2)
m = -6
2
m=-3
Find the slope of a line perpendicular to the
line containing points M and N.
M(4, -1) and N(-5, -2)
m = -2 – (-1)
-5 - 4
m = -1
-9
m = 1/9
Write an equation of a line parallel to y=-1/3x+
containing the point (1,1)
m=-1/3
P(1, 1)
y-1=-1/3(x-1)
y-1= -1/3x+1/3
y= -1/3x+4/3
Write an equation of a line perpendicular to y=
1/3x+1 containing the point (1,1)
m=-1/3
P(1, 1)
y-1=-1/3(x-1)
y-1= -1/3x+1/3
y= -1/3x+4/3
SECTION 3-5
Systems of Linear
Equations in Two
Variables
SYSTEM OF EQUATIONS
Two linear equations with
the same two variable
form a system of
equations.
SOLUTION
The ordered pair that
makes both equations
true.
INTERSECTING LINES
The SOLUTION to the
system of equations
is a point (point of
intersection of the
two lines).
PARALLEL LINES
There is NO
SOLUTION to the
system of equations
(no intersection of
the two lines).
COINCIDING LINES
The graph of each
equation is the same.
The lines coincide and
any point on the line is
a solution.
EQUIVALENT SYSTEMS
Systems that have the
same solution.
METHODS FOR SOLVING SYSTEM
OF EQUATIONS
Graphing
Substitution
Linear-
Combination
SOLVE BY GRAPHING
4x + 2y = 8
3y = -6x + 12
SOLVE BY GRAPHING
y = 1/2x + 3
2y = x - 2
SUBSTITUTION
A method for solving a
system of equations by
solving for one variable
in terms of the other
variable.
SOLVE BY SUBSTITUTION
3x – y = 6
x + 2y = 2
Solve for y in terms of x.
3x – y = 6
3x = 6 + y
3x – 6 = y then
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
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
SOLVE BY SUBSTITUTION
2x + y = 0
x – 5y = -11
Solve for y in terms of x.
2x + y = 0
y = -2x
then
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
SOLVE BY SUBSTITUTION
Substitute the value of x
into the first equation
2x + y = 0
y = -2x
y = -2(-1)
y=2
LINEAR-COMBINATION
Another method for solving a
system of equations where
one of the variables is
eliminated by adding or
subtracting the two
equations.
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.
LINEAR COMBINATION
Solve the resulting
equation for the remaining
variable.
LINEAR COMBINATION
Substitute the value for the
variable in one of the
original equations and
solve for the unknown
variable.
LINEAR
COMBINATION
Check the solution in both
of the original equations.
LINEAR
COMBINATION
This method combines the
multiplication property of
equations with the
addition/subtraction
method.
SOLVE BY LINEARCOMBINATION
3x – 4y = 10
3y = 2x – 7
SOLUTION
3x – 4y = 10
-2x +3y = -7
Multiply equation 1 by 2
Multiply equation 2 by 3
SOLUTION
6x – 8y = 20
-6x +9y = -21
Add the two equations.
y = -1
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
CONSISTENT SYSTEM
The system of equations
has at least one solution.
INCONSISTENT SYSTEM
The system of equations
has no solution.
DEPENDENT SYSTEM
The graph of each
equation is the same.
The lines coincide and
any point on the line is a
solution.
SECTION 3-6
Problem Solving:
Using Systems
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?
SOLUTION
Let x = cost of a pen
y = cost of a pencil
8x + 11y = 3.37
5x + 11y = 3.10
SOLUTION
Solve the system of
equations.
x = 29
y = 15
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.
SOLUTION
Let x = number of hrs at
daytime rate
y = number of hrs at
nighttime rate
30x + 10.50y = 411
x + y = 28
SOLUTION
Solve the system of
equations.
x=6
y = 22
SECTION 3-7
Linear
Inequalities in
Two Variables
SYSTEM OF INEQUALITIES
Linear equations that have
the equal sign replaced
by one of these symbols
<, ≤, ≥, >
SYSTEM OF LINEAR
INEQUALITIES
The SOLUTION of an
inequality in two variables
is an ordered pair of
numbers that satisfies the
inequality.
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.
HALF-PLANE
Graphically its the
region on either side
of the line.
BOUNDARY
is the line separating
the half-planes
OPEN HALF-PLANE
is the region on either
side of the boundary
line (illustrated by a
dashed-line).
CLOSED HALF-PLANE
is the solution that
includes the boundary
line.
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.
TEST POINT
is a point that does not
lie on the boundary,
but rather above or
below it.
GRAPHING INEQUALITIES
x+y≥4
(0,4),(4,0)
Shade the region above
a dashed line if y > mx
+ b.
Shade the region above
a solid line if
y  mx + b.
Shade the region below
a dashed line if y < mx
+ b.
Shade the region
below a solid line
if
y ≤ mx + b.
SOLVE BY GRAPHING THE
INEQUALITIES
x + 2y < 5
2x – 3y ≤ 1
SOLVE BY GRAPHING THE
INEQUALITIES
4x - y  5
8x + 5y ≤ 3
SECTION 3-8
Functions
MAPPING DIAGRAM
A picture showing a
correspondence
between two sets
MAPPING – the
relationship between the
elements of the domain
and range
DOMAIN – the set of all
possible x-coordinates
RANGE – the set of all
possible y-coordinates
FUNCTION
A correspondence
between two sets, D
and R, that assigns
to each member of D
exactly one member
of R.
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
Characteristics of a Function
1. Each element in D must be matched with
an element in R.
2. Some elements in R may not be matched
with any element in D.
3. Two or more elements in D may be
matched with the same element in R.
4. An element in D (domain) cannot be match
with two different elements in R.
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)}
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
SOLUTION
x
1
2
3
4
5
4x-x2
4(1) –
4(2) –
4(3) –
4(4) –
4(5) –
1=3
4=4
9=3
16 = 0
25 = -5
(x,y)
(1,3)
(2,4)
(3,3)
(4,0)
(5,-5)
FUNCTIONAL NOTATION
f(x) denotes the
value of f at x
VALUES of a
FUNCTION
The members of its
range.
SECTION 3-9
Linear Functions
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.
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
Rate of Change m
Rate of Change m =
change in f(x)
Change in x
EXAMPLE
Find equations of the
linear function f using
the given information.
f(4) = 1 and f(8) = 7
SOLUTION
m = f(8) – f(4)
8–4
m = 3/2
SECTION 3-10
Relations
RELATION
Is any set of ordered pairs.
The set of first coordinates
in the ordered pairs is the
domain of the relation, and
RELATION
and the set of second
coordinates is the range.
FUNCTION
is a relation in which
different ordered pairs
have different first
coordinates.
VERTICAL LINE TEST
a relation is a function if and
only if no vertical line
intersects its graph more
than once.
Determine if Relation is a
Function
{2,1),(1,-2), (1,2)}
{(x,y): x  + y = 3}
END