Transcript Ch. 3.1 power point

Chapter 3
Section 1
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3.1
1
2
3
4
5
6
Reading Graphs; Linear Equations in
Two Variables
Interpret graphs.
Write a solution as an ordered pair.
Decide whether a given ordered pair is a
solution of a given equation.
Complete ordered pairs for a given equation.
Complete a table of values.
Plot ordered pairs.
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Objective 1
Interpret graphs.
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Interpret graphs.
Recall that a bar graph is used to show comparisons. It
consists of a series of bars (or simulations of bars) arranged either
vertically or horizontally. In a bar graph, values from two
categories are paired with each other.
A line graph is used to show changes or trends in data over
time. To form a line graph, we connect a series of points
representing data with line segments.
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EXAMPLE 1
Interpreting a Bar Graph
Compare sales of motor scooters in 1999 and 2001.
Solution:
Sales were about 25
thousand in 1999 and
about 50 thousand in
2001, so sales doubled
over those years.
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EXAMPLE 2
Interpreting a Line Graph
Estimate the number of households that purchased a
real tree in 2004.
Solution: about 27 million
About how much did the
number of households
purchasing real trees increase
from 2002 to 2004?
Solution: about 5 million
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Interpret Graphs. (cont’d)
A linear equation in two variables is an equation that can be
written in the form
Ax  By  C,
where A, B, and C are real numbers and A and B are not both 0.
Some examples of linear equations in two variables in this
form, called standard form, are
equations
3x  4 y  9, x  y  0, and x  2 y  8. Linear
in two variables
Other linear equations in two variables, such as
and
y  4x  5
3 x  7  2 y,
are not written in standard form, but could be. We discuss the
forms of linear equations in more detail in Section 3.4.
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Objective 2
Write a solution as an ordered pair.
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Write a solution as an ordered pair.
A solution of a linear equation in two variables requires two
numbers, one for each variable. For example, a true statement
results when we replace x with 2 and y with 13 in the equation
y  4 x  5, since 13  4  2  5.
The pair of numbers x = 2 and y = 13 gives a solution of the
equation y  4 x  5. The phrase “x = 2 and y = 13” is abbreviated
y-value
x-value
 2,13
The x-value is always given first. A pair of numbers such as
(2,13) is called an ordered pair, since the order in which the
numbers are written is important.
The ordered pairs (2,13) and (13,2) are not the same. The
second pair indicates that x = 13 and y = 2. For ordered pairs to
be equal, their x-coordinates must be equal and their y-coordinates
must be equal.
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Objective 3
Decide whether a given ordered
pair is a solution of a given
equation.
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Slide 3.1 - 10
Decide whether a given ordered pair is a
solution of a given equation.
We substitute the x- and y-values of an ordered pair into a
linear equation in two variables to see whether the ordered pair
is a solution. An ordered pair that is a solution of an equation is
said to satisify the equation.
Infinite numbers of ordered pairs can satisfy a linear equation in two
variables.
When listing ordered pairs, be sure to always list the x-value first.
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EXAMPLE 3
Deciding Whether Ordered Pairs
Are Solutions of an Equation
Decide whether each ordered pair is a solution of the
equation 5x  2 y  20.
 2, 5 Solution: 5 2  2  5  20
10   10  20
0  20
 4, 20
5  4  2  20  20
20  40  20
20  20
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No
Yes
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Objective 4
Complete ordered pairs for a given
equation.
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EXAMPLE 4
Completing Ordered Pairs
Complete each ordered pair for the equation y  2 x  9.
 2, 
 ,7
Solution: y  2  2  9
y  49
y  5
7  9  2x  9  9
16 2x

2
2
x8
 2, 5
8, 7 
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Objective 5
Complete a table of values.
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EXAMPLE 5
Completing Tables of Value
Complete the table of values for the equation
2 x  3 y  12. Then write the results as ordered pairs.
Solution:
4
6
2
3
2
2  0  3 y  12
3 y 12

3 3
 0, 4 
y  4
2  3  3 y  12
6  3 y  6  12  6
3 y 6

 3 3
3,

2
 
y  2
2x  3 0  12
2 x 12

2
2
x6
 6, 0 
2x  3 3  12
2x  9  9  12  9
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2x 3

2
2
x
3
2
3

,

3


2


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Objective 6
Plot ordered pairs.
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Slide 3.1 - 17
Plot ordered pairs.
Every linear in two variables equation has an infinite number
of ordered pairs as solutions. Each choice of a number for one
variable leads to a particular real number for the other variable.
To graph these solutions, represented as ordered pairs (x,y),
we need two number lines, one for each variable. The two
number lines are drawn as shown
below. The horizontal number line is
called the x-axis and the vertical line
is called the y-axis. Together, these
axes form a rectangular coordinate
system, also called the Cartesian
coordinate system.
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Slide 3.1 - 18
Plot ordered pairs. (cont’d)
The coordinate system is divided into four regions, called
quadrants. These quadrants are numbered counterclockwise,
starting with the one in the top right quadrant.
Points on the axes themselves are not in any quadrant.
The point at which the x-axis and y-axis meet is called the
origin, labeled 0 on the previous diagram. This is the point
corresponding to (0, 0).
The x-axis and y-axis determine a plane— a flat surface
illustrated by a sheet of paper. By referring to the two axes, we
can associate every point in the plane with an ordered pair. The
numbers in the ordered pair are called the coordinates of the
point.
In a plane, both numbers in the ordered pair are needed to
locate a point. The ordered pair is a name for the point.
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Plot ordered pairs. (cont’d)
For example, locate the point associated with the ordered pair
(2,3) by starting at the origin.
Since the x-coordinate is 2, go 2 units to the right along the
x-axis.
Since the y-coordinate is 3, turn
and go up 3 units on a line parallel to
the y-axis.
The point (2,3) is plotted in the
figure to the right. From now on
the point with x-coordinate 2 and
y-coordinate 3 will be referred to as
point (2,3).
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EXAMPLE 6
Plotting Ordered Pairs
Plot the given points in a coordinate system:
3,5 ,  2,6 ,  4,0 ,  5, 2 , 5, 2 , 0, 6.
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EXAMPLE 7
Completing Ordered Pairs to
Estimate the Number of Twin
Births
Complete the table of ordered pairs for the equation,
y  3.563x  7007.7., where x = year and y = number
of twin births in thousands. Round answers to the
nearest whole number. Interpret the results for 2002.
115
122
125
Solution:
There were about 125
thousand twin births in
the U.S. in 2002.
y  3.5631999  7007.7
y  7122.4  7007.7
y  115
y  3.563 2001  7007.7
y  7.129.6  7007.7
y  122
y  3.563 2002  7007.7
y  7133.1  7007.7
y  125
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Plot ordered pairs. (cont’d)
The ordered pairs of twin births in the U.S. for 1998, 2000,
and 2003 are graphed to the right. This graph of ordered pairs
of data is called a scatter diagram.
Notice how the how the axes are labeled:
x represents the year, and y represents
the number of twin births in thousands.
A scatter diagram enables us to tell whether two quantities
are related to each other. These plotted points could be
connected to form a straight line, so the variables x (years) and
y (number of births have a linear relationship.
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Slide 3.1 - 23
Plot ordered pairs. (cont’d)
Think of ordered pairs as representing an input value x and an
output value y. If we input x into the equation, the output is y.
We encounter many examples of this type of relationship every
day.



The cost to fill a tank with gasoline depends on how many gallons
are needed; the number of gallons is the input, and the cost is the
output
The distance traveled depends on the traveling time; input a time
and the output is a distance.
The growth of a plant depends on the amount of sun it gets; the
input is the amount of sun, and the output is growth.
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