Transcript Complete a table of values.

3.1
Linear Equations in Two Variables; The Rectangular Coordinate System
1
Interpret graphs.
2
Write a solution as an ordered pair.
3
Decide whether a given ordered pair is a solution of a given
equation.
4
Complete ordered pairs for a given equation.
5
Complete a table of values.
6
Plot ordered pairs.
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.
Slide 3.1-4
EXAMPLE 1 Interpreting a Line Graph
Refer to the line graph below.
Estimate the average price of a gallon of gasoline in 2001.
Solution:
about $1.45
About how much did the
average price of a gallon
of gasoline decrease
from 2001 to 2002?
about $0.10
Slide 3.1-5
Interpret graphs. (cont’d)
Linear Equation in Two Variables
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
3 x  4 y  9, x  y  0, and x  2 y  8. Linear equations
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.
Slide 3.1-6
Objective 2
Write a solution as an ordered pair.
Slide 3.1-7
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.
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. 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.
Slide 3.1-8
Objective 3
Decide whether a given ordered
pair is a solution of a given
equation.
Slide 3.1-9
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.
Slide 3.1-10
EXAMPLE 2
Deciding Whether Ordered Pairs Are Solutions of an Equation
Decide whether each ordered pair is a solution of the equation.
5 x  2 y  20
Solution:
 2, 5
 4, 20
5  2  2  5  20
10   10   20
0  20
No
5  4  2  20  20
20  40  20
20  20
Yes
Slide 3.1-11
Objective 4
Complete ordered pairs for a given
equation.
Slide 3.1-12
EXAMPLE 3 Completing Ordered Pairs
Complete each ordered pair for the equation.
y  2x  9
Solution:
 2, 
 , 7 
y  2  2  9
y  49
y  5
7  9  2x  9  9
16 2x

2
2
x 8
 2, 5
8,7 
Slide 3.1-13
Objective 5
Complete a table of values.
Slide 3.1-14
Complete a table of values.
Ordered pairs are often displayed in a table of values. Although we
usually write tables of values vertically, they may be written
horizontally.
Slide 3.1-15
EXAMPLE 4 Completing Tables of Values
Complete the table of values for the equation. Write the results as
ordered pairs.
2 x  3 y  12
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, 2 3 3
y  2
2 x  3  0   12
2 x 12

2
2
x6
 6, 0 
2 x  3  3  12
2x  9  9  12  9
2x 3

2
2
x
3
2
3

,

3


2


Slide 3.1-16
Objective 6
Plot ordered pairs.
Slide 3.1-17
Plot ordered pairs.
Every linear in two variables equation has an infinite number of
ordered pairs (x, y) 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 axis
form a rectangular coordinate
system, also called the Cartesian
coordinate system.
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.
Slide 3.1-19
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).
Slide 3.1-20
EXAMPLE 5 Plotting Order Pairs
Plot the given points in a coordinate system:
 3,5 ,  2,6 ,  4,0 ,  5, 2 , 5, 2 ,  0, 6 .
Slide 3.1-21
EXAMPLE 6
Completing Ordered Pairs to Estimate the Number of Twin Births
Complete the table of ordered pairs for the equation,
y  3.049 x  5979,
where x = year and y = number of twin births in thousands. Round
answers to the nearest whole number. Interpret the results for 2005.
125
134
Solution:
There were about 134
thousand twin births in the
U.S. in 2005.
y  3.049  2002  5979
y  6104.1  5979
y  125.1
y  3.049  2005  5979
y  6113.2  5979
y  134.2
Slide 3.1-22
Plot ordered pairs. (cont’d)
The ordered pairs of twin births in
the U.S. for 2001 until 2006 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.
Do not assume that this scatter diagram or resulting equation would provide
reliable data for other years, since the data for those years may not follow the
same pattern.
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.
Slide 3.1-24