Download Ch3-Sec 3.1

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3.1 – Slide 1
Chapter 3
Graphs of Linear Equations
and Inequalities; Functions
Copyright © 2010 Pearson Education, Inc. All rights reserved.
3.1 – Slide 2
3.1
Reading Graphs; Linear
Equations in Two Variables
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3.1 – Slide 3
3.1 Reading Graphs; Linear Equations
in Two Variables
Objectives
1.
2.
3.
4.
5.
6.
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
3.1 – Slide 4
3.1 Reading Graphs; Linear Equations
in Two Variables
Interpreting Graphs
Example 1
(a) Which quiz had the highest class
average? Quiz #3
The line graph below
shows the class averages
on the first eight quizzes
in a college math course.
(b) Which quiz experienced the
biggest jump in class average from
the previous quiz? Quiz #5
1010
(c) Estimate the difference
between the class average
on Quiz #2 and Quiz #3.
7.5
Score
5
Difference ≈ 9 – 7
2.5
0
Difference ≈ 2
Quiz #1
Quiz #2
Quiz #3
Quiz #4 Quiz #5
Quiz #6
Quiz #7 Quiz #8
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3.1 – Slide 5
3.1 Reading Graphs; Linear Equations
in Two Variables
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.
Note
Other linear equations in two variables, such as
y = 4x + 5 and 3x = 7 – 2y,
are not written in standard form but could be. We
discuss the forms of linear equations in Section 3.4.
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3.1 – Slide 6
3.1 Reading Graphs; Linear Equations
in Two Variables
Writing 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 = 4x + 5 since
13 = 4(2) + 5.
Let x = 2, y = 13.
The pair of numbers x = 2 and y = 13 gives one solution of the
equation y = 4x + 5. The phrase “x = 2 and y = 13” is abbreviated
x-value
y-value
(2,13)
Ordered Pair
The x-value is always given first. A pair of numbers such as (2,13)
is called an ordered pair.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
3.1 – Slide 7
3.1 Reading Graphs; Linear Equations
in Two Variables
Deciding Whether an Ordered Pair is a Solution
Example 2
Decide whether each ordered pair is a solution to 5x – 2y = 4.
(a) (2,3)
To see whether (2,3) is a
solution, substitute 2 for x
and 3 for y.
?
5(2) – 2(3) = 4
?
10 – 6 = 4
4=4
True
Thus, (2,3) is a solution.
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(b) (–2,–3)
To see whether (–2,–3) is a
solution, substitute –2 for x
and –3 for y.
?
5(–2) – 2(–3) = 4
?
–10 + 6 = 4
–4 = 4
False
Thus, (–2,–3) is not a solution.
3.1 – Slide 8
3.1 Reading Graphs; Linear Equations
in Two Variables
Completing Ordered Pairs
Example 3
Complete each ordered pair for the equation –3x + y = 4.
(a) (3, )
Substitute 3 for x and solve
for y.
(b) ( ,1)
Substitute 1 for y and solve
for x.
–3(3) + y = 4
–3x + 1 = 4
–1 –1
–9 + y = 4
+9
+9
y = 13
The ordered pair is (3,13).
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–3x = 3
3 3
x = –1
The ordered pair is (–1,1).
3.1 – Slide 9
3.1 Reading Graphs; Linear Equations
in Two Variables
Completing a Table of Values
Example 4
Complete the table of values for each equation.
(a) x – 3y = 6
x
y
–1
x –3(–1) = 6
x +3 = 6
–3 –3
x=3
12
For the first
ordered pair, let
y = –1.
For the second
ordered pair, let
x = 12.
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12 – 3y = 6
– 12
– 12
– 3y = –6
3
3
y=2
x y
Thus, the
completed 3 –1
table is:
12 2
3.1 – Slide 10
3.1 Reading Graphs; Linear Equations
in Two Variables
Plotting Ordered Pairs
To graph solutions, represented as the ordered pairs (x, y),
we need two number lines, one for each variable, as drawn
below. The horizontal number line is called the x-axis, and
the vertical line is called the y-axis.
Together, the x-axis and the y-axis
form a rectangular coordinate
system, also called the Cartesian
coordinate system, in honor of
René Descartes, the French
mathematician who is
credited with its invention.
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3.1 – Slide 11
3.1 Reading Graphs; Linear Equations
in Two Variables
Plotting Ordered Pairs
The coordinate system is divided into four regions, called
quadrants. These quadrants are numbered counterclockwise, as shown on the previous slide. Points on the
axes themselves are not in any quadrant. The point at
which the x-axis and the y-axis meet is called the origin.
The origin, which is labeled 0 in the previous figure, is the
point corresponding to (0,0).
The x-axis and y-axis determine a plane. By referring to the
two axes, every point in the plane can be associated with an
ordered pair. The numbers in the ordered pair are called the
coordinates of the point.
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3.1 – Slide 12
3.1 Reading Graphs; Linear Equations
in Two Variables
Plotting Ordered Pairs
For example, we locate the point associated with the ordered
pair (2,3) by starting at the origin. Since the x-coordinate is 2,
we go 2 units to the right along the x-axis. Then, since the ycoordinate is 3, we turn and go up 3 units on a line parallel to
the y-axis. Thus, the point (2,3) is plotted.
Note
When we graph on a
number line, one number
corresponds to each point.
On a plane, however, both
numbers in an ordered
pair are needed to locate
a point. The ordered pair
is a name for the point.
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3.1 – Slide 13
3.1 Reading Graphs; Linear Equations
in Two Variables
Plotting Ordered Pairs
Example 5
Plot each ordered pair on a coordinate system.
(a) (1,5)
(b) (–2,3)
(c) (–1, –4)
3 
(e)  , 2 
2 
(f) (5,0)
(g) (0, –3)
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(d) (3, –2)
3.1 – Slide 14