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Chapter 3
Section 2
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3.2
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Graphing Linear Equations in Two
Variables
Graph linear equations by plotting ordered
pairs.
Find intercepts.
Graph linear equations of the form Ax + By = 0.
Graph linear equations of the form y = k or
x = k.
Use a linear equation to model data.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 1
Graph linear equations by
plotting ordered pairs.
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Slide 3.2 - 3
Graph linear equations by plotting ordered
pairs.
Infinitely many ordered pairs satisfy a linear equation in two
variables. We find these ordered-pair solutions by choosing as
many values of x (or y) as we wish and then completing each
ordered pair.
Some solutions of the equation x + 2y = 7 are graphed below.
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Slide 3.2 - 4
Graph linear equations by plotting ordered
pairs. (cont’d)
Notice that the points plotted in the previous graph all appear
to lie on a straight line, as shown below.
Every point on the line represents a solution of the equation
x + 2y = 7, and every solution of the equation corresponds to a
point on the line.
The line gives a “picture” of all the solutions of the equation
x + 2y = 7. Only a portion of the line is shown, but it extends
indefinitely in both directions, suggested by the arrowheads.
The line is called the graph of the
equation, and the process of plotting the
ordered pairs and drawing the line
through the corresponding points is
called graphing.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3.2 - 5
Graph linear equations by plotting ordered
pairs. (cont’d)
In summary, the graph of any linear equation in two variables
is a straight line.
Notice the word line appears in the name “lineear equation.”
Since two distinct points determine a line, we can graph a
straight line by finding any two different points on the line.
However, it is a good idea to plot a third point as a check.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3.2 - 6
EXAMPLE 1
Graphing a Linear Equation
Graph 5 x  2 y  10.
Solution: 5  0  2 y  10
5x  2  0  10
2 y 1 0

2
2 0, 5


y  5
5 x 10

5
5
x  2
 2, 0
5  4  2 y  10
20  2 y  20  10  20
2 y 10

2
2
4, 2
y2


When graphing a linear equation, all three points should lie on the same
straight line. If they don’t, double-check the ordered pairs you found.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3.2 - 7
EXAMPLE 2
2
Graph y  x  2.
3
Solution:
Graphing a Linear Equation
2
y   0  2
3
y  2
 0, 2
2
02  x22
3
2 3
3
 2  x 
3 2
2
2
4  2  x  2  2
3
x 3
3,0
2
3
 2  x
3
2
x  3
 3, 4
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Slide 3.2 - 8
Objective 2
Find intercepts.
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Slide 3.2 - 9
Find intercepts.
In the previous example, the graph
intersects (crosses) the y-axis at (0,−2)
and the x-axis at (3,0). For this reason
(0,−2) is called the y-intercept and (3,0)
is called the x-intercept of the graph.
The intercepts are particularly useful for graphing linear
equations. The intercepts are found by replacing, in turn, each
variable with 0 in the equation and solving for the value of the
other variable.
To find the x-intercept, let y = 0 and solve for x. Then (x,0) is
the x-intercept.
To find the y-intercept, let x = 0 and solve for y. Then (0, y) is
the y-intercept.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3.2 - 10
EXAMPLE 3
Finding Intercepts
Find the intercepts for 5x + 2y = 10. Then draw the
graph.
Solution:
5  0  2 y  10
5x  2  0  10
2 y 10
5 x 10


2
2
5
5
y 5
x2
x-intercept:
y-intercept:
 2, 0 
 0,5
When choosing x- or y-values to find ordered pairs to plot, be careful to
choose so that the resulting points are not too close together. This may
result in an inaccurate line.
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Slide 3.2 - 11
Objective 3
Graph linear equations of the
form Ax + By = 0.
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Slide 3.2 - 12
Graph linear equations of the form Ax + By = 0.
If A and B are nonzero real numbers, the graph of a linear
equation of the form
Ax  By  0
passes through the origin (0,0).
A second point for a linear equation that passes through the origin
can be found as follows:
1. Find a multiple of the coefficients of x and y.
2. Substitute this multiple for x.
3. Solve for y.
4. Use these results as a second ordered pair.
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EXAMPLE 4
Graphing an Equation of the
Form Ax + By = 0
Graph 4x − 2 = 0.
Solution:
12
1
4  6  2 y  0
24  2 y  24  0  24
2 y 2 4

2
2
y  12
4x  2  2  0
4x  4  4  0  4
4 x 4

4
4
x  1
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Slide 3.2 - 14
Objective 4
Graph linear equations of the
form y = k or x = k.
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Slide 3.2 - 15
Graphing linear equations of the form
y = k or x = k.
The equation y = −4 is the linear equation in which the
coefficient of x is 0. Also, x = 3 is a linear equation in which
the coefficient of y is 0. These equations lead to horizontal
and vertical straight lines, respectively.
The graph of the linear equation y = k, where k is a real
number, is a horizontal line with y-intercept (0, k) and no
x-intercept.
The graph of the linear equation x = k, where k is a real
number, is a vertical line with x-intercept (k ,0) and no
y-intercept.
The equations of horizontal and vertical lines are often confused with
each other. Remember that the graph of y = k is parallel to the x-axis
and that of x = k is parallel to the y-axis.
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Slide 3.2 - 16
EXAMPLE 5
Graphing an Equation of the
Form y = k
Graph y = −5.
Solution:
The equation states that every value of y = −5.
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Slide 3.2 - 17
EXAMPLE 6
Graphing an Equation of the
Form x = k
Graph x − 2 = 0.
Solution:
After 2 is added to each side the equation states that
every value of x = 2.
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Objective 5
Use a linear equation to model
data.
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Slide 3.2 - 19
The different forms of linear equations from this section and the
methods of graphing them are given in the following summary.
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Slide 3.2 - 20
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EXAMPLE 7
Use a Linear Equation to Model
Credit Card Debt
Use a) the graph and b) the equation to approximate
credit card debt in 1997, where x = 2.
Solution:
a) about 525 billion dollars
b) y  38.7  2  450
y  77.4  450
1997
y  527.4
527.4 billion dollars.
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