11.2 Graphing Linear Equations in Two Variables

1. Slide - 1Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
2
Graphs of Linear
Equations and
Inequalities in Two
Variables
11

2. Slide - 2Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
1. Graph linear equations by plotting ordered
pairs.
2. Find intercepts.
3. Graph linear equations of the form
Ax + By = 0.
4. Graph linear equations of the form y = b or
x = a.
5. Use a linear equation to model data.
Objectives
11.2 Graphing Linear Equations in Two
Variables

3. Slide - 3Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
y = 2(0) – 1
Example Graph the linear equation y = 2x – 1.
Note that although this
equation is not of the
form Ax + By = C, it
could be. Therefore, it
is linear. To graph it,
we will first find two
points by letting x = 0
and then y = 0.
Graph by Plotting Ordered Pairs
If x = 0, then
The graph of any linear equation in two variables is a
straight line.
y = – 1
So, we have
the ordered
pair (0,–1).
0 = 2x – 1
If y = 0, then
1 = 2x
So, we have the
ordered pair (½,0).
+ 1 + 1
2 2
½ = x

4. Slide - 4Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
y = 2(1) – 1
Example (cont) Graph the linear equation y = 2x – 1.
Now we will find a
third point (just as a
check) by letting x = 1.
Graph by Plotting Ordered Pairs
If x = 1, then
y = 1
So, we have the
ordered pair (1,1).
When we graph, all three points,
(0,–1), (½,0), and (1,1), should
lie on the same straight line.
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5. Slide - 5Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Finding Intercepts
To find the x-intercept, let y = 0 in the given equation and
solve for x. Then (x, 0) is the x-intercept.
To find the y-intercept, let x = 0 in the given equation and
solve for y. Then (0, y) is the y-intercept.
Find Intercepts

6. Slide - 6Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
0 + y = 2
Example
Find the intercepts for the graph of x + y = 2. Then draw the
graph.
To find the y-intercept,
let x = 0; to find the
x-intercept, let y = 0.
Graphing a Linear Equation Using Intercepts
y = 2
The y-intercept
is (0, 2).
x + 0 = 2
x = 2
The x-intercept
is (2, 0).
Plotting the intercepts gives
the graph.
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7. Slide - 7Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
–6(0) + 2y = 0
Example
Graph the linear equation –6x + 2y = 0.
First, find the intercepts.
Graphing an Equation with x- and y-Intercepts (0, 0)
2y = 0
The y-intercept
is (0, 0).
The x-intercept
is (0, 0).
Since the x and y
intercepts are the
same (the origin),
choose a different
value for x or y.
2 2
y = 0
–6x + 2(0) = 0
–6x = 0
6 6
x = 0

8. Slide - 8Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
–6(1) + 2y = 0
Example (cont)
Graph the linear equation –6x + 2y = 0.
Let x = 1.
Graphing an Equation with x- and y-Intercepts (0, 0)
2y = 6
A second point is (1, 3).
2 2
y = 3
–6 + 2y = 0
+6 +6
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9. Slide - 9Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Line through the Origin
The graph of a linear equation of the form
Ax + By = 0
where A and B are nonzero real numbers, passes
through the origin (0,0).
Graph Linear Equations of the Form Ax + By = 0

10. Slide - 10Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Note that this is the graph
of a horizontal line with
y-intercept (0,–2).
Example
Graph y = –2.
Graphing a Horizontal Line
The expanded version of this
linear equation would be
0 · x + y = –2. Here, the
y-coordinate is unaffected by
the value of the x-coordinate.
Whatever x-value we choose,
the y-value will be –2. Thus,
we could plot the points
(–1, –2), (2,–2), (4,–2), etc.
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11. Slide - 11Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
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Example
Graph x – 1 = 0.
Graphing a Vertical Line
Add 1 to each side of the
equation. x = 1.
The x-coordinate is unaffected
by the value of the y-
coordinate.
Thus, we could plot the points
(1, –3), (1, 0), (1, 2), etc.
Note that this is the graph
of a vertical line with no
y-intercept.

12. Slide - 12Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Horizontal and Vertical Lines
The graph of y = b, where b is a real number, is a
horizontal line with y-intercept (0, b) and no x-intercept
(unless the horizontal line is the x-axis itself).
The graph of x = a, where a is a real number, is a
vertical line with x-intercept (a, 0) and no y-intercept
(unless the vertical line is the y-axis itself).
Graph Linear Equations of the Form y = k or x = k

13. Slide - 13Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Bob has owned and managed Bob’s Bagels for the past 5 years
and has kept track of his costs over that time. Based on his
figures, Bob has determined that his total monthly costs can be
modeled by C = 0.75x + 2500, where x is the number of bagels
that Bob sells that month.
(a) Use Bob’s cost equation to determine his costs if he sells
1000 bagels next month, 4000 bagels next month.
Use a Linear Equation to Model Data
C = 0.75(1000) + 2500
C = $3250
C = 0.75(4000) + 2500
C = $5500

14. Slide - 14Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example 7 (cont)
(b) Write the information from part (a) as two ordered pairs and
use them to graph Bob’s cost equation.
Use a Linear Equation to Model Data
From part (a) we have (1000, 3250) and (4000, 5500).
Cost ($)
# of bagels
6000
1000
3000
5000
2000
4000
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Note that we did not
extend the graph to the
left beyond the vertical
axis. That area would
correspond to a negative
number of bagels, which
does not make sense.