Download Graph linear equations by plotting ordered pairs.

Chapter 3
Section 2
3.2 Graphing Linear Equations in Two Variables
Objectives
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 = k or x = k.
5
Use a linear equation to model data.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 1
Graph linear equations by plotting
ordered pairs.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 © 2012, 2008, 2004 Pearson Education, Inc.
Slide 3.2-5
Graph linear equations by plotting ordered pairs. (cont’d)
Graph of a Linear Equation
The graph of any linear equation in two variables is a straight line.
Notice the word line appears in the name “linear 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 © 2012, 2008, 2004 Pearson Education, Inc.
Slide 3.2-6
EXAMPLE 1 Graphing a Linear Equation
Graph
5 x  2 y  10.
Solution:
5  0  2 y  10
2 y 1 0

2
2
y  5
5x  2  0  10
 0, 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
y2
 4, 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 © 2012, 2008, 2004 Pearson Education, Inc.
Slide 3.2-7
EXAMPLE 2 Graphing a Linear Equation
Graph
2
y  x  2.
3
Solution:
2
y   0  2
3
y  2
 0, 2
2
02  x22
3
2 3
3
 2  x 
3 2
2
x 3
2
4  2  x  2  2
3
3,0
2
3
 2  x
3
2
x  3
 3, 4
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 yintercept and (3,0) is called the x-intercept of the graph.
The intercepts are particularly useful for graphing linear equations.
They are found by replacing, in turn, each variable with 0 in the
equation and solving for the value of the other variable.
Finding Intercepts
To find the x-intercept, let y = 0 and
solve for x. Then (x,0) is the xintercept.
To find the y-intercept, let x = 0 and
solve for y. Then (0, y) is the yintercept.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 3.2-10
EXAMPLE 3 Finding Intercepts
Find the intercepts for the graph of 5x + 2y = 10. Then draw the graph.
Solution:
5x  2  0  10
5 x 10

5
5
x2
x-intercept:
 2, 0 
5  0  2 y  10
2 y 10

2
2
y 5
y-intercept:
 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.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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.
Line through the Origin
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.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 3.2-13
EXAMPLE 4 Graphing an Equation with x- and y-Intercepts (0, 0)
Graph 4x − 2y = 0.
Solution:
12
1
4  6  2 y  0
24  2 y  24  0  24
2 y 2 4

2
2
y  12
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
4x  2  2  0
4x  4  4  0  4
4 x 4

4
4
x  1
Slide 3.2-14
Objective 4
Graph linear equations of the form
y = k or x = k.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 straight lines and vertical
straight lines, respectively.
Horizontal Line
The graph of the linear equation y = k, where k is a real number, is the
horizontal line with y-intercept (0, k). There is no y-intercept (unless
the vertical line is the y-axis itself).
Vertical Line
The graph of the linear equation x = k, where k is a real number, is the
vertical line with x-intercept (k, 0). There is no x-intercept (unless the
horizontal line is the x-axis itself).
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 (for k ≠ 0).
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 3.2-16
EXAMPLE 5 Graphing an Equation of the Form y = k (Horizontal Line)
Graph y = − 5.
Solution:
The equation states that every value of y = − 5.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 3.2-17
EXAMPLE 6 Graphing an Equation of the Form x = k (Vertical Line)
Graph x − 2 = 0.
Solution:
After 2 is added to each side the equation states that every value
of x = 2.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 3.2-18
Graphing linear equations of the form y = k or x = k. (cont’d)
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Slide 3.2-19
Graphing linear equations of the form y = k or x = k. (cont’d)
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 3.2-20
Objective 5
Use a linear equation to model data.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 3.2-21
Example 7
Using a Linear Equation to Model Credit Card Debt
Use (a) the graph and (b) the equation from Example 7 to approximate
credit card debt in 2005.
y  32.0 x  684
Solution:
2005
About 850 billion dollars using the
graph to estimate.
y  32.0  5  684
y  160  684
y  844
Exactly 844 billion dollars using the equation.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 3.2-22