Download Solving Systems of Linear Equations by Graphing

Systems of Linear Equations
and Inequalities in
Two Variables
Copyright © Cengage Learning. All rights reserved.
7
Section
7.1
Solving Systems of Linear
Equations by Graphing
Copyright © Cengage Learning. All rights reserved.
Objectives
1 Determine whether an ordered pair is a
solution to a given system of linear equations.
2 Solve a system of linear equations by
graphing.
3 Recognize that an inconsistent system has no
solution.
4 Express the infinitely many solutions of a
dependent system as a general ordered pair.
3
Solving Systems of Linear Equations by Graphing
The lines graphed below approximate the percentages of
American households with only a landline phone and those
with only a cell phone for the years 2005 to 2010. We can
see that over this period, the percentage of those with only
a landline decreased while those with only a cell phone
increased.
4
Solving Systems of Linear Equations by Graphing
By graphing this information on the same coordinate
system, it appears that the percentage of households with
a cell phone only was the same as those with only a
landline in October 2008—about 19.3% each.
In this section, we will work with pairs of linear equations
whose graphs often will be intersecting lines.
5
1.
Determine whether an ordered pair is a
solution to a given system of linear equations
6
Determine whether an ordered pair is a solution to a given
system of linear equations
Recall that we have previously graphed equations such as
x + y = 3 that contain two variables. Because there are
infinitely many pairs of numbers whose sum is 3, there are
infinitely many pairs (x, y) that will satisfy this equation.
Some of these pairs are listed in Table 7-1(a).
(a)
Table 7-1
7
Determine whether an ordered pair is a solution to a given
system of linear equations
Likewise, there are infinitely many pairs (x, y) that will
satisfy the equation 3x – y = 1.
Some of these pairs are listed in Table 7-1(b).
Although there are infinitely many
pairs that satisfy each of these
equations, only the pair (1, 2) satisfies
both equations.
(b)
Table 7-1
8
Determine whether an ordered pair is a solution to a given
system of linear equations
The pair of equations
x+y=3
3x – y = 1
is called a system of equations. Because the ordered pair
(1, 2) satisfies both equations, it is called a simultaneous
solution or just a solution of the system of equations.
We will discuss three methods for finding the solution of a
system of two linear equations. In this section, we consider
the graphing method.
9
2.
Solve a system of linear equations by
graphing
10
Solve a system of linear equations by graphing
To use the method of graphing to solve the system
x+y=3
3x – y = 1
we will graph both equations on one set of coordinate axes.
Using the intercept method, Recall that to find the
y-intercept, we let x = 0 and solve for y and to find the
x-intercept, we let y = 0 and solve for x.
11
Solve a system of linear equations by graphing
We will also plot one extra point as a check. See Figure 7-2.
Figure 7-2
12
Solve a system of linear equations by graphing
Although there are infinitely many pairs (x, y) that satisfy
x + y = 3 and infinitely pairs (x, y) that satisfy 3x – y = 1,
only the coordinates of the point where their graphs
intersect satisfy both equations. The solution of the system
is the ordered pair (1, 2).
To check the solution, we substitute 1 for x and 2 for y in
each equation and verify that the pair (1, 2) satisfies each
equation.
13
Solve a system of linear equations by graphing
First equation
x+y=3
1+2≟3
3=3
Second equation
3x – y = 1
3(1) – 2 ≟ 1
3–2≟1
1=1
When the graphs of two equations in a system are different
lines, the equations are called independent equations.
When a system of equations has a solution, the system is
called a consistent system.
14
Solve a system of linear equations by graphing
To solve a system of equations in two variables by
graphing, we follow these steps.
The Graphing Method
1. Graph each equation on one set of coordinate axes.
2. Find the coordinates of the point where the graphs
intersect, if applicable.
3. Check the solution in the equations of the original
system, if applicable.
15
Example
Solve the system
2x + 3y = 2
.
3x = 2y + 16
Solution:
Using the intercept method, we graph both equations on
one set of coordinate axes, as shown in Figure 7-3. We
also plot a third point as a check.
Figure 7-3
16
Example – Solution
cont’d
Although there are infinitely many pairs (x, y) that satisfy
2x + 3y = 2 and infinitely many pairs (x, y) that satisfy
3x = 2y + 16, only the coordinates of the point where the
graphs intersect satisfy both equations.
The solution is the ordered pair (4, –2).
To check, we substitute 4 for x and –2 for y in each
equation and verify that the pair (4, –2) satisfies each
equation.
2x + 3y = 2
3x = 2y + 16
2(4) + 3(–2) ≟ 2
3(4) ≟ 2(–2) + 16
17
Example – Solution
8–6≟2
2=2
cont’d
12 ≟ –4 + 16
12 = 12
The equations in this system are independent equations,
and the system is a consistent system of equations.
18
Solve a system of linear equations by graphing
Comment
Always check your answer in the original equations. If you
made an error in simplifying one of the equations, your
answer would check in the simplified equations but not in
the original.
19
3.
Recognize that an inconsistent
system has no solution
20
Recognize that an inconsistent system has no solution
Sometimes a system of equations will have no solution.
These systems are called inconsistent systems.
21
Example
Solve the system by graphing:
2x + y = –6
.
4x + 2y = 8
Solution:
We graph both equations on one set of coordinate axes, as
in Figure 7-5.
Figure 7-5
22
Example – Solution
cont’d
Since the graphs are different lines, the equations of the
system are independent. The lines in the figure appear to
be parallel. To be sure, we can find their slopes. Recall that
if two lines have the same slope but different y-intercepts,
they will be parallel.
To determine the slope of each line, we write each
equation in slope-intercept form, y = mx + b.
2x + y = –6
y = –2x – 6
4x + 2y = 8
2y = –4x + 8
y = –2x + 4
23
Example – Solution
cont’d
Because both equations have the same slope (–2) but
different y-intercepts (0, –6) and (0, 4), they are parallel.
Since parallel lines do not intersect, the system is
inconsistent. Its solution set is ∅.
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4.
Express the infinitely many solutions of a
dependent system as a general ordered pair
25
Express the infinitely many solutions of a dependent
system as a general ordered pair
Sometimes a system will have infinitely many solutions. In
this case, we say that the equations of the system are
dependent equations.
26
Example
Solve the system by graphing:
y – 2x = 4 .
4x + 8 = 2y
Solution:
We graph each equation on one set of axes, as in
Figure 7-6.
Figure 7-6
27
Example – Solution
cont’d
The lines in the figure appear to be the same line. To be
sure, we can find their slopes and y-intercepts by writing
each equation in slope-intercept form.
If two lines have the same slope and the same y-intercept,
they will be the same line.
y – 2x = 4
y = 2x + 4
4x + 8 = 2y
2x + 4 = y
28
Example – Solution
cont’d
Since the lines in the figure are the same line, they
intersect at infinitely many points and there are infinitely
many solutions. Every solution to the first equation is a
solution to the second equation. To describe these
solutions, we can solve either equation for y.
Because 2x + 4 is equal to y, every solution (x, y) of the
system will have the form (x, 2x + 4). This solution can also
be written in set-builder notation, {(x, y) | y = 2x + 4}.
29
Example – Solution
cont’d
To find some specific solutions, we can substitute
0, –3, and –1 for x in the general ordered pair (x, 2x + 4) to
obtain (0, 4), (–3, –2), and (–1, 2). From the graph, we can
see that each point lies on the one line that is the graph of
both equations.
30
Express the infinitely many solutions of a dependent
system as a general ordered pair
Table 7-2 summarizes the possibilities that can occur when
two nonvertical linear equations, each with two variables,
are graphed.
Table 7-2
31