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Transcript
Chapter 4
Section 1
4.1 Solving Systems of Linear Equations by Graphing
Objectives
1
Decide whether a given ordered pair is a solution of a system.
2
Solve linear systems by graphing.
3
Solve special systems by graphing.
4
Identify special systems without graphing.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solving Systems of Linear Equations by Graphing
A system of linear equations, often called a linear system, consists
of two or more linear equations with the same variables. Examples of
systems include
2x  3y  4
3x  y  5
x  3y  4
 y  4  2x
x  y 1
y  3.
Linear systems
In the system on the right, think of y = 3 as an equation in two variables by
writing it as 0x + y = 3.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 4.1-3
Objective 1
Decide whether a given ordered
pair is a solution of a system.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 4.1-4
Decide whether a given ordered pair is a solution of a
system.
A solution of a system of a linear equations is an ordered pair that
makes both equations true at the same time. A solution is said to
satisfy the equation.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 4.1-5
EXAMPLE 1 Determining Whether an Ordered Pair is a Solution
Decide whether the ordered pair (4,−1) is a solution of each system.
5 x  6 y  14
2x  5 y  3
 x  y  3
x  y  3
5  4   6  1  14
  4    1  3
Solution:
2  4   5  1  3
20  6  14
85  3
14  14
3  3 Yes
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
 4    1  3
4  1  3
4  1  3
3  3
3  3
No
Slide 4.1-6
Objective 2
Solve linear systems by graphing.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 4.1-7
Solve linear systems by graphing.
The set of all ordered pairs that are solutions of a system is its
solution set.
One way to find the solution set of a system of two linear equations is
to graph both equations on the same axes. The graph of each line
shows points whose coordinates satisfy the equation of that line.
Any intersection point would be on both lines and would therefore be a
solution of both equations. Thus, the coordinates of any point at
which the lines intersect give a solution of the system.
Because the two different straight lines can intersect at no more then
one point, there can never be more than one solution set for such a
system.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 4.1-8
Solve linear systems by graphing. (cont’d)
Solving a Linear System by Graphing
Step 1: Graph each equation of the system on the same
coordinate axes.
Step 2: Find the coordinates of the point of intersection of
the graphs if possible. This is the solution of the system.
Step 3: Check the solution in both of the original equations. Then
write the solution set.
A difficulty with the graphing method is that it may not be possible to
determine from the graph the exact coordinates of the point that
represents the solution, particularly if those coordinates are not integers.
The graphing method does, however, show geometrically how solutions
are found and is useful when approximate answer will do.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 4.1-9
EXAMPLE 2 Solving a System by Graphing
Solve the system by graphing.
5x  3 y  9
x  2y  7
Solution:
{(3,2)}
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 4.1-10
Objective 3
Solve special systems by graphing.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 4.1-11
Solve special systems by graphing.
Sometimes the graphs of the two equations in a system either do not
intersect at all or are the same line.
When a system has an infinite number of solutions, either equation of the
system could be used to write the solution set. It’s best to use the
equation (in standard form) with coefficients that are integers having
no common factor (except 1).
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 4.1-12
EXAMPLE 3 Solving Special Systems by Graphing
Solve each system by graphing
3x  y  4
6 x  2 y  12
Solution:

Copyright © 2012, 2008, 2004 Pearson Education, Inc.
2x  5 y  8
4 x  10 y  16
 x, y  2 x  5 y  8
Slide 4.1-13
Solve special systems by graphing. (cont’d)
Three Cases for Solutions of Systems
1. The graphs intersect at exactly one point, which gives the (single)
ordered pair solution of the system. The system is consistent and
the equations are independent. See below left.
2. The graphs are parallel lines, so there is no solution and the solution
set is Ø. The system is inconsistent and the equations are
independent. See below middle.
3. The graphs are the same line. There is an infinite number of
solutions, and the solution set is written in set-builder notation as
{(x,y)|_________}, where one of the equations is written after the |
symbol. The system is consistent and equations are dependent.
See below right.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 4.1-14
Objective 4
Identify special systems without
graphing.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 4.1-15
Identify special systems without graphing.
Example 3 showed that the graphs of an inconsistent system are
parallel lines and the graphs of a system of dependent equations are
the same line. We can recognize these special kinds of systems
without graphing by using slopes.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 4.1-16
EXAMPLE 4 Identifying the Three Cases by Using Slopes
Describe each system without graphing. State the number of solutions.
2x  3y  5
3y  2x  7
1
2
y 
x
3
3
1
2
y 
x
3
3
Solution:
The equations
represent parallel
lines. The system
has no solution.
x  3y  2
2 x  6 y  4
2
y 
x
3
2
y 
x
3
6x  y  3
2 x  y  11
5
3
7
3
The equations
represent the
same line. The
system has an
infinite number of
solutions.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
y  6 x  3
y  2 x  11
The equations
represent lines that
are neither parallel
nor the same line.
The system has
exactly one
solution.
Slide 4.1-17