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7.2 TWO-VARIABLE LINEAR SYSTEMS
Copyright © Cengage Learning. All rights reserved.
2
• Use the method of elimination to solve systems
of linear equations in two variables.
• Interpret graphically the numbers of solutions of
systems of linear equations in two variables.
• Use systems of linear equations in two variables
to model and solve real-life problems.
What You Should Learn
3
The Method of Elimination
4
The Method of Elimination
5
Example 2 – Solving a System of Equations by Elimination
Solve the system of linear equations.
Solution:
Multiplying Equation 2 by 4.
Equation 1
Equation 2
Write Equation 1.
Multiply Equation 2 by 4.
Add equations.
Solve for x.
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By back-substituting into Equation 1, you can solve
for y.
2x – 4y = –7
–4y = –7
The solution is
cont’d Example 2 – Solution
Write Equation 1.
Substitute for x.
Combine like terms.
Solve for y.
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Check this in the original system, as follows.
Check
2x – 4y = –7
–1 – 6 = –7
5x + y = –1
cont’d Example 2 – Solution
Write original Equation 1.
Substitute into Equation 1.
Equation 1 checks.
Write original Equation 2.
Substitute into Equation 2.
Equation 2 checks.
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The Method of Elimination
Equivalent systems: have precisely the same solution set.
9
The Method of Elimination
The operations that can be performed on a system of linear
equations to produce an equivalent system are
(1) interchanging any two equations,
(2) multiplying an equation by a nonzero constant, and
(3) adding a multiple of one equation to any other equation
in the system.
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Graphical Interpretation of Solutions
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Graphical Interpretation of Solutions
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Graphical Interpretation of Solutions
A system of linear equations is consistent if it has at least
one solution.
A consistent system with exactly one solution is
independent, whereas a consistent system with infinitely
many solutions is dependent.
A system is inconsistent if it has no solution.
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Example
5𝑥 + 3𝑦 = 92𝑥 − 4𝑦 = 14
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Example
𝑥 − 2𝑦 = 3
−2𝑥 + 4𝑦 = 1
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Example
2𝑥 − 𝑦 = 14𝑥 − 2𝑦 = 3
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In Class Assignment
0.2𝑥 − 0.5𝑦 = −27.80.3𝑥 + 0.4𝑦 = 68.7
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Example 4 – Recognizing Graphs of Linear Systems
Match each system of linear equations with its graph
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Example 4 – Solution
a. The graph of system (a) is a pair of parallel lines (ii).
The lines have no point of intersection, so the system
has no solution. The system is inconsistent.
b. The graph of system (b) is a pair of intersecting
lines (iii). The lines have one point of intersection, so
the system has exactly one solution. The system is
consistent.
c. The graph of system (c) is a pair of lines that
coincide (i). The lines have infinitely many points of
intersection, so the system has infinitely many solutions.
The system is consistent.
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Applications
20
Applications
At this point, you may be asking the question “How can I
tell which application problems can be solved using a
system of linear equations?” The answer comes from
the following considerations.
1. Does the problem involve more than one unknown
quantity?
2. Are there two (or more) equations or conditions to be
satisfied?
If one or both of these situations occur, the appropriate
mathematical model for the problem may be a system of
linear equations.
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Example 8 – An Application of a Linear System
An airplane flying into a headwind travels the 2000-mile
flying distance between Chicopee, Massachusetts and
Salt Lake City, Utah in 4 hours and 24 minutes. On the
return flight, the same distance is traveled in 4 hours.
Find the airspeed of the plane and the speed of the wind,
assuming that both remain constant.
Solution:
The two unknown quantities are the speeds of the wind and
the plane.
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If r1 is the speed of the plane and r2 is the speed of the
wind, then
r1 – r2 = speed of the plane against the wind
r1 + r2 = speed of the plane with the wind
as shown in Figure 7.10.
cont’d Example 8 – Solution
Figure 7.10
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Using the formula distance = (rate)(time) for these two
speeds, you obtain the following equations.
2000 = (r1 – r2)
2000 = (r1 + r2)(4)
These two equations simplify as follows.
cont’d Example 8 – Solution
Equation 1
Equation 2
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To solve this system by elimination, multiply Equation 2 by
11.
cont’d Example 8 – Solution
Write Equation 1.
Multiply Equation 2 by 11.
Add equations.
25
So,
miles per hour
and
miles per hour.
cont’d Example 8 – Solution
Speed of plane
Speed of wind