Transcript Section 9.1

9.1 Systems of Equations in
Two Variables
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S O LV E A S Y S T E M O F T W O L I N E A R E Q U AT I O N S I N T W O
VA R I A B L E S B Y G R A P H I N G .
S O LV E A S Y S T E M O F T W O L I N E A R E Q U AT I O N S I N T W O
VA R I A B L E S U S I N G T H E S U B S T I T U T I O N A N D T H E
E L I M I N AT I O N M E T H O D S .
U S E S Y S T E M S O F T W O L I N E A R E Q U AT I O N S T O S O LV E
APPLIED
PROBLEMS.
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Systems of Equations
A system of equations is composed of two or more
equations considered simultaneously.
Example: 5x  y = 5
4x  y = 3
This is a system of two linear equations in two
variables. The solution set of this system consists of all
ordered pairs that make both equations true. The ordered
pair (2, 5) is a solution of this system.
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Solving Systems of Equations Graphically
When we graph a system of linear equations, each point at
which the graphs intersect is a solution of both equations
and therefore a solution of the system of equations.
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Solving Systems of Equations Graphically
Let’s solve the previous system graphically.
5x  y = 5
4x  y = 3
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Types of Solutions
Graphs of linear equations may be related to each
other in one of three ways.
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Practice
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Method 1 – Substitution
The substitution method is a technique that gives
accurate results when solving systems of equations. It is
most often used when a variable is alone on one side of
an equation or when it is easy to solve for a variable.
One equation is used to express one variable in terms of
the other, then it is substituted in the other equation.
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Example
Use substitution to solve the system
5x  y = 5,
4x  y = 3.
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Method 2 – Elimination
Using the elimination method, we eliminate one
variable by adding the two equations. If the coefficients
of a variable are opposites, that variable can be
eliminated by simply adding the original equations. If the
coefficients are not opposites, it is necessary to multiply
one or both equations by suitable constants, before we
add.
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Example
Solve the system using the elimination method.
6x + 2y = 4
10x + 7y =  8
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Another Example
Solve the system.
x  3y = 9
2x  6y = 3
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Another Example
Solve the system.
9x + 6y = 48
3x + 2y = 16
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Practice
Substitution
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Elimination
Application--Example
Ethan and Ian are twins. They have decided to save all of the
money they earn, at their part-time jobs, to buy a car to share at
college. One week, Ethan worked 8 hours and Ian worked 14
hours. Together they saved $256. The next week, Ethan worked
12 hours and Ian worked 16 hours and they earned $324. How
much does each twin make per hour?
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Example
At Max’s Munchies, caramel corn worth $2.50 per pound is
mixed with honey roasted mixed nuts worth $7.50 per pound in
order to get 20 lb. of a mixture worth $4.50 per pound. How
many of each snack is used?
Carmel
corn
Nuts
Mixture
Price per
pound
$2.50
$7.50
$4.50
Number
of pounds
x
y
20
Value of
Mixture
2.50x
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7.50y 4.50(20) = 90
Example
An airplane flies the 3000-mi distance from Los Angeles to New
York, with a tailwind, in 5 hr. The return trip, against the wind,
takes 6 hr. Find the speed of the airplane and the speed of the
wind.
Distance
Rate
Time
With Tailwind
3000
p+w
5
With headwind
3000
p–w
6
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