Systems of Equations - §4.1
Fall 2013 - Math 1010
Ready your clicker!
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Roadmap
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Class activity: Exam 1
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Section 4.1: Systems of Equations and Clicker Questions
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Homework for §3.7 due at the end of class.
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M 1010 §4.1
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Assignment
Assignment:
For Wednesday:
1. Exercises from §4.1 due Wednesday, October 9.
2. Wednesday - Quiz # 4: §3.6 and 3.7 and Cumulative portion
3. Read §4.2. Read over break §5.1 - 5.3, and review material from
previous weeks.
4. Next Exam: October 30
Tip: Use the textbook’s cumulative reviews (end of each section), review
exercises (end of each chapter), mid-chapter quizzes, and chapter tests
(end of each chapter) for study materials. Practice these for quizzes,
exams, and finals.
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M 1010 §4.1
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Systems of Equations
Systems of equations involve two or more equations with two or more
variables. A solution of a system is an ordered pair (x, y ) satisfying each
equation.
(Math 1010)
M 1010 §4.1
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Systems of Equations
Systems of equations involve two or more equations with two or more
variables. A solution of a system is an ordered pair (x, y ) satisfying each
equation.
Example The system of equations
(
x +y
2x − 5y
= 10
= −8
has (7, 4) as a valid solution.
(Math 1010)
M 1010 §4.1
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Systems of Equations
Systems of equations involve two or more equations with two or more
variables. A solution of a system is an ordered pair (x, y ) satisfying each
equation.
Example The system of equations
(
x +y
2x − 5y
= 10
= −8
has (7, 4) as a valid solution.
(clicker)
(Math 1010)
M 1010 §4.1
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Graphing Systems of Equations
Graphs of equations aid identifying the location and number of valid
solutions to systems of equations. Solutions correspond to points of
intersection.
12 y
10
x + y = 10
8
2x − 5y = −8
6
(6, 4)
•
4
2
x
2
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6
M 1010 §4.1
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10
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Linear Systems of Equations
8 y
6
4
•
2
x
−4
−2
2
4
6
8
−2
−4
The two lines intersect at exactly one point. The slopes are not equal.
This is a consistent system.
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Linear Systems of Equations
8 y
6
4
2
x
−4
−2
2
4
6
8
−2
−4
The two lines are identical. They intersect at infinitely many points. The
slopes are equal. This is a dependent (consistent) system.
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M 1010 §4.1
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Linear Systems of Equations
8 y
6
4
2
x
−4
−2
2
4
6
8
−2
−4
The two lines are parallel. There is no point of intersection. Their slopes
are equal. This is an inconsistent system.
(Math 1010)
M 1010 §4.1
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Linear Systems of Equations
8 y
6
4
2
x
−4
−2
2
4
6
8
−2
−4
The two lines are parallel. There is no point of intersection. Their slopes
are equal. This is an inconsistent system. (clicker)
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Solving Systems of Equations
Accurate solutions are found algebraically. One technique to finding
solutions is substitution. (By the way, this is referred to as an ”analytic”
method.)
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M 1010 §4.1
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Solving Systems of Equations
Accurate solutions are found algebraically. One technique to finding
solutions is substitution. (By the way, this is referred to as an ”analytic”
method.)
Example To solve
(
−x + y
3x + y
= −4
= −2
begin by solving the top equation for y , then substitute that expression for
y in the bottom equation.
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Example Continued
(
y
=x −4
3x + (x − 4) = −2
Solve 3x + (x − 4) = −2 for x.
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Example Continued
(
y
=x −4
3x + (x − 4) = −2
Solve 3x + (x − 4) = −2 for x.
x = − 12 You now know the x-coordinate. You need also the y -coordinate.
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Example Continued
(
y
=x −4
3x + (x − 4) = −2
Solve 3x + (x − 4) = −2 for x.
x = − 12 You now know the x-coordinate. You need also the y -coordinate.
Back-substitute the x-value into y


y
y


y
(Math 1010)
=x −4
=x −4
= − 21 − 4
= − 29
M 1010 §4.1
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Example Continued
(
y
=x −4
3x + (x − 4) = −2
Solve 3x + (x − 4) = −2 for x.
x = − 12 You now know the x-coordinate. You need also the y -coordinate.
Back-substitute the x-value into y


y
y


y
=x −4
=x −4
= − 21 − 4
= − 29
The solution is − 12 , − 29 . Check this in the original system of equations.
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Method of Substitution (page 222)
1. Solve one of the equations for one variable in terms of the other
variable. (Ex. y in terms of x)
2. Substitute the algebraic expression obtained in (1) into the other
equation to obtain an equation in one variable.
3. Solve the equation in (2).
4. Back-substitute the solution obtained in (3) in the expression
obtained in (1) to find the value other variable.
5. The found pair (x, y ) must be checked for validity in both of the
original equations.
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M 1010 §4.1
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Method of Substitution (page 222)
1. Solve one of the equations for one variable in terms of the other
variable. (Ex. y in terms of x)
2. Substitute the algebraic expression obtained in (1) into the other
equation to obtain an equation in one variable.
3. Solve the equation in (2).
4. Back-substitute the solution obtained in (3) in the expression
obtained in (1) to find the value other variable.
5. The found pair (x, y ) must be checked for validity in both of the
original equations.
(clicker)
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M 1010 §4.1
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