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Warm-UpWrite each system as a matrix equation. Then solve the system, if possible, by using the matrix equation.
6 minutes
x 7y 51)
3x 2y 8
3x 6y 32)
4x 8y 4
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4.5.1 Using Matrix Row Operations4.5.1 Using Matrix Row Operations4.5.1 Using Matrix Row Operations4.5.1 Using Matrix Row OperationsObjectives: •Represent a system of equations as an augmented matrix•Perform elementary row operations on matrices
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Matrix Row OperationsThe row-reduction method of solving a system allows you to determine whether the system is independent, dependent, or inconsistent.
m a n 21
2m a 23
a 3n 25
The row-reduction method of solving a system is performed on an augmented matrix.
An augmented matrix consists of the coefficients and constant terms in the system of equations.
System
1 1 1 21
2 1 0 23
0 1 3 25
Augmented Matrix
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Matrix Row OperationsThe goal of the row-reduction method is to transform, if possible, the coefficient columns into columns that form an identity matrix.
1 0 0 8
0 1 0 7
0 0 1 6
This is called the reduced row-echelon form of an augmented matrix if the matrix represents an independent system.
The resulting constants will represent the unique solution to the system.
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Elementary Row Operations
The following operations produce equivalent matrices, and may be used in any order and as many times as necessary to obtain reduced row-echelon form.
-Interchange two rows
-Multiply all entries in one row by a nonzero number -Add a multiple of one row to another row
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Row Operations and their Notations
-Interchange rows 1 and 2
-Multiply each entry in row 3 by -2
-Replace row 1 with the sum of row 1 and 4 times each entry in row 2
1 2R R
3 32R R
2 1 14R R R
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Example 1Perform the indicated row operations on matrix A.
1 4 3 3
A 2 0 0 8
7 3 8 2
1 3 2a) 4R R R
3 1 1b) 2R 3R R
2R 11
19 20 14
1R -11
6 -7 5
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PracticePerform the indicated row operations on matrix A.
1 4 3 3
A 2 0 0 8
7 3 8 2
2 3 3a) R 3R R
2 1 3b) 3R R R
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Homework
p.256 #8-12
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Warm-Up6 minutesPerform the indicated row operations on matrix A.
2 4 7 4
A 5 0 6 5
7 3 8 2
1 2 2a) R 2R R
2 1 3b) 3R 4R R
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4.5.2 Using Matrix Row Operations4.5.2 Using Matrix Row Operations4.5.2 Using Matrix Row Operations4.5.2 Using Matrix Row OperationsObjectives: •Solve a system of linear equations by using elementary row operations
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Example 1Solve the system of equations by using the row-reduction method. Then classify the system.
x 2y 16
2x y 11
12R
13R
3 0 6
0 3 21
1 1
1R R
3
1 0 2
0 3 21
2 2
1R R
3
1 0 2
0 1 7
2R 2R1 2 1
60 -3 -21
1 2 16
2 1 11
22R 1R
x = 2; y = 7
independent
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PracticeSolve the system of equations by using the row-reduction method. Then classify the system.
x 4y 3z 13
2y z 1
6z 30
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Example 2Solve the system of equations by using the row-reduction method. Then classify the system.
4x 12y 8z 2
2x 6y 4z 8
4x 2y 6z 14
x – 1.4z = 0y – 0.2z = 00 = 1
no solution, inconsistent
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Example 3Solve the system of equations by using the row-reduction method. Then classify the system.
x y z 2
3x 2y z 3
6x 4y 2z 6
x – z = -1
y + 2z = 3
0 = 0
infinitely many solutions
dependent
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PracticeSolve the system of equations by using the row-reduction method. Then classify the system.
4x 4y 3z 2
4x 3z 3
4y 6z 3
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Homework
worksheet