warm-up write each system as a matrix equation. then solve the system, if possible, by using the...
TRANSCRIPT
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
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
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
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.
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
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
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
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
Homework
p.256 #8-12
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
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
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
PracticeSolve the system of equations by using the row-reduction method. Then classify the system.
x 4y 3z 13
2y z 1
6z 30
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
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
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
Homework
worksheet