Transcript 6 -6 Factoring by Grouping

6 minutes
Warm-Up
Write each system as a matrix equation. Then solve the
system, if possible, by using the matrix equation.
x  7y  5
1) 
3x  2y  8
3x  6y  3
2) 
4x  8y  4
4.5.1 Using Matrix Row Operations
Objectives:
•Represent a system of equations as an augmented matrix
•Perform elementary row operations on matrices
Matrix Row Operations
The row-reduction method of solving a system allows
you to determine whether the system is independent,
dependent, or inconsistent.
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
m  a  n  21

2m  a  23
a  3n  25

Augmented Matrix
1 1 1
2 1 0

0 1 3
21 
23
25 
Matrix Row Operations
The goal of the row-reduction method is to transform,
if possible, the coefficient columns into columns that
form an identity matrix.
This is called the reduced row-echelon form of an
augmented matrix if the matrix represents an independent
system.
8
1 0 0
0 1 0

7


0 0 1
6 
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
R1  R2
-Multiply each entry in row 3 by -2
2R3  R3
-Replace row 1 with the sum of row 1
and 4 times each entry in row 2
4R2  R1  R1
Example 1
Perform the indicated row operations on matrix A.
a) 4R1  R3  R2
R2   11
19 20
14 
b)  2R3  3R1  R1
R1   -11 6
-7
5 
1 4 3
A  2 0 0
7 3 8
3
8
2
Practice
Perform the indicated row operations on matrix A.
a) R2  3R3  R3
b)  3R2  R1  R3
1 4 3
A  2 0 0
7 3 8
3
8
2
Homework
p.256 #8-12
6 minutes
Warm-Up
Perform the indicated row operations on matrix A.
a) R1  2R2  R2
b)  3R2  4R1  R3
 2 4 7
A   5 0 6
 7 3 8
4
5 
2
4.5.2 Using Matrix Row Operations
Objectives:
•Solve a system of linear equations by using
elementary row operations
Example 1
Solve the system of equations by using the rowreduction method. Then classify the system.
x  2y  16
1 2


2x  y  11
2 1
3R1 2R2  R1
3 0
0 3

6 
21
16
11 
 2R1 R2  R2
1

0
1
R1  R1
3
2 
1 0
0 3


21


2
16 
-3
-21 

1
 R2  R2
3
2
1 0
0 1

7


x = 2; y = 7
independent
Practice
Solve the system of equations by using the rowreduction method. Then classify the system.
x  4y  3z  13

2y  z  1
6z  30

Example 2
Solve the system of equations by using the rowreduction method. Then classify the system.
4x  12y  8z  2

2x  6y  4z  8
4x  2y  6z  14

 x – 1.4z = 0
 y – 0.2z = 0

0 = 1

no solution, inconsistent
Example 3
Solve the system of equations by using the rowreduction 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
Practice
Solve the system of equations by using the rowreduction method. Then classify the system.
4x  4y  3z  2

4x  3z  3
4y  6z  3

Homework
worksheet