Row Operations and

1
Row Operations and Augmented Matrices
4-6
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
2
Warm Up Solve. 1. 2. 3. What are the
three types of linear systems?
(4, 3)
(8, 5)
consistent independent, consistent dependent,
inconsistent
3
Objective
Use elementary row operations to solve systems of
equations.
4
Vocabulary
augmented matrix row operation row
reduction reduced row-echelon form
5
In previous lessons, you saw how Cramers rule
and inverses can be used to solve systems of
equations. Solving large systems requires a
different method using an augmented matrix. An
augmented matrix consists of the coefficients and
constant terms of a system of linear equations.
A vertical line separates the coefficients from
the constants.
6
Example 1A Representing Systems as Matrices
Write the augmented matrix for the system of
equations.
Step 2 Write the augmented matrix, with
coefficients and constants.
Step 1 Write each equation in the ax by c
form.
6x 5y 14 2x 11y 57
7
Example 1B Representing Systems as Matrices
Write the augmented matrix for the system of
equations.
Step 2 Write the augmented matrix, with
coefficients and constants.
Step 1 Write each equation in the Ax By Cz D
8
Check It Out! Example 1a
Write the augmented matrix.
Step 2 Write the augmented matrix, with
coefficients and constants.
Step 1 Write each equation in the ax by c
form.
x y 0 x y 2
9
Check It Out! Example 1b
Write the augmented matrix.
Step 2 Write the augmented matrix, with
coefficients and constants.
Step 1 Write each equation in the Ax By Cz D
10
You can use the augmented matrix of a system to
solve the system. First you will do a row
operation to change the form of the matrix. These
row operations create a matrix equivalent to the
original matrix. So the new matrix represents a
system equivalent to the original system.
For each matrix, the following row operations
produce a matrix of an equivalent system.
11
(No Transcript)
12
Row reduction is the process of performing
elementary row operations on an augmented matrix
to solve a system. The goal is to get the
coefficients to reduce to the identity matrix on
the left side.
This is called reduced row-echelon form.
1x 5 1y 2
13
Example 2A Solving Systems with an Augmented
Matrix
Write the augmented matrix and solve.
Step 1 Write the augmented matrix.
Step 2 Multiply row 1 by 3 and row 2 by 2.
14
Example 2A Continued
Step 3 Subtract row 1 from row 2. Write the
result in row 2.
Although row 2 is now 7y 21, an equation
easily solved for y, row operations can be used
to solve for both variables
15
Example 2A Continued
Step 4 Multiply row 1 by 7 and row 2 by 3.
Step 5 Subtract row 2 from row 1. Write the
result in row 1.
16
Example 2A Continued
Step 6 Divide row 1 by 42 and row 2 by 21.
The solution is x 4, y 3. Check the result in
the original equations.
17
Example 2B Solving Systems with an Augmented
Matrix
Write the augmented matrix and solve.
Step 1 Write the augmented matrix.
Step 2 Multiply row 1 by 5 and row 2 by 8.
18
Example 2B Continued
Step 3 Subtract row 1 from row 2.
Step 4 Multiply row 1 by 89 and row 2 by 25.
Step 5 Add row 2 to row 1.
19
Example 2B Continued
Step 6 Divide row 1 by 3560 and row 2 by 2225.
The solution is x 1, y 2.
20
Check It Out! Example 2a
Write the augmented matrix and solve.
Step 1 Write the augmented matrix.
Step 2 Multiply row 2 by 4.
21
Check It Out! Example 2a Continued
Step 3 Subtract row 1 from row 2. Write the
result in row 2.
Step 4 Multiply row 1 by 2.
22
Check It Out! Example 2a Continued
Step 5 Subtract row 2 from row 1. Write the
result in row 1.
Step 6 Divide row 1 and row 2 by 8.
The solution is x 4 and y 4.
23
Check It Out! Example 2b
Write the augmented matrix and solve.
Step 1 Write the augmented matrix.
Step 2 Multiply row 1 by 2 and row 2 by 3.
24
Check It Out! Example 2b Continued
Step 3 Add row 1 to row 2. Write the result in
row 2.
The second row means 0 0 60, which is always
false. The system is inconsistent.
25
On many calculators, you can add a column to a
matrix to create the augmented matrix and can use
the row reduction feature. So, the matrices in
the Check It Out problem are entered as 2 ? 3
matrices.
26
Example 3 Charity Application
A shelter receives a shipment of items worth
1040. Bags of cat food are valued at 5 each,
flea collars at 6 each, and catnip toys at 2
each. There are 4 times as many bags of food as
collars. The number of collars and toys together
equals 100. Write the augmented matrix and solve,
using row reduction, on a calculator. How many of
each item are in the shipment?
27
Example 3 Continued
Use the facts to write three equations.
c flea collars
5f 6c 2t 1040
f 4c 0
f bags of cat food
c t 100
t catnip toys
Enter the 3 ? 4 augmented matrix as A.
28
Example 3 Continued
There are 140 bags of cat food, 35 flea collars,
and 65 catnip toys.
29
Check It Out! Example 3a
Solve by using row reduction on a calculator.
The solution is (5, 6, 2).
30
Check It Out! Example 3b
A new freezer costs 500 plus 0.20 a day to
operate. An old freezer costs 20 plus 0.50 a
day to operate. After how many days is the cost
of operating each freezer equal? Solve by using
row reduction on a calculator.
Let t represent the total cost of operating a
freezer for d days.
The solution is (820, 1600). The costs are equal
after 1600 days.
31
Check It Out! Example 3b Continued
The solution is (820, 1600). The costs are equal
after 1600 days.
32
Lesson Quiz Part I
1. Write an augmented matrix for the system of
equations. 2. Write an augmented matrix for
the system of equations and solve using row
operations.
(5.5, 3)
33
Lesson Quiz Part II
3. Solve the system using row reduction on a
calculator.
(5, 3, 1)