Download 4-6 Row Operations and Augmented Matrices

Row
Operations
and
Row
Operations
and
4-6
4-6 Augmented Matrices
Augmented Matrices
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra 2Algebra 2
Holt
Row Operations and
4-6 Augmented Matrices
Warm Up
Solve.
1.
2.
(4, 3)
(8, 5)
3. What are the three types of linear
systems?
consistent independent, consistent
dependent, inconsistent
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Objective
Use elementary row operations to solve
systems of equations.
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Vocabulary
augmented matrix
row operation
row reduction
reduced row-echelon form
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
In previous lessons, you saw how Cramer’s 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.
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Example 1A: Representing Systems as Matrices
Write the augmented matrix for the system of
equations.
Step 1 Write each equation
in the ax + by = c form.
6x – 5y = 14
2x + 11y = 57
Holt McDougal Algebra 2
Step 2 Write the
augmented matrix, with
coefficients and constants.
Row Operations and
4-6 Augmented Matrices
Example 1B: Representing Systems as Matrices
Write the augmented matrix for the system of
equations.
Step 1 Write each
equation in the
Ax + By + Cz =D
x + 2y + 0z = 12
2x + y + z = 14
0x + y + 3z = 16
Holt McDougal Algebra 2
Step 2 Write the
augmented matrix, with
coefficients and constants.
Row Operations and
4-6 Augmented Matrices
Check It Out! Example 1a
Write the augmented matrix.
Step 1 Write each equation
in the ax + by = c form.
–x – y = 0
–x – y = –2
Holt McDougal Algebra 2
Step 2 Write the
augmented matrix, with
coefficients and constants.
Row Operations and
4-6 Augmented Matrices
Check It Out! Example 1b
Write the augmented matrix.
Step 1 Write each
equation in the
Ax + By + Cz =D
–5x – 4y + 0z = 12
x + 0y + z = 3
0x + 4y + 3z = 10
Holt McDougal Algebra 2
Step 2 Write the
augmented matrix, with
coefficients and constants.
Row Operations and
4-6 Augmented Matrices
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.
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
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
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
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.
31
2 2
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Example 2A Continued
Step 3 Subtract row 1 from row 2. Write the result in
row 2.
2– 1
Although row 2 is now –7y = –21, an equation easily
solved for y, row operations can be used to solve for
both variables
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Example 2A Continued
Step 4 Multiply row 1 by 7 and row 2 by –3.
71
–3 2
Step 5 Subtract row 2 from row 1. Write the result
in row 1.
1 – 2
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Example 2A Continued
Step 6 Divide row 1 by 42 and row 2 by 21.
1  42
2  21
1x = 4
1y = 3
The solution is x = 4, y = 3. Check the result in the
original equations.
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
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.
5 1
8 2
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Example 2B Continued
Step 3 Subtract row 1 from row 2.
2– 1
Step 4 Multiply row 1 by 89 and row 2 by 25.
89 1
25 2
Step 5 Add row 2 to row 1.
1+2
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Example 2B Continued
Step 6 Divide row 1 by 3560 and row 2 by 2225.
1  3560
2  2225
The solution is x = 1, y = –2.
Holt McDougal Algebra 2
1x = 1
1y = –2
Row Operations and
4-6 Augmented Matrices
Check It Out! Example 2a
Write the augmented matrix and solve.
Step 1 Write the augmented matrix.
Step 2 Multiply row 2 by 4.
4 2
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Check It Out! Example 2a Continued
Step 3 Subtract row 1 from row 2. Write the result
in row 2.
2– 1
Step 4 Multiply row 1 by 2.
2 1
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Check It Out! Example 2a Continued
Step 5 Subtract row 2 from row 1. Write the result
in row 1.
1– 2
Step 6 Divide row 1 and row 2 by 8.
1 8
2 8
1x = 4
1y = 4
The solution is x = 4 and y = 4.
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
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.
21
32
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Check It Out! Example 2b Continued
Step 3 Add row 1 to row 2. Write the result in row 2.
2+1
The second row means 0 + 0 = 60, which is always
false. The system is inconsistent.
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
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.
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
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?
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Example 3 Continued
Use the facts to write three equations.
5f + 6c + 2t = 1040 c = flea collars
f – 4c = 0
f = bags of cat food
c + t = 100
t = catnip toys
Enter the 3  4 augmented matrix as A.
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Example 3 Continued
Press
, select MATH, and move down the list
to B:rref( to find the reduced row-echelon form of
the augmented matrix.
There are 140 bags of cat food, 35 flea collars, and
65 catnip toys.
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Check It Out! Example 3a
Solve by using row reduction on a calculator.
The solution is (5, 6, –2).
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
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.
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Check It Out! Example 3b Continued
The solution is (820, 1600). The costs are equal
after 1600 days.
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
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)
Holt McDougal Algebra 2
Row Operations and
4-6 Augmented Matrices
Lesson Quiz: Part II
3. Solve the system using row reduction on a
calculator.
(5, 3, 1)
Holt McDougal Algebra 2