Transcript 4-6 Reduced Row Echelon Form (Rref)

Row Operations and
4-6 Augmented Matrices
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 Algebra 2
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 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 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 Algebra 2
Row Operations and
4-6 Augmented Matrices
Holt 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 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 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 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 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 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 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 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 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 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 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 Algebra 2
Row Operations and
4-6 Augmented Matrices
HW pg. 291
# 14, 15, 16, 18, 22, 23, 24
Holt Algebra 2
Row Operations and
4-6 Augmented Matrices
Homework set #2
HW pg. 291
# 17, 19, 20, 21, 25, 31, 34
Holt Algebra 2