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Holt Algebra 2
4-6 Row Operations andAugmented 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
4-6 Row Operations andAugmented Matrices
Example 1B: Representing Systems as Matrices
Step 1 Write each equation in the Ax + By + Cz =D
Step 2 Write the augmented matrix, with coefficients and constants.
Write the augmented matrix for the system of equations.
x + 2y + 0z = 12
2x + y + z = 14
0x + y + 3z = 16
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
Check It Out! Example 1a
Write the augmented matrix.
Step 1 Write each equation in the ax + by = c form.
Step 2 Write the augmented matrix, with coefficients and constants.
–x – y = 0
–x – y = –2
Holt Algebra 2
4-6 Row Operations andAugmented 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
4-6 Row Operations andAugmented Matrices
Holt Algebra 2
4-6 Row Operations andAugmented 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
4-6 Row Operations andAugmented 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.
3
2
12
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
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
– 12
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
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.
7
–3
12
– 1 2
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
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.
42
21
1
2
1x = 4
1y = 3
Holt Algebra 2
4-6 Row Operations andAugmented 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.
2
3
1
2
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
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.
+ 2 1
Holt Algebra 2
4-6 Row Operations andAugmented 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
4-6 Row Operations andAugmented Matrices
Example 3 Continued
Use the facts to write three equations.
Enter the 3 4 augmented matrix as A.
5f + 6c + 2t = 1040
f – 4c = 0
c + t = 100
f = bags of cat food
c = flea collars
t = catnip toys
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
Example 3 Continued
There are 140 bags of cat food, 35 flea collars, and 65 catnip toys.
Press , select MATH, and move down the list to B:rref( to find the reduced row-echelon form of the augmented matrix.
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
Check It Out! Example 3a
Solve by using row reduction on a calculator.
The solution is (5, 6, –2).
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
HW pg. 291
# 14, 15, 16, 18, 22, 23, 24
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
Homework set #2
HW pg. 291
# 17, 19, 20, 21, 25, 31, 34
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