holt algebra 2 4-6 row operations and augmented matrices 4-6 row operations and augmented matrices...
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Holt Algebra 2
4-6 Row Operations andAugmented Matrices4-6 Row Operations and
Augmented Matrices
Holt Algebra 2
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
Warm UpSolve.
1.
2.
3. What are the three types of linear systems?
consistent independent, consistent dependent, inconsistent
(4, 3)
(8, 5)
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
Use elementary row operations to solvesystems of equations.
Objective
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
augmented matrixrow operationrow reductionreduced row-echelon form
Vocabulary
Holt Algebra 2
4-6 Row Operations andAugmented 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 Algebra 2
4-6 Row Operations andAugmented 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.
Step 2 Write the augmented matrix, with coefficients and constants.
6x – 5y = 14
2x + 11y = 57
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
Check It Out! Example 1b
Write the augmented matrix.
Step 1 Write each equation in the Ax + By + Cz =D
Step 2 Write the augmented matrix, with coefficients and constants.
–5x – 4y + 0z = 12
x + 0y + z = 3
0x + 4y + 3z = 10
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
Example 2B: Solving Systems with an Augmented Matrix
Write the augmented matrix and solve.
Step 1 Write the augmented matrix.
5
8
1
2
Step 2 Multiply row 1 by 5 and row 2 by 8.
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
Example 2B Continued
Step 3 Subtract row 1 from row 2.
– 2 1
89
25
1
2
Step 4 Multiply row 1 by 89 and row 2 by 25.
Step 5 Add row 2 to row 1.
+ 1 2
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
Example 2B Continued
The solution is x = 1, y = –2.
Step 6 Divide row 1 by 3560 and row 2 by 2225.
3560
2225
1
2
1x = 1
1y = –2
Holt Algebra 2
4-6 Row Operations andAugmented 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 Algebra 2
4-6 Row Operations andAugmented Matrices
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.
– 2 1
2 1
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
Check It Out! Example 2a Continued
Step 5 Subtract row 2 from row 1. Write the result in row 1.
The solution is x = 4 and y = 4.
– 1 2
Step 6 Divide row 1 and row 2 by 8.
8
8
1
2
1x = 4
1y = 4
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
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 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
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.
The solution is (820, 1600). The costs are equal after 1600 days.
Let t represent the total cost of operating a freezer for d days.
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
Check It Out! Example 3b Continued
The solution is (820, 1600). The costs are equal after 1600 days.
Holt Algebra 2
4-6 Row Operations andAugmented Matrices
Lesson Quiz: Part I1. 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 Algebra 2
4-6 Row Operations andAugmented Matrices
Lesson Quiz: Part II
3. Solve the system using row reduction on a
calculator.
(5, 3, 1)
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