4-6

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


Use elementary row operations to solvesystems of equations.

Objective

Holt Algebra 2


augmented matrixrow operationrow reductionreduced row-echelon form

Vocabulary

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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


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

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Example 1B: Representing Systems as Matrices

Step 1 Write each equation in the Ax + By + Cz =D


Write the augmented matrix for the system of equations.

x + 2y + 0z = 12

2x + y + z = 14

0x + y + 3z = 16

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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

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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

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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.

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Holt Algebra 2


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

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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

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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

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Step 4 Multiply row 1 by 7 and row 2 by –3.


7

–3

12

– 1 2

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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

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Example 2B: Solving Systems with an Augmented Matrix



5

8

1

2


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Example 2B Continued

Step 3 Subtract row 1 from row 2.

– 2 1

89

25

1

2


Step 5 Add row 2 to row 1.

+ 1 2

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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

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Step 2 Multiply row 2 by 4.

4 2

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Check It Out! Example 2a Continued


Step 4 Multiply row 1 by 2.

– 2 1

2 1

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Check It Out! Example 2a Continued


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

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2

3

1

2

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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

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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.

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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?

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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

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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.

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Solve by using row reduction on a calculator.

The solution is (5, 6, –2).

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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.

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Check It Out! Example 3b Continued

The solution is (820, 1600). The costs are equal after 1600 days.

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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


Lesson Quiz: Part II

3. Solve the system using row reduction on a

calculator.

(5, 3, 1)

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