holt algebra 2 4-6 row operations and augmented matrices an augmented matrix consists of the...

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


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


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


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


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

Holt Algebra 2


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


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



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



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


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


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


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


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


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


Check It Out! Example 3a

Solve by using row reduction on a calculator.

The solution is (5, 6, –2).

Holt Algebra 2


HW pg. 291

# 14, 15, 16, 18, 22, 23, 24

Holt Algebra 2


Homework set #2

HW pg. 291

# 17, 19, 20, 21, 25, 31, 34

holt algebra 2 4-6 row operations and augmented matrices an augmented matrix consists of the...

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