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**4-6 Row Operations and Augmented Matrices Warm Up Lesson Presentation**

Lesson Quiz Holt Algebra 2

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Warm Up Solve. 1. 2. 3. What are the three types of linear systems? (4, 3) (8, 5) consistent independent, consistent dependent, inconsistent

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Objective Use elementary row operations to solve systems of equations.

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**Vocabulary augmented matrix row operation row reduction**

reduced row-echelon form

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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.

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**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**

Write the augmented matrix for the system of equations. Step 2 Write the augmented matrix, with coefficients and constants. Step 1 Write each equation in the Ax + By + Cz =D 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. Step 2 Write the augmented matrix, with coefficients and constants. –x – y = 0 –x – y = –2

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Check It Out! Example 1b Write the augmented matrix. Step 2 Write the augmented matrix, with coefficients and constants. 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**

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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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 1

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Example 2A Continued Step 3 Subtract row 1 from row 2. Write the result in row 2. – 1 2 Although row 2 is now –7y = –21, an equation easily solved for y, row operations can be used to solve for both variables

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Example 2A Continued Step 4 Multiply row 1 by 7 and row 2 by –3. 7 –3 1 2 Step 5 Subtract row 2 from row 1. Write the result in row 1. – 1 2

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Example 2A Continued Step 6 Divide row 1 by 42 and row 2 by 21. 42 21 1 2 1x = 4 1y = 3 The solution is x = 4, y = 3. Check the result in the original equations.

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**Example 2B: Solving Systems with an Augmented Matrix**

Write the augmented matrix and solve. Step 1 Write the augmented matrix. Step 2 Multiply row 1 by 5 and row 2 by 8. 5 8 1 2

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Example 2B Continued Step 3 Subtract row 1 from row 2. – 2 1 Step 4 Multiply row 1 by 89 and row 2 by 25. 89 25 1 2 Step 5 Add row 2 to row 1. + 1 2

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Example 2B Continued Step 6 Divide row 1 by 3560 and row 2 by 2225. 1x = 1 1y = –2 3560 2225 1 2 The solution is x = 1, y = –2.

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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

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**Check It Out! Example 2a Continued**

Step 3 Subtract row 1 from row 2. Write the result in row 2. – 2 1 Step 4 Multiply row 1 by 2. 2 1

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**Check It Out! Example 2a Continued**

Step 5 Subtract row 2 from row 1. Write the result in row 1. – 1 2 Step 6 Divide row 1 and row 2 by 8. 1x = 4 1y = 4 8 1 2 The solution is x = 4 and y = 4.

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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

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**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 = 60, which is always false. The system is inconsistent.

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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 $ 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. 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.

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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.

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Check It Out! Example 3a Solve by using row reduction on a calculator. The solution is (5, 6, –2).

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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. Let t represent the total cost of operating a freezer for d days. The solution is (820, 1600). The costs are equal after 1600 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 I 1. 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)

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Lesson Quiz: Part II 3. Solve the system using row reduction on a calculator. (5, 3, 1)

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