Row Operations and 4 6 Augmented Matrices Warm

Row Operations and 4 -6 Augmented Matrices Warm Up Lesson Presentation Lesson Quiz Holt. Mc. Dougal Algebra 2 Holt

Row Operations and 4 -6 Augmented Matrices Warm Up Solve. 1. 2. (4, 3) (8, 5) 3. What are three types of linear systems? consistent independent, consistent dependent, inconsistent Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Objective Use elementary row operations to solve systems of equations. Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Vocabulary augmented matrix row operation row reduction reduced row-echelon form Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented 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 Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Example 1 A: Representing Systems as Matrices Write the augmented matrix for the system of equations. Step 1 Write each equation in the ax + by = c form. 6 x – 5 y = 14 2 x + 11 y = 57 Holt Mc. Dougal Algebra 2 Step 2 Write the augmented matrix, with coefficients and constants.

Row Operations and 4 -6 Augmented Matrices Example 1 B: Representing Systems as Matrices Write the augmented matrix for the system of equations. Step 1 Write each equation in the Ax + By + Cz =D x + 2 y + 0 z = 12 2 x + y + z = 14 0 x + y + 3 z = 16 Holt Mc. Dougal Algebra 2 Step 2 Write the augmented matrix, with coefficients and constants.

Row Operations and 4 -6 Augmented Matrices Check It Out! Example 1 a Write the augmented matrix. Step 1 Write each equation in the ax + by = c form. –x – y = 0 –x – y = – 2 Holt Mc. Dougal Algebra 2 Step 2 Write the augmented matrix, with coefficients and constants.

Row Operations and 4 -6 Augmented Matrices Check It Out! Example 1 b Write the augmented matrix. Step 1 Write each equation in the Ax + By + Cz =D – 5 x – 4 y + 0 z = 12 x + 0 y + z = 3 0 x + 4 y + 3 z = 10 Holt Mc. Dougal Algebra 2 Step 2 Write the augmented matrix, with coefficients and constants.

Row Operations and 4 -6 Augmented 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 Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented 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. 1 x = 5 1 y = 2 Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Example 2 A: 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. 31 2 2 Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Example 2 A Continued Step 3 Subtract row 1 from row 2. Write the result in row 2. 2– 1 Although row 2 is now – 7 y = – 21, an equation easily solved for y, row operations can be used to solve for both variables Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Example 2 A Continued Step 4 Multiply row 1 by 7 and row 2 by – 3. 71 – 3 2 Step 5 Subtract row 2 from row 1. Write the result in row 1. 1 – 2 Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Example 2 A Continued Step 6 Divide row 1 by 42 and row 2 by 21. 1 42 2 21 1 x = 4 1 y = 3 The solution is x = 4, y = 3. Check the result in the original equations. Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Example 2 B: 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 1 8 2 Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Example 2 B Continued Step 3 Subtract row 1 from row 2. 2– 1 Step 4 Multiply row 1 by 89 and row 2 by 25. 89 1 25 2 Step 5 Add row 2 to row 1. 1+2 Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Example 2 B Continued Step 6 Divide row 1 by 3560 and row 2 by 2225. 1 3560 2 2225 The solution is x = 1, y = – 2. Holt Mc. Dougal Algebra 2 1 x = 1 1 y = – 2

Row Operations and 4 -6 Augmented Matrices Check It Out! Example 2 a Write the augmented matrix and solve. Step 1 Write the augmented matrix. Step 2 Multiply row 2 by 4. 4 2 Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Check It Out! Example 2 a 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 Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Check It Out! Example 2 a 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. 1 8 2 8 1 x = 4 1 y = 4 The solution is x = 4 and y = 4. Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Check It Out! Example 2 b Write the augmented matrix and solve. Step 1 Write the augmented matrix. Step 2 Multiply row 1 by 2 and row 2 by 3. 21 32 Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Check It Out! Example 2 b Continued Step 3 Add row 1 to row 2. Write the result in row 2. 2+1 The second row means 0 + 0 = 60, which is always false. The system is inconsistent. Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented 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 Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented 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 Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Example 3 Continued Use the facts to write three equations. 5 f + 6 c + 2 t = 1040 c = flea collars f – 4 c = 0 f = bags of cat food c + t = 100 t = catnip toys Enter the 3 4 augmented matrix as A. Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices 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. Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Check It Out! Example 3 a Solve by using row reduction on a calculator. The solution is (5, 6, – 2). Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Check It Out! Example 3 b 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. Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Check It Out! Example 3 b Continued The solution is (820, 1600). The costs are equal after 1600 days. Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices 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) Holt Mc. Dougal Algebra 2

Row Operations and 4 -6 Augmented Matrices Lesson Quiz: Part II 3. Solve the system using row reduction on a calculator. (5, 3, 1) Holt Mc. Dougal Algebra 2