Presentation on theme: "An augmented matrix consists of the coefficients and constant terms of a system of linear equations. A vertical line separates the coefficients from the."— Presentation transcript:
1 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.
2 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 theAx + By + Cz =Dx + 2y + 0z = 122x + y + z = 140x + y + 3z = 16
3 Check It Out! Example 1aWrite 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
4 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.
6 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 = 51y = 2
7 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 2.321
8 Example 2A ContinuedStep 3 Subtract row 1 from row 2. Write the result in row 2.–12Although row 2 is now –7y = –21, an equation easily solved for y, row operations can be used to solve for both variables
9 Example 2A ContinuedStep 4 Multiply row 1 by 7 and row 2 by –3.7–312Step 5 Subtract row 2 from row 1. Write the result in row 1.–12
10 Example 2A ContinuedStep 6 Divide row 1 by 42 and row 2 by 21. 42 21121x = 41y = 3The solution is x = 4, y = 3. Check the result in the original equations.
11 Check It Out! Example 2bWrite the augmented matrix and solve.Step 1 Write the augmented matrix.Step 2 Multiply row 1 by 2 and row 2 by 3.231
12 Check It Out! Example 2b Continued Step 3 Add row 1 to row 2. Write the result in row 2.+21The second row means = 60, which is always false. The system is inconsistent.
13 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?
14 Example 3 ContinuedUse the facts to write three equations.5f + 6c + 2t = 1040c = flea collarsf – 4c = 0f = bags of cat foodc + t = 100t = catnip toysEnter the 3 4 augmented matrix as A.
15 Example 3 ContinuedPress , 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.
16 Check It Out! Example 3aSolve by using row reduction on a calculator.The solution is (5, 6, –2).