4-8 Augmented Matrices & Systems

Objectives

Solving Systems Using Cramer’s Rule

Solving Systems Using Augmented Matrices

Vocabulary

Cramer’s Rule

ax + by = mcx + dy = n dc

baD

dn

bmDx

nc

maDy

SystemUse the x- and y-coefficients.

Replace the x-coefficients with the constants

Replace the y-coefficients with the constants

D

Dx x

D

Dy yThen, &

Use Cramer’s rule to solve the system .

Evaluate three determinants. Then find x and y.

7x – 4y = 153x + 6y = 8

D = = 547 –43 6

Dx = = 12215 –4 8 6

Dy = = 117 153 8

x = = Dx

D6127

1154

Dy

Dy = =

The solution of the system is , .6127

1154

Using Cramer’s Rule

Find the y-coordinate of the solution of the

system .–2x + 8y + 2z = –3–6x + 2z = 1–7x – 5y + z = 2

D = = –24 Evaluate the determinant.–2 8 2–6 0 2–7 –5 1

Dy = = 20 Replace the y-coefficients with theconstants and evaluate again.

–2 –3 2–6 1 2–7 2 1

y = = – = – Find y.2024

Dy

D56

The y-coordinate of the solution is – .56

Using Cramer’s Rule with

Vocabulary

An augmented matrix contains the coefficients and the constants from a system of equations. Each row represents an equation.

-6x + 2y = 10 4x = -20

2004

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System of Equations

Augmented Matrix

Write an augmented matrix to represent the

system –7x + 4y = –3 x + 8y = 9

System of equations –7x + 4y = –3 x + 8y = 9

x-coefficients y-coefficients constants

Augmented matrix –7 4 –3 1 8 9

Draw a vertical bar to separate the coefficients from constants.

Writing an Augmented Matrix

Write a system of equations for the augmented

matrix .9 –7 –12 5 –6

Augmented matrix 9 –7 –1 2 5 –6

x-coefficients y-coefficients constants

System of equations 9x – 7y = –12x + 5y = –6

Writing a System From an Augmented Matrix

Vocabulary

Row Operations

To solve a system of equations using an augmented matrix, you can use one or more of the following row operations.

•Switch any two rows•Multiply a row by a constant•Add one row to another•Combine one or more of these steps

The goal is to get the matrix to the left of the line into the identity matrix. The values to the right of the line will be your solutions.

y

x

10

01 Number here will be x-value

Number here will be y-value

Use an augmented matrix to solve the system

x – 3y = –174x + 2y = 2

1 –3 –174 2 2

Write an augmented matrix.

Multiply Row 1 by –4 and add it to Row 2.Write the new augmented matrix.

1 –3 –17

0 14 70

–4(1 –3 –17) 4 2 2 0 14 70

1141 –3 –17

0 1 5

Multiply Row 2 by .

Write the new augmented matrix.

(0 14 70) 0 1 5

114

Using an Augmented Matrix

(continued)

1141 –3 –17

0 1 5 (0 14 70) 0 1 5

1 0 –20 1 5

1 –3 –173(0 1 5) 1 0 –2

Multiply Row 2 by 3 and add it to Row 1.Write the final augmented matrix.

The solution to the system is (–2, 5).

Check: x – 3y = –17 4x + 2y = 2 Use the original equations. (–2) – 3(5) –17 4(–2) + 2(5) 2 Substitute. –2 – 15 –17 –8 + 10 2 Multiply. –17 = –17 2 = 2

Continued

Use the rref feature on a graphing calculator to solve the

system 4x + 3y + z = –1–2x – 2y + 7z = –10. 3x + y + 5z = 2

Step 1: Enter theaugmented matrixas matrix A.

Step 2: Use the rref featureof your graphingcalculator.

The solution is (7, –9, –2).

Using a Graphing Calculator

(continued)

Partial Check: 4x + 3y + z = –1 Use the original equation.

4(7) + 3(–9) + (–2) –1 Substitute.

28 – 27 – 2 –1 Multiply.

–1 = –1 Simplify.

Continued

Homework

4-8 pg 220 # 2-20 even