4-8 Augmented Matrices and Systems

Cramer’s RuleSystem

ndycx

mbyaxUse the x- and y- coefficients.

dc

baD

dn

bmDx

nc

maDy

Replace the x-coefficient with the constants

Replace the y-coefficient with the constants

The solution of the system is :

D

D

D

D yx ,

Use Cramer’s rule to solve the system .

Evaluate three determinants. Then find x and y.

7x – 4y = 153x + 6y = 8

The solution of the system is , .6127

1154

5463

47

D 122

68

415

xD 11

83

157

yD

27

61

D

Dx x

54

11

D

Dy y

Find the y-coordinate of the solution of the

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

24

157

206

282

D

2 3 2

6 1 2 20

7 2 1yD

6

5

D

Dy y

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.

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

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

(continued)

1 –3 –170 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).

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).

Homework

Worksheet