4-8 Augmented Matrices &amp; Systems

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4-8 Augmented Matrices &amp; Systems . Objectives. Solving Systems Using Cramer’s Rule Solving Systems Using Augmented Matrices. Vocabulary. Cramer’s Rule. ax + by = m cx + dy = n. System. Replace the y-coefficients with the constants. Use the x- and y-coefficients.

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### 4-8 Augmented Matrices & Systems

Objectives

Solving Systems Using Cramer’s Rule

Solving Systems Using Augmented Matrices

Vocabulary

Cramer’s Rule

ax + by = m

cx + dy = n

System

Replace the y-coefficients with the constants

Use the x- and y-coefficients.

Replace the x-coefficients with the constants

Then, &

7 –4

3 6

15 –4

8 6

7 15

3 8

D = = 54

Dx = = 122

Dy = = 11

61

27

11

54

Dx

D

Dy

D

x = =

y = =

61

27

11

54

The solution of the system is , .

Using Cramer’s Rule

7x – 4y = 15

3x + 6y = 8

Use Cramer’s rule to solve the system .

Evaluate three determinants. Then find x and y.

–2 8 2

–6 0 2

–7 –5 1

D = = –24 Evaluate the determinant.

–2 –3 2

–6 1 2

–7 2 1

Dy = = 20 Replace the y-coefficients with the

constants and evaluate again.

Dy

D

20

24

5

6

y = = – = – Find y.

5

6

The y-coordinate of the solution is – .

Using Cramer’s Rule with

Find the y-coordinate of the solution of the

system .

–2x + 8y + 2z = –3

–6x + 2z = 1

–7x – 5y + z = 2

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

System of Equations

Augmented Matrix

–7x + 4y = –3

x + 8y = 9

System of equations

x-coefficients

y-coefficients

constants

–74 –3

189

Augmented matrix

Draw a vertical bar to separate the coefficients from constants.

Writing an Augmented Matrix

Write an augmented matrix to represent the

system

–7x + 4y = –3

x + 8y = 9

9x – 7y = –1

2x + 5y = –6

9–7–1

25–6

System of equations

Augmented matrix

x-coefficients

y-coefficients

constants

Writing a System From an Augmented Matrix

Write a system of equations for the augmented

matrix .

9 –7 –1

2 5 –6

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.

Number here will be x-value

Number here will be y-value

1 –3 –17

4 2 2

Write an augmented matrix.

1 –3 –17

0 14 70

–4(1 –3 –17)

4 2 2

0 14 70

Multiply Row 1 by –4 and add it to Row 2.

Write the new augmented matrix.

1

14

1

14

Multiply Row 2 by .

Write the new augmented matrix.

1 –3 –17

0 1 5

(0 14 70)

0 1 5

Using an Augmented Matrix

Use an augmented matrix to solve the system

x – 3y = –17

4x + 2y = 2

1 –3 –17

3(0 1 5)

1 0 –2

Multiply Row 2 by 3 and add it to Row 1.

Write the final augmented matrix.

1 0 –2

0 1 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

(continued)

1

14

1 –3 –17

0 1 5

(0 14 70)

0 1 5

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

Step 1: Enter the

augmented matrix

as matrix A.

Step 2: Use the rref feature

calculator.

Using a Graphing Calculator

Use the rref feature on a graphing calculator to solve the system

4x + 3y + z = –1

–2x – 2y + 7z = –10.

3x + y + 5z = 2

The solution is (7, –9, –2).

4(7) + 3(–9) + (–2) –1 Substitute.

28 – 27 – 2 –1 Multiply.

–1 = –1 Simplify.

Continued

(continued)

Homework

4-8 pg 224 & 225 # 1, 4, 6, 7, 9, 12, 13, 18, 19

You must do by hand and show your work for all the steps.