3 3 b. 0 -9 000 -1 -2. In each part of Exercises 7-8, find the augmented matrix for the lin- ear system. 7. a. -2x₁ = 6 b. 6x₁ - x₂ + 3x3 = 4 5x₂ - x₂ = 1 3x1 = 8 9x₁ = -3 C. - 3x4 + x₂ = 0 2x2 -3x₁ - x₂ + x3 6x₁ + 2x2x3 + 2x4 - 3x5 = 6 b. 2x₁ + 2x3 = 1 3x²₁x₂ + 4x3 = 7 1. ошос O O AL 8. a. 3x₁2x₂ = -1 4x. + 3 -
3 3 b. 0 -9 000 -1 -2. In each part of Exercises 7-8, find the augmented matrix for the lin- ear system. 7. a. -2x₁ = 6 b. 6x₁ - x₂ + 3x3 = 4 5x₂ - x₂ = 1 3x1 = 8 9x₁ = -3 C. - 3x4 + x₂ = 0 2x2 -3x₁ - x₂ + x3 6x₁ + 2x2x3 + 2x4 - 3x5 = 6 b. 2x₁ + 2x3 = 1 3x²₁x₂ + 4x3 = 7 1. ошос O O AL 8. a. 3x₁2x₂ = -1 4x. + 3 -
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Questions7 and 15 please on paper.
![### Augmented Matrices for Linear Systems
In each part of Exercises 7-8, find the augmented matrix for the linear system.
#### Exercise 7
##### Part (a)
Given the system of linear equations:
1. \( -2x_1 = 6 \)
2. \( 3x_1 = 8 \)
3. \( 9x_1 = -3 \)
The augmented matrix for this system is:
\[
\begin{bmatrix}
-2 & | & 6 \\
3 & | & 8 \\
9 & | & -3
\end{bmatrix}
\]
##### Part (b)
The stated equations and corresponding augmented matrix are missing in the image provided. Ensure to cross-check and provide the equations to form the augmented matrix manually.
#### Exercise 8
##### Part (a)
Given the system of linear equations:
1. \( 3x_1 + 2x_2 = 3 \)
2. \( 6x_1 + 2x_2 - x_3 = -1 \)
3. \( 4x_1 + 5x_2 = 7 \)
The augmented matrix for this system is:
\[
\begin{bmatrix}
3 & 2 & 0 & | & 3 \\
6 & 2 & -1 & | & -1 \\
4 & 5 & 0 & | & 7
\end{bmatrix}
\]
##### Part (b)
Given the system of linear equations:
1. \( 6x_1 - x_2 + 3x_3 = 4 \)
2. \( 5x_2 - x_3 = 1 \)
The augmented matrix for this system is:
\[
\begin{bmatrix}
6 & -1 & 3 & | & 4 \\
0 & 5 & -1 & | & 1
\end{bmatrix}
\]
##### Part (c)
Given the system of linear equations:
1. \( -3x_1 + 3x_5 = 0 \)
2. \( 3x_3 + 4x_4 = -1 \)
3. \( -2x_1 + 4x_3 - 3x_5 = 6 \)
The augmented matrix for this system is:
\[
\begin{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48922dd6-b6f8-4fac-84c6-05a7db5750f6%2Ff9829c80-9a3f-4455-aac2-9d82c473b214%2F1f1a5rp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Augmented Matrices for Linear Systems
In each part of Exercises 7-8, find the augmented matrix for the linear system.
#### Exercise 7
##### Part (a)
Given the system of linear equations:
1. \( -2x_1 = 6 \)
2. \( 3x_1 = 8 \)
3. \( 9x_1 = -3 \)
The augmented matrix for this system is:
\[
\begin{bmatrix}
-2 & | & 6 \\
3 & | & 8 \\
9 & | & -3
\end{bmatrix}
\]
##### Part (b)
The stated equations and corresponding augmented matrix are missing in the image provided. Ensure to cross-check and provide the equations to form the augmented matrix manually.
#### Exercise 8
##### Part (a)
Given the system of linear equations:
1. \( 3x_1 + 2x_2 = 3 \)
2. \( 6x_1 + 2x_2 - x_3 = -1 \)
3. \( 4x_1 + 5x_2 = 7 \)
The augmented matrix for this system is:
\[
\begin{bmatrix}
3 & 2 & 0 & | & 3 \\
6 & 2 & -1 & | & -1 \\
4 & 5 & 0 & | & 7
\end{bmatrix}
\]
##### Part (b)
Given the system of linear equations:
1. \( 6x_1 - x_2 + 3x_3 = 4 \)
2. \( 5x_2 - x_3 = 1 \)
The augmented matrix for this system is:
\[
\begin{bmatrix}
6 & -1 & 3 & | & 4 \\
0 & 5 & -1 & | & 1
\end{bmatrix}
\]
##### Part (c)
Given the system of linear equations:
1. \( -3x_1 + 3x_5 = 0 \)
2. \( 3x_3 + 4x_4 = -1 \)
3. \( -2x_1 + 4x_3 - 3x_5 = 6 \)
The augmented matrix for this system is:
\[
\begin{
![**Linear Systems and Parametric Equations**
_In Exercises 15-16, each linear system has infinitely many solutions. Use parametric equations to describe its solution set._
---
### **Exercise 15:**
**a.**
\[
2x - 3y = 1 \\
6x - 9y = 3
\]
**b.**
\[
\begin{aligned}
& x_{1} + 3x_{2} - x_{3} = 4 \\
& x_{1} + 9x_{2} - 3x_{3} = 12 \\
& -x_{1} - 3x_{2} + x_{3} = -4
\end{aligned}
\]
---
### **Exercise 16:**
**a.**
\[
6x_{1} + 2x_{2} = -8
\]
**b.**
\[
\begin{aligned}
& 2x - y + 2z = -4 \\
& 3x + 6z = -12
\end{aligned}
\]
**c.**
\[
4x_{1} + 2x_{2} + 3x_{3} + x_{4} = 3
\]
**d.**
\[
v + w + x - 5y + 7z = 0
\]
---
In each of these exercises, we are given systems of linear equations which have infinitely many solutions. This implies that there are free variables, and thus, the solutions can be expressed using parametric equations.
- **For Exercise 15a:**
Two equations are provided. To describe the solution set parametrically, solve the equations in terms of a free variable.
- **For Exercise 15b:**
Three equations and three variables are given. Convert to row echelon form (or any simplified form) and express in terms of free variables.
- **For Exercise 16a:**
A straightforward equation in terms of two variables. Express one variable in terms of the other.
- **For Exercise 16b:**
Here, the presence of a third variable suggests introducing a parameter for the z variable and solving for x and y in term of z.
- **For Exercise 16c and 16d:**
Linear systems with four and five variables respectively. As there are fewer equations than variables, parametric](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48922dd6-b6f8-4fac-84c6-05a7db5750f6%2Ff9829c80-9a3f-4455-aac2-9d82c473b214%2Foc0tr45_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Linear Systems and Parametric Equations**
_In Exercises 15-16, each linear system has infinitely many solutions. Use parametric equations to describe its solution set._
---
### **Exercise 15:**
**a.**
\[
2x - 3y = 1 \\
6x - 9y = 3
\]
**b.**
\[
\begin{aligned}
& x_{1} + 3x_{2} - x_{3} = 4 \\
& x_{1} + 9x_{2} - 3x_{3} = 12 \\
& -x_{1} - 3x_{2} + x_{3} = -4
\end{aligned}
\]
---
### **Exercise 16:**
**a.**
\[
6x_{1} + 2x_{2} = -8
\]
**b.**
\[
\begin{aligned}
& 2x - y + 2z = -4 \\
& 3x + 6z = -12
\end{aligned}
\]
**c.**
\[
4x_{1} + 2x_{2} + 3x_{3} + x_{4} = 3
\]
**d.**
\[
v + w + x - 5y + 7z = 0
\]
---
In each of these exercises, we are given systems of linear equations which have infinitely many solutions. This implies that there are free variables, and thus, the solutions can be expressed using parametric equations.
- **For Exercise 15a:**
Two equations are provided. To describe the solution set parametrically, solve the equations in terms of a free variable.
- **For Exercise 15b:**
Three equations and three variables are given. Convert to row echelon form (or any simplified form) and express in terms of free variables.
- **For Exercise 16a:**
A straightforward equation in terms of two variables. Express one variable in terms of the other.
- **For Exercise 16b:**
Here, the presence of a third variable suggests introducing a parameter for the z variable and solving for x and y in term of z.
- **For Exercise 16c and 16d:**
Linear systems with four and five variables respectively. As there are fewer equations than variables, parametric
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