Write an augmented matrix for the following system of equations. 5x- 4y + 3z = - 3 8x- 5y+3z 1 4y-5z = - 4 The entries in the matrix are
Write an augmented matrix for the following system of equations. 5x- 4y + 3z = - 3 8x- 5y+3z 1 4y-5z = - 4 The entries in the matrix are
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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![**Writing an Augmented Matrix for a System of Equations**
To represent the given system of linear equations as an augmented matrix, follow the steps outlined below. Let's start with the system of equations:
\[
\begin{aligned}
5x - 4y + 3z &= -3 \\
8x - 5y + 3z &= 1 \\
4y - 5z &= -4
\end{aligned}
\]
### Steps to Form the Augmented Matrix:
1. **Identify the coefficients and constants**:
- The coefficients of \( x \), \( y \), and \( z \) from each equation
- Constants on the right side of each equation
2. **Build the Matrix**:
- Write the coefficients in a matrix format
- Separate the constants using a vertical line to form the augmented part
### Coefficient Extraction and Matrix Formation:
Let's extract the coefficients from the equations above:
- Equation 1: \(5x - 4y + 3z = -3\) ⟶ Coefficients: [5, -4, 3, -3]
- Equation 2: \(8x - 5y + 3z = 1\) ⟶ Coefficients: [8, -5, 3, 1]
- Equation 3: \(0x + 4y - 5z = -4\) ⟶ Coefficients: [0, 4, -5, -4] (Note: \( x\) term is missing, assume 0 coefficient)
### Augmented Matrix:
Now, write the above coefficients into a matrix form:
\[
\left[
\begin{array}{ccc|c}
5 & -4 & 3 & -3 \\
8 & -5 & 3 & 1 \\
0 & 4 & -5 & -4
\end{array}
\right]
\]
### Summary:
The entries in the augmented matrix for the given system of equations are:
\[
\left[
\begin{array}{ccc|c}
5 & -4 & 3 & -3 \\
8 & -5 & 3 & 1 \\
0 & 4 & -5 & -4
\end{array}
\right]
\]
This augmented matrix format will allow you to apply matrix operations effectively to solve the system](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57708f6a-cbd4-4295-9f28-330268fec57a%2F2b1f6c84-2d18-45e7-b4aa-da5064070722%2F1mvcnsf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Writing an Augmented Matrix for a System of Equations**
To represent the given system of linear equations as an augmented matrix, follow the steps outlined below. Let's start with the system of equations:
\[
\begin{aligned}
5x - 4y + 3z &= -3 \\
8x - 5y + 3z &= 1 \\
4y - 5z &= -4
\end{aligned}
\]
### Steps to Form the Augmented Matrix:
1. **Identify the coefficients and constants**:
- The coefficients of \( x \), \( y \), and \( z \) from each equation
- Constants on the right side of each equation
2. **Build the Matrix**:
- Write the coefficients in a matrix format
- Separate the constants using a vertical line to form the augmented part
### Coefficient Extraction and Matrix Formation:
Let's extract the coefficients from the equations above:
- Equation 1: \(5x - 4y + 3z = -3\) ⟶ Coefficients: [5, -4, 3, -3]
- Equation 2: \(8x - 5y + 3z = 1\) ⟶ Coefficients: [8, -5, 3, 1]
- Equation 3: \(0x + 4y - 5z = -4\) ⟶ Coefficients: [0, 4, -5, -4] (Note: \( x\) term is missing, assume 0 coefficient)
### Augmented Matrix:
Now, write the above coefficients into a matrix form:
\[
\left[
\begin{array}{ccc|c}
5 & -4 & 3 & -3 \\
8 & -5 & 3 & 1 \\
0 & 4 & -5 & -4
\end{array}
\right]
\]
### Summary:
The entries in the augmented matrix for the given system of equations are:
\[
\left[
\begin{array}{ccc|c}
5 & -4 & 3 & -3 \\
8 & -5 & 3 & 1 \\
0 & 4 & -5 & -4
\end{array}
\right]
\]
This augmented matrix format will allow you to apply matrix operations effectively to solve the system
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