D. Determine the solution for the following augmented matrix for a 3x3 linear system of equations. y r3 -9 18 (x, y, z) = (_) 7 | 14]
D. Determine the solution for the following augmented matrix for a 3x3 linear system of equations. y r3 -9 18 (x, y, z) = (_) 7 | 14]
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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Question
![6).
Determine the solution for the following augmented matrix for a 3x3 linear system of equations.
\[
\begin{bmatrix}
3 & 0 & 0 & | & 6 \\
0 & -9 & 0 & | & 18 \\
0 & 0 & 7 & | & 14 \\
\end{bmatrix}
\]
\((x, y, z) = (\_\_, \_\_, \_\_)\)
### Explanation of the Augmented Matrix
- This matrix represents a system of linear equations with three variables \(x\), \(y\), and \(z\).
- The first column corresponds to the coefficients of \(x\), the second to \(y\), and the third to \(z\).
- The numbers to the right of the vertical line represent the constants on the right side of each equation.
### System of Equations
From the augmented matrix, the system of equations is:
1. \(3x = 6\)
2. \(-9y = 18\)
3. \(7z = 14\)
### Solving the System
Solve each equation:
1. \(3x = 6\) \\
Divide both sides by 3: \\
\(x = 2\)
2. \(-9y = 18\) \\
Divide both sides by -9: \\
\(y = -2\)
3. \(7z = 14\) \\
Divide both sides by 7: \\
\(z = 2\)
### Solution
The solution to the system of equations is:
\((x, y, z) = (2, -2, 2)\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd920f800-7cb4-4708-a86f-fdc64006663e%2Fdc1534c9-0b78-4b4e-bda5-c7b52428b0cf%2Fozai7ei_processed.png&w=3840&q=75)
Transcribed Image Text:6).
Determine the solution for the following augmented matrix for a 3x3 linear system of equations.
\[
\begin{bmatrix}
3 & 0 & 0 & | & 6 \\
0 & -9 & 0 & | & 18 \\
0 & 0 & 7 & | & 14 \\
\end{bmatrix}
\]
\((x, y, z) = (\_\_, \_\_, \_\_)\)
### Explanation of the Augmented Matrix
- This matrix represents a system of linear equations with three variables \(x\), \(y\), and \(z\).
- The first column corresponds to the coefficients of \(x\), the second to \(y\), and the third to \(z\).
- The numbers to the right of the vertical line represent the constants on the right side of each equation.
### System of Equations
From the augmented matrix, the system of equations is:
1. \(3x = 6\)
2. \(-9y = 18\)
3. \(7z = 14\)
### Solving the System
Solve each equation:
1. \(3x = 6\) \\
Divide both sides by 3: \\
\(x = 2\)
2. \(-9y = 18\) \\
Divide both sides by -9: \\
\(y = -2\)
3. \(7z = 14\) \\
Divide both sides by 7: \\
\(z = 2\)
### Solution
The solution to the system of equations is:
\((x, y, z) = (2, -2, 2)\)
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