Solve the following system of linear equations: 3x1-3x2-3x3 = -3 -3x₁+6x₂ = 0 -x1-2x₂+4x3 = 5 If the system has no solution, demonstrate this by giving a row-echelon form of the augmented matrix for the system You can resize a matrix (when appropriate) by clicking and dragging the bottom-right corner of the matrix. The system has no solution [000] Row-echelon form of augmented matrix: 0 0 0
Solve the following system of linear equations: 3x1-3x2-3x3 = -3 -3x₁+6x₂ = 0 -x1-2x₂+4x3 = 5 If the system has no solution, demonstrate this by giving a row-echelon form of the augmented matrix for the system You can resize a matrix (when appropriate) by clicking and dragging the bottom-right corner of the matrix. The system has no solution [000] Row-echelon form of augmented matrix: 0 0 0
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
![### Solving a System of Linear Equations
Given the following system of linear equations:
\[
3x_1 - 3x_2 - 3x_3 = -3
\]
\[
-3x_1 + 6x_2 = 0
\]
\[
-x_1 - 2x_2 + 4x_3 = 5
\]
If the system has **no solution**, this can be demonstrated by giving the row-echelon form of the augmented matrix for the system.
### Augmented Matrix Method
An augmented matrix represents the coefficients and constants of a system of linear equations in matrix form. For this system, the augmented matrix is:
\[
\left[ \begin{array}{ccc|c}
3 & -3 & -3 & -3 \\
-3 & 6 & 0 & 0 \\
-1 & -2 & 4 & 5
\end{array} \right]
\]
Upon performing row operations to convert this matrix into row-echelon form, if it results in a row that suggests inconsistency (e.g., a row of the form \([0\ 0\ 0\ |\ c]\) where \(c \neq 0\)), it implies that the system has no solution.
### Example of Row-Echelon Form:
For this particular system, the row-echelon form is:
\[
\left[ \begin{array}{ccc|c}
1 & -1 & -1 & -1 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array} \right]
\]
This can be simplified further if necessary, but this form already shows that the system potentially does not contribute to a unique solution. It ultimately leads to:
\[
\left[ \begin{array}{ccc|c}
0 & 0 & 0 & 0
\end{array} \right]
\]
### Conclusion
From the row-echelon form, it is evident that the system has **no solution** since the third row essentially gives us a contradiction (i.e., \(0 = 1\) in another equivalent form).
Thus, **the system has no solution.**
This kind of result often happens when the given equations describe parallel planes that do not intersect each other, thereby having no common points of intersection.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd936c6c0-8e3b-4a92-9646-2c460fec57ae%2Ff0f14904-7fef-42c1-bd7c-ed901bfba583%2F8p5n83h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving a System of Linear Equations
Given the following system of linear equations:
\[
3x_1 - 3x_2 - 3x_3 = -3
\]
\[
-3x_1 + 6x_2 = 0
\]
\[
-x_1 - 2x_2 + 4x_3 = 5
\]
If the system has **no solution**, this can be demonstrated by giving the row-echelon form of the augmented matrix for the system.
### Augmented Matrix Method
An augmented matrix represents the coefficients and constants of a system of linear equations in matrix form. For this system, the augmented matrix is:
\[
\left[ \begin{array}{ccc|c}
3 & -3 & -3 & -3 \\
-3 & 6 & 0 & 0 \\
-1 & -2 & 4 & 5
\end{array} \right]
\]
Upon performing row operations to convert this matrix into row-echelon form, if it results in a row that suggests inconsistency (e.g., a row of the form \([0\ 0\ 0\ |\ c]\) where \(c \neq 0\)), it implies that the system has no solution.
### Example of Row-Echelon Form:
For this particular system, the row-echelon form is:
\[
\left[ \begin{array}{ccc|c}
1 & -1 & -1 & -1 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array} \right]
\]
This can be simplified further if necessary, but this form already shows that the system potentially does not contribute to a unique solution. It ultimately leads to:
\[
\left[ \begin{array}{ccc|c}
0 & 0 & 0 & 0
\end{array} \right]
\]
### Conclusion
From the row-echelon form, it is evident that the system has **no solution** since the third row essentially gives us a contradiction (i.e., \(0 = 1\) in another equivalent form).
Thus, **the system has no solution.**
This kind of result often happens when the given equations describe parallel planes that do not intersect each other, thereby having no common points of intersection.
Transcribed Image Text:### Solving a System of Linear Equations
**Given System:**
Solve the following system of linear equations:
1. \(3x_1 - 3x_2 - 3x_3 = -3\)
2. \(-3x_1 + 6x_2 = 0\)
3. \(-x_1 - 2x_2 + 4x_3 = 5\)
**Instructions:**
If the system has no solution, demonstrate this by giving a row-echelon form of the augmented matrix for the system.
You can resize a matrix (when appropriate) by clicking and dragging the bottom-right corner of the matrix.
**Answer Choices:**
- The system has a unique solution.
- The system has no solution.
- The system has a unique solution.
- The system has infinitely many solutions.
**Matrix Representation:**
Below, in the region that looks like an answer box found on an educational quiz platform, the systems' possible solutions are listed, and an empty matrix labeled \(x_3\) is shown with a filled value of 0 in the box.
The solutions include:
- The system has a unique solution.
- The system has no solution.
- The system has a unique solution.
- The system has infinitely many solutions.
**Analysis Approach:**
To determine the nature of the solutions (whether unique, none, or infinitely many), you may perform the following steps:
1. Write the system of equations in matrix form, using the coefficients of the variables and the constants on the right side.
2. Transform the system into its augmented matrix.
3. Apply Gaussian elimination to the augmented matrix to achieve row-echelon form.
4. Analyze the resulting matrix to conclude the nature of the system's solutions.
By following these steps, you will be able to clarify if the system has unique solutions, no solutions, or infinitely many solutions.
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