Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Solving Systems of Linear Equations
Consider the following system of linear equations:
\[ \begin{aligned}
5x_1 - 5x_2 &= 5 \\
x_1 - 5x_3 &= -4
\end{aligned} \]
If the system has infinitely many solutions, select: "The system has at least one solution." Your answer may use expressions involving the parameters \( r \), \( s \), and \( t \).
#### The system has at least one solution:
\[
\begin{aligned}
x_1 &= 0 \\
x_2 &= 0 \\
x_3 &= 0
\end{aligned}
\]
In this problem, the equations form a linear system which can be analyzed for solutions. The steps to determine the solution typically involve simplifying and solving the equations using methods such as substitution, elimination, or matrix operations (such as Gaussian elimination).
In this case, the given solution suggests that for the system to have at least one solution, the variables \( x_1 \), \( x_2 \), and \( x_3 \) are all equal to 0.
### Explanation of Graphs or Diagrams
The image does not contain any graphs or diagrams. Therefore, there are no graphical elements to explain. If there were diagrams, they would usually aid in visualizing the problem, such as plotting equations on a graph to find the intersection points, which represent the solutions to the system.
### Additional Notes
- **Linear Equations**: Equations of the first order where each term is either a constant or the product of a constant and (a single) variable.
- **Infinitely Many Solutions**: For a system of linear equations, it means that there are numerous solutions, and typically the equations are dependent.
- **Consistent System**: A system that has at least one solution.
- **Gaussian Elimination**: A method for solving a system of linear equations which transforms the system into an upper triangular matrix form.
Understanding and solving systems of linear equations is a fundamental skill in algebra and is widely applicable in various fields including engineering, physics, computer science, economics, and more.](https://content.bartleby.com/qna-images/question/d936c6c0-8e3b-4a92-9646-2c460fec57ae/c4407ec3-48e4-4730-9a1b-dc9f13ff7b31/tft0b3l_processed.jpeg)
Transcribed Image Text:### Solving Systems of Linear Equations
Consider the following system of linear equations:
\[ \begin{aligned}
5x_1 - 5x_2 &= 5 \\
x_1 - 5x_3 &= -4
\end{aligned} \]
If the system has infinitely many solutions, select: "The system has at least one solution." Your answer may use expressions involving the parameters \( r \), \( s \), and \( t \).
#### The system has at least one solution:
\[
\begin{aligned}
x_1 &= 0 \\
x_2 &= 0 \\
x_3 &= 0
\end{aligned}
\]
In this problem, the equations form a linear system which can be analyzed for solutions. The steps to determine the solution typically involve simplifying and solving the equations using methods such as substitution, elimination, or matrix operations (such as Gaussian elimination).
In this case, the given solution suggests that for the system to have at least one solution, the variables \( x_1 \), \( x_2 \), and \( x_3 \) are all equal to 0.
### Explanation of Graphs or Diagrams
The image does not contain any graphs or diagrams. Therefore, there are no graphical elements to explain. If there were diagrams, they would usually aid in visualizing the problem, such as plotting equations on a graph to find the intersection points, which represent the solutions to the system.
### Additional Notes
- **Linear Equations**: Equations of the first order where each term is either a constant or the product of a constant and (a single) variable.
- **Infinitely Many Solutions**: For a system of linear equations, it means that there are numerous solutions, and typically the equations are dependent.
- **Consistent System**: A system that has at least one solution.
- **Gaussian Elimination**: A method for solving a system of linear equations which transforms the system into an upper triangular matrix form.
Understanding and solving systems of linear equations is a fundamental skill in algebra and is widely applicable in various fields including engineering, physics, computer science, economics, and more.
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