Suppose the system + 5x3 7x4 %3D X2 - - x3 X4 2x2 + + 16x4 has infinitely many solutions. Can you say anything about the number of solutions to the following system? + 5æ3 7x4 10 %3D x2 X4 7 %3D x2 x3 2x2 16x4 20 I| |||| + + +
Suppose the system + 5x3 7x4 %3D X2 - - x3 X4 2x2 + + 16x4 has infinitely many solutions. Can you say anything about the number of solutions to the following system? + 5æ3 7x4 10 %3D x2 X4 7 %3D x2 x3 2x2 16x4 20 I| |||| + + +
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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![The problem presents two systems of linear equations.
**System 1** is a homogeneous system given by:
\[
\begin{align*}
x_1 - x_2 + 5x_3 - 7x_4 &= 0 \\
x_2 + x_3 - x_4 &= 0 \\
x_1 - 2x_2 + x_3 + 16x_4 &= 0 \\
\end{align*}
\]
This system is said to have infinitely many solutions.
**System 2** is a non-homogeneous system structured as follows:
\[
\begin{align*}
x_1 - x_2 + 5x_3 - 7x_4 &= 10 \\
x_2 + x_3 - x_4 &= 7 \\
x_1 - 2x_2 + x_3 + 16x_4 &= 20 \\
\end{align*}
\]
**Explanation:**
System 1, being homogeneous and having infinitely many solutions, suggests that the equations are linearly dependent. This means the system's coefficient matrix does not have full rank.
For System 2, which is non-homogeneous, if the coefficient matrix remains the same as System 1 but with different constant terms, the system will have no solution or exactly one solution, depending on the compatibility of the new constants with the solution space determined by the homogeneous equations.
To analyze the number of solutions for System 2, one could:
1. Determine if the augmented matrix of System 2 remains consistent.
2. Use Gaussian elimination or row reduction to see if a contradiction arises.
3. If System 2 has a consistent augmented matrix with dependent rows, it could have a unique solution considering the infinite solutions of System 1 do not satisfy these specific constants.
The outcome hinges on whether the change in constants still allows a solution within the established infinite solution space of the homogeneous equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b1372ae-e668-4971-89ee-29da7ac7c466%2Fee9f3a22-d105-48aa-bd59-837977ed3b22%2Frdpx5ih_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The problem presents two systems of linear equations.
**System 1** is a homogeneous system given by:
\[
\begin{align*}
x_1 - x_2 + 5x_3 - 7x_4 &= 0 \\
x_2 + x_3 - x_4 &= 0 \\
x_1 - 2x_2 + x_3 + 16x_4 &= 0 \\
\end{align*}
\]
This system is said to have infinitely many solutions.
**System 2** is a non-homogeneous system structured as follows:
\[
\begin{align*}
x_1 - x_2 + 5x_3 - 7x_4 &= 10 \\
x_2 + x_3 - x_4 &= 7 \\
x_1 - 2x_2 + x_3 + 16x_4 &= 20 \\
\end{align*}
\]
**Explanation:**
System 1, being homogeneous and having infinitely many solutions, suggests that the equations are linearly dependent. This means the system's coefficient matrix does not have full rank.
For System 2, which is non-homogeneous, if the coefficient matrix remains the same as System 1 but with different constant terms, the system will have no solution or exactly one solution, depending on the compatibility of the new constants with the solution space determined by the homogeneous equations.
To analyze the number of solutions for System 2, one could:
1. Determine if the augmented matrix of System 2 remains consistent.
2. Use Gaussian elimination or row reduction to see if a contradiction arises.
3. If System 2 has a consistent augmented matrix with dependent rows, it could have a unique solution considering the infinite solutions of System 1 do not satisfy these specific constants.
The outcome hinges on whether the change in constants still allows a solution within the established infinite solution space of the homogeneous equations.
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