Find the set of solutions for the given linear system. (I -8x1 + X2 + 8x3 = 1 7x3 + X4 = -3 (X1, X2, X3, X4) )-
Find the set of solutions for the given linear system. (I -8x1 + X2 + 8x3 = 1 7x3 + X4 = -3 (X1, X2, X3, X4) )-
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 the Linear System**
Consider the given set of linear equations:
\[
-8x_1 + x_2 + 8x_3 = 1
\]
\[
-7x_3 + x_4 = -3
\]
Our goal is to determine the set of solutions for the variables \(x_1, x_2, x_3,\) and \(x_4\). If there are infinitely many solutions, use \(s_1\) and \(s_2\) as your parameters.
The objective is to express the solution as:
\[
(x_1, x_2, x_3, x_4) = \begin{pmatrix}
\boxed{}
\end{pmatrix}
\]
Given that the system might have free variables, the general solution can include parameters related to these free variables. In this case, \(s_1\) and \(s_2\) may be used to express these parameters.
**Steps and Detailed Explanation**
1. **Identify Free Variables and Parameters:**
- If there are fewer equations than unknowns, then the system might have infinitely many solutions. Identify which variables can be considered as free variables (parameters \(s_1, s_2\)).
2. **Rewrite the System with Parameters:**
- Substitute the values of free variables into the equations to express other variables in terms of these parameters.
3. **Construct the General Solution:**
- Combine the results to write the general solution vector in terms of the free parameters.
By following these steps, the goal is to solve for the set \( (x_1, x_2, x_3, x_4) \) which satisfies both equations.
Feel free to apply matrix row operations or substitution methods to determine the relationship between the variables and express the final solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Facdec92e-028d-4f37-af60-841922a4cee5%2F8855de8d-799c-4471-93b6-bdc085de2d12%2Fcdvzf48_processed.png&w=3840&q=75)
Transcribed Image Text:**Solving the Linear System**
Consider the given set of linear equations:
\[
-8x_1 + x_2 + 8x_3 = 1
\]
\[
-7x_3 + x_4 = -3
\]
Our goal is to determine the set of solutions for the variables \(x_1, x_2, x_3,\) and \(x_4\). If there are infinitely many solutions, use \(s_1\) and \(s_2\) as your parameters.
The objective is to express the solution as:
\[
(x_1, x_2, x_3, x_4) = \begin{pmatrix}
\boxed{}
\end{pmatrix}
\]
Given that the system might have free variables, the general solution can include parameters related to these free variables. In this case, \(s_1\) and \(s_2\) may be used to express these parameters.
**Steps and Detailed Explanation**
1. **Identify Free Variables and Parameters:**
- If there are fewer equations than unknowns, then the system might have infinitely many solutions. Identify which variables can be considered as free variables (parameters \(s_1, s_2\)).
2. **Rewrite the System with Parameters:**
- Substitute the values of free variables into the equations to express other variables in terms of these parameters.
3. **Construct the General Solution:**
- Combine the results to write the general solution vector in terms of the free parameters.
By following these steps, the goal is to solve for the set \( (x_1, x_2, x_3, x_4) \) which satisfies both equations.
Feel free to apply matrix row operations or substitution methods to determine the relationship between the variables and express the final solution.
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