[4] Use Gaussian elimination with back substitution to solve the linear system of equations 4x1 − X2 + X3 = 8 2x1 +5x₂ + 2x3 3 x₁ + 2x₂ + 4x3 = 11 =
[4] Use Gaussian elimination with back substitution to solve the linear system of equations 4x1 − X2 + X3 = 8 2x1 +5x₂ + 2x3 3 x₁ + 2x₂ + 4x3 = 11 =
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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Gaussian elim
![**Gaussian Elimination and Back Substitution Method**
In this section, we will use Gaussian elimination followed by back substitution to solve the given linear system of equations:
\[
\begin{align*}
4x_1 - x_2 + x_3 &= 8 \\
2x_1 + 5x_2 + 2x_3 &= 3 \\
x_1 + 2x_2 + 4x_3 &= 11
\end{align*}
\]
Gaussian elimination is an algorithm for solving systems of linear equations. It transforms the system into an upper triangular matrix, making it easier to solve through back substitution.
**Steps to Solve:**
1. **Gaussian Elimination:**
- Convert the system of equations into an augmented matrix.
- Use row operations to get zeros below the leading coefficients (the first non-zero number from the left in each row) of each row.
2. **Back Substitution:**
- Start with the last equation (in simplest form) and solve for the variables in reverse order (from last to first).
**Example Application:**
Apply the Gaussian elimination on the augmented matrix. Once the matrix has been reduced to an upper triangular form, use back substitution to find the values of \( x_1 \), \( x_2 \), and \( x_3 \).
By following these steps diligently, you will obtain the solution to the system, ultimately validating the precision of your results through back substitution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2Fb8afd36e-8230-45b2-b502-eeeca7599caf%2F029pew_processed.png&w=3840&q=75)
Transcribed Image Text:**Gaussian Elimination and Back Substitution Method**
In this section, we will use Gaussian elimination followed by back substitution to solve the given linear system of equations:
\[
\begin{align*}
4x_1 - x_2 + x_3 &= 8 \\
2x_1 + 5x_2 + 2x_3 &= 3 \\
x_1 + 2x_2 + 4x_3 &= 11
\end{align*}
\]
Gaussian elimination is an algorithm for solving systems of linear equations. It transforms the system into an upper triangular matrix, making it easier to solve through back substitution.
**Steps to Solve:**
1. **Gaussian Elimination:**
- Convert the system of equations into an augmented matrix.
- Use row operations to get zeros below the leading coefficients (the first non-zero number from the left in each row) of each row.
2. **Back Substitution:**
- Start with the last equation (in simplest form) and solve for the variables in reverse order (from last to first).
**Example Application:**
Apply the Gaussian elimination on the augmented matrix. Once the matrix has been reduced to an upper triangular form, use back substitution to find the values of \( x_1 \), \( x_2 \), and \( x_3 \).
By following these steps diligently, you will obtain the solution to the system, ultimately validating the precision of your results through back substitution.
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