Given a Linear System of equations below. The matrix equation of the linear system is given by: Ax=b. The determinant of A is 8. Using Cramer's Rule find the value for х. x + 3y+ 4z = 3 2x + 6y + 9z = 5 Зх + у — 2х 3 7
Given a Linear System of equations below. The matrix equation of the linear system is given by: Ax=b. The determinant of A is 8. Using Cramer's Rule find the value for х. x + 3y+ 4z = 3 2x + 6y + 9z = 5 Зх + у — 2х 3 7
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Given a linear system of equations below. The matrix equation of the linear system is given by: (see image)
![### Linear Systems and Cramer's Rule
Given a linear system of equations below, we are dealing with a matrix equation of the form \( Ax = b \). The determinant of matrix \( A \) is given as 8. Using Cramer's Rule, our goal is to find the value for \( x \).
#### System of Linear Equations
\[
\begin{align*}
x + 3y + 4z &= 3 \\
2x + 6y + 9z &= 5 \\
3x + y - 2z &= 7
\end{align*}
\]
1. **First Equation**: \( x + 3y + 4z = 3 \)
2. **Second Equation**: \( 2x + 6y + 9z = 5 \)
3. **Third Equation**: \( 3x + y - 2z = 7 \)
#### Applying Cramer's Rule
To find \( x \) using Cramer's Rule, we replace the \( x \)-column of matrix \( A \) with the column vector \( b \):
1. **Matrix \( A \)**:
\[
A = \begin{pmatrix}
1 & 3 & 4 \\
2 & 6 & 9 \\
3 & 1 & -2
\end{pmatrix}
\]
2. **Column Vector \( b \)**:
\[
b = \begin{pmatrix}
3 \\
5 \\
7
\end{pmatrix}
\]
3. **Matrix \( A_x \)** (Matrix \( A \) with the \( x \)-column replaced by \( b \)):
\[
A_x = \begin{pmatrix}
3 & 3 & 4 \\
5 & 6 & 9 \\
7 & 1 & -2
\end{pmatrix}
\]
Finally, \( x \) is calculated as:
\[
x = \frac{\det(A_x)}{\det(A)}
\]
Given \( \det(A) = 8 \), calculate \( \det(A_x) \) and then use it in the above formula to find \( x \).
#### Detailed Explanation
1. **Step-by-Step Method**
- Compute the determinant of matrix \( A_x \).
- Use \( x = \frac{\det(A_x)}{\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc17e2a4-76f7-46f2-97e3-7d367a16a370%2F804b06b5-159b-4090-91ed-6381c5e0dc20%2F4wkv82.png&w=3840&q=75)
Transcribed Image Text:### Linear Systems and Cramer's Rule
Given a linear system of equations below, we are dealing with a matrix equation of the form \( Ax = b \). The determinant of matrix \( A \) is given as 8. Using Cramer's Rule, our goal is to find the value for \( x \).
#### System of Linear Equations
\[
\begin{align*}
x + 3y + 4z &= 3 \\
2x + 6y + 9z &= 5 \\
3x + y - 2z &= 7
\end{align*}
\]
1. **First Equation**: \( x + 3y + 4z = 3 \)
2. **Second Equation**: \( 2x + 6y + 9z = 5 \)
3. **Third Equation**: \( 3x + y - 2z = 7 \)
#### Applying Cramer's Rule
To find \( x \) using Cramer's Rule, we replace the \( x \)-column of matrix \( A \) with the column vector \( b \):
1. **Matrix \( A \)**:
\[
A = \begin{pmatrix}
1 & 3 & 4 \\
2 & 6 & 9 \\
3 & 1 & -2
\end{pmatrix}
\]
2. **Column Vector \( b \)**:
\[
b = \begin{pmatrix}
3 \\
5 \\
7
\end{pmatrix}
\]
3. **Matrix \( A_x \)** (Matrix \( A \) with the \( x \)-column replaced by \( b \)):
\[
A_x = \begin{pmatrix}
3 & 3 & 4 \\
5 & 6 & 9 \\
7 & 1 & -2
\end{pmatrix}
\]
Finally, \( x \) is calculated as:
\[
x = \frac{\det(A_x)}{\det(A)}
\]
Given \( \det(A) = 8 \), calculate \( \det(A_x) \) and then use it in the above formula to find \( x \).
#### Detailed Explanation
1. **Step-by-Step Method**
- Compute the determinant of matrix \( A_x \).
- Use \( x = \frac{\det(A_x)}{\
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