Write the given system of equations as a matrix equation and solve by using inverses. = k, 4x1 - 12x2 + X3 = k2 + X3 = k3 X1 - X2 - 9x1 a. What are x,, x2, and x3 when k, = 9, k2 = - 8, and kg = - 7? X1 = X2 = X3 =
Write the given system of equations as a matrix equation and solve by using inverses. = k, 4x1 - 12x2 + X3 = k2 + X3 = k3 X1 - X2 - 9x1 a. What are x,, x2, and x3 when k, = 9, k2 = - 8, and kg = - 7? X1 = X2 = X3 =
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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![### Solving a System of Equations Using Matrices
**Problem Statement:**
Write the given system of equations as a matrix equation and solve by using inverses.
\[
\begin{align*}
x_1 - x_2 &= k_1 \\
4x_1 - 12x_2 + x_3 &= k_2 \\
-9x_1 + x_3 &= k_3
\end{align*}
\]
**Question:**
a. What are \(x_1\), \(x_2\), and \(x_3\) when \(k_1 = 9\), \(k_2 = -8\), and \(k_3 = -7\)?
\[
\begin{align*}
x_1 &= \_\_\_ \\
x_2 &= \_\_\_ \\
x_3 &= \_\_\_
\end{align*}
\]
**Instruction:**
Transform the system of equations into a matrix equation of the form \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the column matrix of variables, and \(B\) is the column matrix of constants. Use matrix inversion to solve for the variables.
### Detailed Explanation:
1. **Write the system of equations in matrix form:**
The system of equations can be written as:
\[
\begin{pmatrix}
1 & -1 & 0 \\
4 & -12 & 1 \\
-9 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
=
\begin{pmatrix}
k_1 \\
k_2 \\
k_3
\end{pmatrix}
\]
2. **Substitute the given values \(k_1 = 9\), \(k_2 = -8\), and \(k_3 = -7\):**
\[
\begin{pmatrix}
1 & -1 & 0 \\
4 & -12 & 1 \\
-9 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
x_](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb3f5949-3440-4952-a068-9f8891699419%2F517ffdb5-acaf-4981-bd03-4250dc71dfc8%2Fj071me5.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving a System of Equations Using Matrices
**Problem Statement:**
Write the given system of equations as a matrix equation and solve by using inverses.
\[
\begin{align*}
x_1 - x_2 &= k_1 \\
4x_1 - 12x_2 + x_3 &= k_2 \\
-9x_1 + x_3 &= k_3
\end{align*}
\]
**Question:**
a. What are \(x_1\), \(x_2\), and \(x_3\) when \(k_1 = 9\), \(k_2 = -8\), and \(k_3 = -7\)?
\[
\begin{align*}
x_1 &= \_\_\_ \\
x_2 &= \_\_\_ \\
x_3 &= \_\_\_
\end{align*}
\]
**Instruction:**
Transform the system of equations into a matrix equation of the form \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the column matrix of variables, and \(B\) is the column matrix of constants. Use matrix inversion to solve for the variables.
### Detailed Explanation:
1. **Write the system of equations in matrix form:**
The system of equations can be written as:
\[
\begin{pmatrix}
1 & -1 & 0 \\
4 & -12 & 1 \\
-9 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
=
\begin{pmatrix}
k_1 \\
k_2 \\
k_3
\end{pmatrix}
\]
2. **Substitute the given values \(k_1 = 9\), \(k_2 = -8\), and \(k_3 = -7\):**
\[
\begin{pmatrix}
1 & -1 & 0 \\
4 & -12 & 1 \\
-9 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
x_
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