12. X1 – 5x2 + 4x3 = -3 %3D 2x1 - 7x2 + 3x3 =-2 MAAN -2x1 + x2 + 7x3 = -1 |3D
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Number 12 1.1 show all work
![**Linear Algebra Systems of Equations**
In this lesson, we will be handling systems of equations and determining their solutions. Below are some problems and exercises to solidify your understanding.
### Problem Set
**10.** Consider the augmented matrix:
\[
\begin{bmatrix}
1 & 3 & 0 & -2 & -7 \\
0 & 1 & 0 & 3 & 6 \\
0 & 0 & 1 & 0 & 2 \\
0 & 0 & 0 & 1 & -2
\end{bmatrix}
\]
Transpose this matrix into a system of linear equations and solve it accordingly.
### Exercises 11-14: Solve the Following Systems of Equations
**11.**
\[
\begin{align*}
x_2 + 5x_3 &= -4 \\
x_1 + 4x_2 + 3x_3 &= -2 \\
2x_1 + 7x_2 + x_3 &= -2
\end{align*}
\]
**12.**
\[
\begin{align*}
x_1 - 5x_2 + 4x_3 &= -3 \\
2x_1 - 7x_2 + 3x_3 &= -2 \\
-2x_1 + x_2 + 7x_3 &= -1
\end{align*}
\]
**13.**
\[
\begin{align*}
x_1 - 3x_3 &= 8 \\
x_1 + 2x_2 + 9x_3 &= 7 \\
x_2 + 5x_3 &= -2
\end{align*}
\]
**14.**
\[
\begin{align*}
2x_1 - 6x_3 &= -8 \\
x_2 + 2x_3 &= 3 \\
3x_1 + 6x_2 - 2x_3 &= -4
\end{align*}
\]
### Exercises 15-16: Determine if the Systems are Consistent
Without fully solving the systems, use methods such as Gaussian elimination or matrix rank to determine whether the following systems of equations have any solutions (i.e., they are consistent).
**15.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee22cf2f-b974-4b00-a3cf-09b388e7d65d%2F06982c6f-8bda-42db-95de-21dbbc4d9ee4%2Fz6ba2kr.jpeg&w=3840&q=75)
Transcribed Image Text:**Linear Algebra Systems of Equations**
In this lesson, we will be handling systems of equations and determining their solutions. Below are some problems and exercises to solidify your understanding.
### Problem Set
**10.** Consider the augmented matrix:
\[
\begin{bmatrix}
1 & 3 & 0 & -2 & -7 \\
0 & 1 & 0 & 3 & 6 \\
0 & 0 & 1 & 0 & 2 \\
0 & 0 & 0 & 1 & -2
\end{bmatrix}
\]
Transpose this matrix into a system of linear equations and solve it accordingly.
### Exercises 11-14: Solve the Following Systems of Equations
**11.**
\[
\begin{align*}
x_2 + 5x_3 &= -4 \\
x_1 + 4x_2 + 3x_3 &= -2 \\
2x_1 + 7x_2 + x_3 &= -2
\end{align*}
\]
**12.**
\[
\begin{align*}
x_1 - 5x_2 + 4x_3 &= -3 \\
2x_1 - 7x_2 + 3x_3 &= -2 \\
-2x_1 + x_2 + 7x_3 &= -1
\end{align*}
\]
**13.**
\[
\begin{align*}
x_1 - 3x_3 &= 8 \\
x_1 + 2x_2 + 9x_3 &= 7 \\
x_2 + 5x_3 &= -2
\end{align*}
\]
**14.**
\[
\begin{align*}
2x_1 - 6x_3 &= -8 \\
x_2 + 2x_3 &= 3 \\
3x_1 + 6x_2 - 2x_3 &= -4
\end{align*}
\]
### Exercises 15-16: Determine if the Systems are Consistent
Without fully solving the systems, use methods such as Gaussian elimination or matrix rank to determine whether the following systems of equations have any solutions (i.e., they are consistent).
**15.**
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