### Describing Solutions to the Linear System $Ax = 0$Given the matrix $A$ is row equivalent to \[\begin{bmatrix}1 & 3 & 0 & -1 & 0 & 4 \\0 & 0 & 1 & 4 & 0 & -2 \\0 & 0 & 0 & 0 & 1 & 5\end{bmatrix},\]describe all solutions to $Ax = 0$.\[\mathbf{x} = \begin{bmatrix}\\ \\\\ \\\\ \\\\\end{bmatrix}= x_2 \begin{bmatrix}\\ \\\\ \\\\ \\\\\end{bmatrix}+ x_4 \begin{bmatrix}\\ \\\\ \\\\ \\\\\end{bmatrix}+ x_6 \begin{bmatrix}\\ \\\\ \\\\ \\\\\end{bmatrix}\]This matrix represents a system of linear equations. To describe all solutions to the equation $Ax = 0$, you need to express the variable vector $\mathbf{x}$ in terms of the basic variables and the free variables. Here, columns with no leading 1s in the row-reduced echelon form (columns 2, 4, and 6) correspond to free variables $x_2$, $x_4$, and $x_6$.In the image, there are placeholders for filling in the expressions for the free variables $x_2, x_4,$ and $x_6$ which combine to describe the general solution to the system. The solutions are vectors that will be combinations of the free variables.Beyond this explanation, students will typically need to solve the system to express $\mathbf{x}$ explicitly in terms of $x_2, x_4,$ and $x_6$.

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1 30-1 0 4 Given the matrix A is row equivalent to 0 0 1 -4 0-2 describe all solutions to Ax = 0. 0 0 0 0 1 X = X2 + X4 + X6 > Next Question

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

### Describing Solutions to the Linear System $Ax = 0$

Given the matrix $A$ is row equivalent to
\[
\begin{bmatrix}
1 & 3 & 0 & -1 & 0 & 4 \\
0 & 0 & 1 & 4 & 0 & -2 \\
0 & 0 & 0 & 0 & 1 & 5
\end{bmatrix},
\]
describe all solutions to $Ax = 0$.

\[
\mathbf{x} =
\begin{bmatrix}
\\ \\
\\ \\
\\ \\
\\
\end{bmatrix}
= x_2
\begin{bmatrix}
\\ \\
\\ \\
\\ \\
\\
\end{bmatrix}
+ x_4
\begin{bmatrix}
\\ \\
\\ \\
\\ \\
\\
\end{bmatrix}
+ x_6
\begin{bmatrix}
\\ \\
\\ \\
\\ \\
\\
\end{bmatrix}
\]

This matrix represents a system of linear equations. To describe all solutions to the equation $Ax = 0$, you need to express the variable vector $\mathbf{x}$ in terms of the basic variables and the free variables. Here, columns with no leading 1s in the row-reduced echelon form (columns 2, 4, and 6) correspond to free variables $x_2$, $x_4$, and $x_6$.

In the image, there are placeholders for filling in the expressions for the free variables $x_2, x_4,$ and $x_6$ which combine to describe the general solution to the system. The solutions are vectors that will be combinations of the free variables.

Beyond this explanation, students will typically need to solve the system to express $\mathbf{x}$ explicitly in terms of $x_2, x_4,$ and $x_6$.

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