Title: Solve Linear Systems in Parametric Form

Q: Created by: Dr. Pang Chung Yin Amy, Dr. Chiang Shun Ling, and Naw Sai.
Department of Mathematics,
Hong Kong Baptist University (HKBU) © 2025.
A: These decks may be freely shared and distributed among students and educators. We hope they help in your linear algebra journey!

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} 0 & 3 & -3 \\ -1 & -7 & 7 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 6 \\ -14 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} -2 & -2 & 0 \\ -2 & -3 & -2 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 6 \\ 8 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} -1 \\ -2 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{ccc|c} -2 & 2 & 0 & -2 \\ -1 & 1 & 1 & 1 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} 0 \\ -1 \\ 2 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} -2 & -2 & 0 \\ -2 & 0 & 1 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 2 \\ 0 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} 0 \\ -1 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{ccc|c} 1 & 0 & -2 & 1 \\ -1 & 3 & 2 & -1 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} -1 & 0 & -1 \\ 9 & 3 & 3 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 1 \\ -12 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} -1 \\ -1 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -1 \\ 2 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{ccc|c} -4 & 4 & 0 & 14 \\ 3 & -4 & -1 & -14 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 7/2 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -1 \\ -1 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} 0 & -6 & 12 \\ 4 & -5 & 2 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} -6 \\ -13 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 2 \\ 2 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} 1 & 0 & 1 \\ 2 & -4 & -6 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 1 \\ -6 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -1 \\ -2 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{ccc|c} 1 & 0 & 1 & 1 \\ 4 & 2 & 8 & 2 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -1 \\ -2 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & 2 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} -2 \\ 2 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -2 \\ -2 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{cccc|c} -1 & 3 & 5 & 8 & -5 \\ 1 & -2 & -2 & -2 & 2 \\ -1 & 4 & 5 & 8 & -5 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 0 \\ -1 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -2 \\ 0 \\ -2 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} 2 & -1 & 0 & 0 \\ 2 & 0 & 1 & 0 \\ 0 & 0 & 2 & 2 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 0 \\ -2 \\ -8 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 0 \\ -2 \\ -2 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ 2 \\ -2 \\ 2 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{cccc|c} 1 & 0 & -1 & 0 & -1 \\ 0 & 1 & -1 & 0 & 2 \\ 0 & 1 & -5 & 2 & -2 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} -1 \\ 2 \\ 0 \\ -2 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ 1 \\ 1 \\ 2 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} 2 & 2 & 0 & 0 \\ -2 & 0 & -1 & 0 \\ -2 & 0 & 0 & 1 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 4 \\ 0 \\ 1 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 2 \\ 0 \\ 1 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ -1 \\ -2 \\ 2 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} -2 & 3 & 9 & -11 \\ 0 & 2 & 0 & -4 \\ 4 & -6 & -9 & 13 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 8 \\ -2 \\ -7 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} -1 \\ -1 \\ 1 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 2 \\ 2 \\ 1 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} 1 & 0 & 0 & 2 \\ 0 & -1 & 0 & -2 \\ 0 & 0 & 2 & 2 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} 0 \\ -1 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -2 \\ -2 \\ -1 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{cccc|c} 1 & 0 & 0 & -1 & 1 \\ 0 & -1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 2 & 2 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} 1 \\ -1 \\ 2 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ 1 \\ -2 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} 1 & 0 & 0 & 1 \\ -1 & -1 & 1 & -2 \\ 1 & 0 & -1 & 0 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} 0 \\ -1 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -1 \\ -2 \\ -1 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} -2 & 0 & -2 & 13 & -6 \\ -1 & -1 & 0 & 0 & 1 \\ 0 & 0 & 0 & -3 & 0 \\ 4 & 1 & 2 & -13 & 6 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 13 \\ -2 \\ -3 \\ -9 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} 2 \\ 0 \\ -2 \\ 1 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -1 \\ 2 \\ -2 \\ 0 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} 1 & 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 2 & 0 \\ 0 & 0 & -2 & 4 & 0 \\ 0 & 0 & 1 & -4 & 1 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 2 \\ 4 \\ -4 \\ 3 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} -2 \\ 2 \\ 2 \\ 0 \\ 1 \end{bmatrix} + s_1\begin{bmatrix} 2 \\ -1 \\ 2 \\ 1 \\ 2 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{ccccc|c} -1 & 0 & -2 & 0 & 0 & -2 \\ 0 & 1 & 2 & 0 & 0 & -2 \\ 0 & 0 & -1 & 2 & 1 & -2 \\ 0 & 0 & 2 & 0 & 2 & 4 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} 2 \\ -2 \\ 0 \\ -2 \\ 2 \end{bmatrix} + s_1\begin{bmatrix} -2 \\ -2 \\ 1 \\ 1 \\ -1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & -1 & 1 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 2 \\ 2 \\ -1 \\ 1 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 2 \\ -1 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -1 \\ 0 \\ 0 \\ 1 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} -6 & 4 & 4 & 8 & 10 \\ -4 & 3 & 2 & 5 & 4 \\ 0 & 4 & 0 & 0 & -8 \\ 2 & 0 & -2 & -3 & -8 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 14 \\ 10 \\ 8 \\ -2 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} -1 \\ 2 \\ 0 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -1 \\ 2 \\ -2 \\ -2 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} 2 & 2 & 0 & -4 & 2 \\ -6 & -4 & 0 & 12 & -10 \\ 2 & 8 & -1 & -4 & -10 \\ -3 & -10 & 1 & 6 & 11 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} -2 \\ 4 \\ -10 \\ 12 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} -2 \\ -1 \\ 2 \\ -1 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -1 \\ 2 \\ 0 \\ 1 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} 2 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 2 & 0 & 0 & 1 & 1 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 4 \\ -3 \\ -1 \\ 2 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} 2 \\ -1 \\ -1 \\ -2 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 0 \\ 1 \\ 0 \\ -1 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & 2 & -2 \\ 0 & 1 & 0 & 1 & 0 & -3 \\ 0 & 1 & 1 & 2 & -3 & -4 \\ 0 & 0 & 0 & 1 & -2 & -1 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} -2 \\ -2 \\ 0 \\ -1 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -2 \\ -2 \\ 1 \\ 2 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} -1 & 2 & 0 & 0 & 3 \\ 0 & 2 & 0 & 0 & 4 \\ 0 & 0 & -1 & 0 & 1 \\ 0 & 0 & 0 & 2 & 4 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 0 \\ 0 \\ 2 \\ -4 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 0 \\ -2 \\ -2 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -1 \\ -2 \\ 1 \\ -2 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & -1 & -1 & 0 & 2 \\ 0 & 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & -2 & -4 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} -2 \\ 0 \\ -2 \\ -2 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} -2 \\ 2 \\ -2 \\ 1 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 0 \\ 1 \\ 1 \\ -2 \\ 1 \end{bmatrix} \mid s_1 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{cccc|c} -3 & 1 & 8 & 2 & 5 \\ -6 & -2 & 8 & -4 & 2 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} -1 \\ 2 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 2 \\ -2 \\ 1 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} 0 \\ -2 \\ 0 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{cccc|c} 1 & 0 & 1 & 1 & 0 \\ 1 & 2 & 5 & 3 & 4 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 2 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -1 \\ -2 \\ 1 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} -1 \\ -1 \\ 0 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{cccc|c} 2 & 2 & 2 & 2 & -2 \\ 0 & 0 & -2 & -1 & 1 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 0 \\ 0 \\ -1 \end{bmatrix} + s_1\begin{bmatrix} -1 \\ 1 \\ 0 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} 1 \\ 0 \\ 1 \\ -2 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} -2 & -4 & 2 & 0 \\ -2 & -5 & 2 & 1 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 0 \\ -1 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 0 \\ 0 \\ -1 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} 0 \\ 1 \\ 2 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{cccc|c} 2 & 1 & 1 & -1 & 1 \\ 8 & 4 & 8 & -4 & 4 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ -2 \\ 0 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} 0 \\ 1 \\ 0 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} 1 & 0 & -1 & -1 \\ 3 & 2 & -5 & -1 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} -1 \\ -3 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} -1 \\ 0 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} 1 \\ -1 \\ 0 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{cccc|c} -1 & 0 & -1 & 1 & 0 \\ 0 & 1 & -2 & -2 & 0 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -1 \\ 2 \\ 1 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} 1 \\ 2 \\ 0 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{cccc|c} 3 & 1 & 8 & 3 & -4 \\ 2 & 1 & 6 & 2 & -2 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} -2 \\ 2 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -2 \\ -2 \\ 1 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} -1 \\ 0 \\ 0 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} -3 & -2 & -5 & -2 \\ 2 & 2 & 4 & 0 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 4 \\ -2 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -1 \\ -1 \\ 1 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} -2 \\ 2 \\ 0 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{cccc|c} 4 & 1 & -5 & 0 & -8 \\ 0 & 1 & -1 & 0 & 0 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} -2 \\ 0 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{ccccc|c} 1 & 2 & 0 & 2 & 2 & 1 \\ 0 & 2 & 0 & 2 & 4 & 2 \\ 0 & 0 & 4 & -4 & 8 & 4 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} -1 \\ 1 \\ 1 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 0 \\ -1 \\ 1 \\ 1 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} 2 \\ -2 \\ -2 \\ 0 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{ccccc|c} 8 & -8 & 4 & 0 & 0 & -4 \\ 2 & 1 & 0 & -1 & 0 & -1 \\ -2 & 1 & 0 & 0 & 1 & 0 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 0 \\ -1 \\ 1 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ 0 \\ -2 \\ 2 \\ 2 \end{bmatrix} + s_2\begin{bmatrix} 0 \\ 1 \\ 2 \\ 1 \\ -1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} 2 & -1 & 0 & 0 & 1 \\ 6 & -1 & 2 & 0 & 5 \\ 0 & 0 & 0 & 1 & -1 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} -1 \\ 3 \\ -1 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 1 \\ 2 \\ -1 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ 2 \\ -2 \\ 0 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} 0 \\ 1 \\ -2 \\ 1 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{ccccc|c} -12 & 1 & 0 & 5 & -12 & 6 \\ -2 & 0 & 2 & 0 & 2 & 2 \\ -8 & 0 & 0 & 4 & -8 & 4 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 1 \\ 1 \\ 1 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ 2 \\ 1 \\ 2 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} 0 \\ 2 \\ -1 \\ 2 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} -1 & 0 & 0 & 1 & -2 \\ 4 & 2 & 0 & -8 & 4 \\ 0 & 0 & 2 & -4 & 4 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 2 \\ -6 \\ 2 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} -2 \\ 1 \\ 1 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ 2 \\ 2 \\ 1 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} -2 \\ 2 \\ -2 \\ 0 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{ccccc|c} 1 & 0 & 0 & -1 & 0 & -2 \\ 0 & -1 & 0 & 2 & 0 & 0 \\ 0 & -1 & -1 & 4 & 0 & -1 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} -2 \\ 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ 2 \\ 2 \\ 1 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{ccccc|c} 1 & 1 & 0 & 1 & 3 & 1 \\ 1 & 3 & 0 & 5 & 5 & 3 \\ 1 & 1 & 1 & -1 & 3 & -1 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} 0 \\ 1 \\ -2 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 1 \\ -2 \\ 2 \\ 1 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} -2 \\ -1 \\ 0 \\ 0 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Consider the matrix $A$ and vector $\mathbf{b}$ given by $A = \begin{bmatrix} -1 & 0 & 1 & -1 & -4 \\ 0 & -1 & 0 & -2 & -2 \\ 0 & 0 & 1 & -1 & -2 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} -3 \\ 1 \\ -1 \end{bmatrix}$. Give, in parametric form, the solution set to $A\mathbf{x} = \mathbf{b}$.
A: The solution set is $\left\{\begin{bmatrix} 2 \\ -1 \\ -1 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} 0 \\ -2 \\ 1 \\ 1 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} -2 \\ -2 \\ 2 \\ 0 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$

Q: Write the solution of the system in parametric form. The augmented matrix is
$\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 2 & 0 & -1 \\ 2 & 1 & 0 & 5 & 2 & -1 \\ 2 & 0 & 2 & 4 & -2 & -4 \end{array}\right]$
A: The solution set is $\left\{\begin{bmatrix} -1 \\ 1 \\ -1 \\ 0 \\ 0 \end{bmatrix} + s_1\begin{bmatrix} -2 \\ -1 \\ 0 \\ 1 \\ 0 \end{bmatrix} + s_2\begin{bmatrix} 0 \\ -2 \\ 1 \\ 0 \\ 1 \end{bmatrix} \mid s_1, s_2 \in \mathbb{R} \right\}.$