a) Fill the corresponding augmented matrix. The rightmost column represents the constant values from the linear equations.

Correct
(b) Use elementary row operations on the augmented matrix in part (a) to make a matrix of the form

[
1
a
b
0
1
c
]
,
[
1
a
b
0
0
1
]
, or
[
1
a
b
0
0
0
]
.*

To address both parts of this problem, let's begin with step (a) by constructing the augmented matrix from the system of linear equations.

### Part (a): Construct the augmented matrix

The augmented matrix is derived from a system of linear equations in the form:
\[
A \mathbf{x} = \mathbf{b}
\]
where \(A\) represents the coefficient matrix, \(\mathbf{x}\) represents the variables, and \(\mathbf{b}\) represents the constant values. 

An augmented matrix is formed by appending the column vector \(\mathbf{b}\) (the constants) to the matrix \(A\). Please provide the system of linear equations so I can fill out the augmented matrix correctly.

### Part (b): Apply row operations to reduce the matrix to one of the specified forms

After constructing the augmented matrix, we'll perform elementary row operations (such as row swapping, multiplying a row by a nonzero constant, or adding/subtracting rows) to reduce the matrix to one of the following forms:
- \[
  \begin{bmatrix}
  1 & a & b \\
  0 & 1 & c \\
  \end{bmatrix}
  \]
- \[
  \begin{bmatrix}
  1 & a & b \\
  0 & 0 & 1 \\
  \end{bmatrix}
  \]
- \[
  \begin{bmatrix}
  1 & a & b \\
  0 & 0 & 0 \\
  \end{bmatrix}
  \]

The goal is to use row operations to achieve this reduced form.

Please provide the system of equations, and I will guide you through the row reduction process!

---

Would you like to provide the system of equations for step (a)? Here are 5 questions that could relate to this process:

1. Are you familiar with how to convert a system of equations into an augmented matrix?
2. Do you want to see examples of different row operations?
3. Should I explain each elementary row operation in detail?
4. Do you need clarification on matrix forms such as row echelon or reduced row echelon form?
5. Are you interested in learning the steps to solve systems of equations using Gaussian elimination?

**Tip**: Always ensure that the leading entry in a row is 1 when reducing a matrix for clarity and consistency.

Fill the corresponding augmented matrix. The rightmost column represents the constant values from the linear equations. Use elementary row operations on the augmented matrix to make a matrix of the form [1 a b 0 1 c], [1 a b 0 0 1], or [1 a b 0 0 0].

Learn how to use elementary row operations on an augmented matrix to transform it into row echelon or reduced row echelon form. This problem covers key concepts in linear algebra, including Gaussian elimination and matrix reduction. Suitable for high school students in Grades 10-12.

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