https://file.aiquickdraw.com/mathbot/file/17253851969420i6rauuj.jpg

To solve the given system of equations, where the augmented matrix is in row echelon form:

\[
\begin{bmatrix}
1 & a & 0 & | & 0 \\
0 & 1 & 1 & | & c \\
0 & 0 & b & | & c
\end{bmatrix}
\]

we need to analyze and find the solutions for all values of \(a\), \(b\), and \(c\).

### Step 1: Interpret the Matrix
The matrix corresponds to the following system of linear equations:

\[
\begin{aligned}
1. & \quad x_1 + a x_2 = 0, \\
2. & \quad x_2 + x_3 = c, \\
3. & \quad b x_3 = c.
\end{aligned}
\]

### Step 2: Solve the System of Equations

#### Equation 3: \(b x_3 = c\)
- If \(b \neq 0\): \(x_3 = \frac{c}{b}\).
- If \(b = 0\): \(x_3\) can only be defined if \(c = 0\), leading to \(x_3\) being a free variable.

#### Equation 2: \(x_2 + x_3 = c\)
- If \(b \neq 0\): Substitute \(x_3 = \frac{c}{b}\) into this equation:
  \[
  x_2 + \frac{c}{b} = c \implies x_2 = c - \frac{c}{b}.
  \]
- If \(b = 0\) and \(c = 0\): The equation reduces to \(x_2 + x_3 = 0\), which implies \(x_2 = -x_3\).

#### Equation 1: \(x_1 + a x_2 = 0\)
- If \(b \neq 0\):
  \[
  x_1 + a\left(c - \frac{c}{b}\right) = 0 \implies x_1 = -a\left(c - \frac{c}{b}\right).
  \]
- If \(b = 0\) and \(c = 0\): Substitute \(x_2 = -x_3\) to get:
  \[
  x_1 - a x_3 = 0 \implies x_1 = a x_3.
  \]

### Step 3: Summary of Solutions

1. **Case \(b \neq 0\):**
   \[
   x_3 = \frac{c}{b}, \quad x_2 = c\left(1 - \frac{1}{b}\right), \quad x_1 = -a c \left(1 - \frac{1}{b}\right).
   \]

2. **Case \(b = 0\) and \(c = 0\):**
   \[
   x_3 = \text{free variable}, \quad x_2 = -x_3, \quad x_1 = a x_3.
   \]

### Conclusion
The solutions to the system depend on the values of \(a\), \(b\), and \(c\) as outlined in the cases above.

Do you need any further details or have any questions?

### Relative Questions:
1. What happens if \(a = 0\) in the solutions derived?
2. How do the solutions change if \(c\) is a non-zero constant and \(b = 1\)?
3. What is the interpretation of a "free variable" in the context of this system?
4. Can the solution be expressed in vector form for both cases of \(b\)?
5. How would the row echelon form change if the coefficient of \(x_1\) was not 1?

### Tip:
Understanding row echelon form is essential for solving linear systems as it helps in identifying free variables and leading variables, simplifying the solution process.

Find all solutions of the set of equations, where the expanded matrix has row echelon form: 
[1 a 0 | 0; 0 1 1 | c; 0 0 b | c] for all values of a, b, and c.

Explore how to find all solutions for a set of linear equations where the matrix is in row echelon form. Learn about solving systems with parameters a, b, and c, considering cases when b = 0 and b ≠ 0, suitable for undergraduate students in linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation