https://file.aiquickdraw.com/mathbot/file/17287519686154axgegix.jpg

Let's solve these problems by analyzing the augmented matrices presented.

### Question 3:

The augmented matrix is:

\[
\left[\begin{array}{ccc|c}
1 & 0 & 0 & 3 \\
0 & 1 & 0 & 4 \\
0 & 0 & 0 & 0 \\
\end{array}\right]
\]

This represents the system of equations:
\[
x = 3
\]
\[
y = 4
\]
\[
0 = 0 \quad \text{(this equation is always true)}
\]

The third row indicates that there is no constraint on \(z\), meaning that \(z\) is a free variable. Therefore, the system has **infinitely many solutions** where \(x = 3\), \(y = 4\), and \(z\) can take any value.

So the correct answer for Question 3 is:
- **B. Infinitely many solutions**

### Question 4:

The augmented matrix is:

\[
\left[\begin{array}{ccc|c}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & -4 \\
0 & 0 & 1 & -2 \\
\end{array}\right]
\]

This represents the system of equations:
\[
x = 0
\]
\[
y = -4
\]
\[
z = -2
\]

Each equation defines a unique value for \(x\), \(y\), and \(z\), meaning the system has a unique solution.

So the correct answer for Question 4 is:
- **E. Unique solution: \(x = 0\), \(y = -4\), \(z = -2\)**

---

Do you want more details or have any further questions?

Here are 5 related questions to consider:
1. How can you tell if a system of equations has no solutions from an augmented matrix?
2. What does a row of all zeros in an augmented matrix indicate?
3. How do you interpret free variables in a system of linear equations?
4. What is the difference between consistent and inconsistent systems of equations?
5. How can Gaussian elimination help solve systems of linear equations?

**Tip:** If the last row of an augmented matrix becomes \([0 \ 0 \ 0 | c]\) where \(c \neq 0\), the system has **no solution** (inconsistent).

Analyze two augmented matrices and select the correct solution from the multiple-choice options provided.

Discover how to solve systems of linear equations using augmented matrices. This problem involves analyzing two matrices—one with infinitely many solutions and another with a unique solution—using linear algebra concepts. Explore Gaussian elimination and understand free variables and consistent systems. Suitable for Grades 10-12.

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