https://file.aiquickdraw.com/mathbot/file/1728900561274o8ujc2ce.jpg

Let's analyze the system of linear equations given in the image:

\[
\begin{aligned}
1. \ & x - 4y + z = 0 \\
2. \ & 2x + y + 4z = 0 \\
3. \ & x + 4y + z = 0 \\
\end{aligned}
\]

To determine the type of solution (trivial, non-trivial, no solution, or \( n-r \) solutions), we can proceed by solving this system.

### Step 1: Convert the system into matrix form

The augmented matrix for this system is:

\[
\begin{pmatrix}
1 & -4 & 1 & | & 0 \\
2 & 1  & 4 & | & 0 \\
1 & 4  & 1 & | & 0 \\
\end{pmatrix}
\]

### Step 2: Perform Gaussian elimination

Start by making the first pivot entry in row 1 equal to 1 (which it already is). Now, eliminate the first column entries in rows 2 and 3.

- Subtract 2 times row 1 from row 2:
  \[
  (R_2 - 2R_1) \to R_2
  \]
  Row 2 becomes:
  \[
  \begin{pmatrix} 0 & 9 & 2 & | & 0 \end{pmatrix}
  \]

- Subtract row 1 from row 3:
  \[
  (R_3 - R_1) \to R_3
  \]
  Row 3 becomes:
  \[
  \begin{pmatrix} 0 & 8 & 0 & | & 0 \end{pmatrix}
  \]

Now, the matrix is:
\[
\begin{pmatrix}
1 & -4 & 1 & | & 0 \\
0 & 9  & 2 & | & 0 \\
0 & 8  & 0 & | & 0 \\
\end{pmatrix}
\]

### Step 3: Further eliminate and solve

Next, we can work on row 3 by eliminating the second column:

- Subtract \( \frac{8}{9} \times \text{Row 2} \) from Row 3 to eliminate the second entry in row 3:
  \[
  R_3 - \frac{8}{9}R_2 \to R_3
  \]
  Row 3 becomes:
  \[
  \begin{pmatrix} 0 & 0 & -\frac{16}{9} & | & 0 \end{pmatrix}
  \]

The matrix now is:
\[
\begin{pmatrix}
1 & -4 & 1 & | & 0 \\
0 & 9  & 2 & | & 0 \\
0 & 0  & -\frac{16}{9} & | & 0 \\
\end{pmatrix}
\]

### Step 4: Analyze the system

We now have a consistent system, with no contradictory equations. The third row indicates that \( z = 0 \). Substituting into the second row, we find \( y = 0 \), and substituting into the first row, we find \( x = 0 \).

Thus, the only solution is the trivial solution: \( x = 0, y = 0, z = 0 \).

### Conclusion:

The system has **only the trivial solution**.

Therefore, the correct answer is:

**A. trivial solution**

---

Would you like more details on any step? Here are some related questions for further practice:

1. How would the solution change if the third equation were different?
2. Can you explain the significance of trivial vs. non-trivial solutions?
3. What is the geometric interpretation of this system of equations?
4. How would you determine if a system has infinitely many solutions?
5. What is the role of Gaussian elimination in solving systems of linear equations?

**Tip:** Always check if the matrix is consistent by reducing it fully and looking for contradictions!

Given system of linear equations: x - 4y + z = 0, 2x + y + 4z = 0, x + 4y + z = 0, have:

Explore how to solve a system of linear equations using Gaussian elimination. This problem involves analyzing consistency and finding the trivial solution for the given system. Suitable for students in Grades 10-12 or early college.

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