https://r2.erweima.ai/mathbot/image/1725699210178e3zqlvvx.jpg

The image provided contains a system of linear equations. Here's the system extracted from the image:

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

The task seems to be solving this system of equations to find the values of \( x \), \( y \), and \( z \), and checking whether the system has a unique solution.

Let's solve this system step-by-step:

1. From the third equation: 
   \[
   4z = 0 \implies z = 0
   \]

2. Substitute \( z = 0 \) into the second equation: 
   \[
   y - 0 = 0 \implies y = 0
   \]

3. Substitute \( y = 0 \) and \( z = 0 \) into the first equation: 
   \[
   x + 3(0) + 0 = 0 \implies x = 0
   \]

Thus, the solution to the system is \( x = 0 \), \( y = 0 \), and \( z = 0 \). This is a unique solution, indicating that the system has a single solution, which is the trivial solution.

Would you like more details on any specific part of the solution? Here are some related questions:

1. What does it mean for a system of equations to have a unique solution?
2. How can you use substitution to solve systems of equations?
3. What are the possible solution types for a system of linear equations (e.g., unique, infinite, or none)?
4. Can the system be solved using matrices and Gaussian elimination?
5. How can the concept of linear independence be applied to this system?

**Tip**: When solving systems of equations, always try to reduce the system step by step, starting from the simplest equation.

Tentukan apakah sistem persamaan linear berikut memiliki solusi unik: x + 3y + z = 0, y - z = 0, 4z = 0. Jelaskan jawaban Anda.

Learn how to solve the system of linear equations x + 3y + z = 0, y - z = 0, 4z = 0 with step-by-step reasoning. Understand the method of substitution and check for unique solutions in systems of equations. Suitable for Grades 10-12.

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