https://file.aiquickdraw.com/mathbot/file/1728693076560m560j1v8.jpg

The system of equations presented in your image is:

1. \( 3x + 3y - z = 0 \)
2. \( 6x + 3y + 6z = 0 \)
3. \( 3x + 9y - 17z = 0 \)

Let me solve this system step by step.

### Step 1: Eliminate one variable
We can try to eliminate one variable (say \(x\)) by subtracting or adding equations.

#### Multiply the first equation by 2 to align it with the second equation:
\[
2(3x + 3y - z) = 6x + 6y - 2z = 0
\]
So the modified first equation is:
\[
6x + 6y - 2z = 0
\]
Now subtract the second equation from this:
\[
(6x + 6y - 2z) - (6x + 3y + 6z) = 0
\]
Simplifying:
\[
6y - 8z = 0
\]
or:
\[
y = \frac{4z}{3}
\]

### Step 2: Substitute \( y = \frac{4z}{3} \) into other equations
We now substitute this value of \(y\) into the first and third equations to simplify the system.

#### First equation:
\[
3x + 3y - z = 0
\]
Substitute \(y = \frac{4z}{3}\):
\[
3x + 3\left(\frac{4z}{3}\right) - z = 0
\]
Simplifying:
\[
3x + 4z - z = 0 \quad \Rightarrow \quad 3x + 3z = 0 \quad \Rightarrow \quad x = -z
\]

#### Third equation:
\[
3x + 9y - 17z = 0
\]
Substitute \(x = -z\) and \(y = \frac{4z}{3}\):
\[
3(-z) + 9\left(\frac{4z}{3}\right) - 17z = 0
\]
Simplifying:
\[
-3z + 12z - 17z = 0 \quad \Rightarrow \quad -8z = 0
\]
This implies \(z = 0\).

### Step 3: Find the values of \(x\) and \(y\)
Since \(z = 0\):
- From \(x = -z\), we have \(x = 0\).
- From \(y = \frac{4z}{3}\), we have \(y = 0\).

### Final Solution:
\[
x = 0, \quad y = 0, \quad z = 0
\]

This system has the trivial solution, where all variables are equal to zero.

Would you like more details, or do you have any questions?

Here are some related questions:
1. How can we identify if a system has only the trivial solution?
2. Can there be non-trivial solutions to a homogeneous system of linear equations?
3. What are the different methods to solve a system of linear equations (e.g., substitution, elimination, matrix method)?
4. How does the determinant of a matrix help in determining the solution of a linear system?
5. What role does linear dependence/independence play in determining the number of solutions?

**Tip**: When working with systems of equations, always check for consistency (if the system has a solution) and whether the solution is unique or not by analyzing the equations carefully.

Solve the system of equations: 3x + 3y - z = 0, 6x + 3y + 6z = 0, 3x + 9y - 17z = 0

Learn how to solve the system of equations: 3x + 3y - z = 0, 6x + 3y + 6z = 0, 3x + 9y - 17z = 0, using the elimination and substitution methods. This solution is ideal for students in Grades 9-12.

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