https://file.aiquickdraw.com/mathbot/file/17298691479625mv22mqt.jpg

The system of equations provided in the image is:

1. \(-3x - 9y + 9z = -18\)
2. \(x + 3y - 3z = 6\)
3. \(2x + 6y - 6z = 12\)

We need to determine if there is one solution, infinitely many solutions, or no solution.

### Step 1: Analyze the system of equations
First, observe that equation 2 and equation 3 are multiples of each other:
- Equation 2 is \(x + 3y - 3z = 6\).
- Equation 3 is \(2x + 6y - 6z = 12\), which is the same as equation 2 multiplied by 2.

This means that equations 2 and 3 are dependent and represent the same plane.

### Step 2: Simplify the system
Since equation 3 is not independent, we can disregard it and focus on the first two equations:
1. \(-3x - 9y + 9z = -18\)
2. \(x + 3y - 3z = 6\)

### Step 3: Solve the simplified system
Multiply equation 2 by 3 to align the coefficients with equation 1:
- \(3(x + 3y - 3z) = 18 \Rightarrow 3x + 9y - 9z = 18\)

Now we have:
1. \(-3x - 9y + 9z = -18\)
2. \(3x + 9y - 9z = 18\)

### Step 4: Add the equations
Adding the two equations:
\[
(-3x + 3x) + (-9y + 9y) + (9z - 9z) = -18 + 18
\]
This simplifies to:
\[
0 = 0
\]
This result indicates that the system is dependent and has infinitely many solutions.

### Conclusion:
The system has infinitely many solutions, which are represented by the set \(\{(x, y, z) \mid x + 3y - 3z = 6\}\).

Thus, the correct answer is **B**: "There are infinitely many solutions. The solution set is \(\{(x, y, z) \mid x + 3y - 3z = 6\}\)."

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Would you like more details on this solution or have any questions?

### Related Questions:
1. How do you determine if a system of equations is dependent?
2. What are the steps to check if a system has no solution?
3. How can you find the parametric form of the solutions for a system of equations?
4. What is the geometric interpretation of dependent equations in a system?
5. How would the system change if a fourth equation was added?

### Tip:
When two or more equations are multiples of each other, it often indicates the system is dependent, leading to infinitely many solutions.

Solve the system of equations: -3x - 9y + 9z = -18, x + 3y - 3z = 6, 2x + 6y - 6z = 12. Determine if there is one solution, infinitely many solutions, or no solution.

Explore how to solve a system of linear equations involving three variables where the system is dependent, leading to infinitely many solutions. This problem focuses on analyzing and simplifying the equations, suitable for Grades 9-12 students.

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