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 $\{(x, y, z) \mid x + 3y - 3z = 6\}$."

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.