https://file.aiquickdraw.com/mathbot/file/1726499720601si19i5bn.jpg

Let's analyze the system of equations labeled as "System B" from the image:

\[
\begin{aligned}
1) \quad & x = 6 + 3y \\
2) \quad & -x + 3y = -6
\end{aligned}
\]

### Step 1: Substitute equation (1) into equation (2)
From the first equation, \( x = 6 + 3y \). Substitute this into the second equation:

\[
- (6 + 3y) + 3y = -6
\]

### Step 2: Simplify the equation
Distribute the negative sign and combine like terms:

\[
-6 - 3y + 3y = -6
\]

This simplifies to:

\[
-6 = -6
\]

### Step 3: Analyze the result
The equation \( -6 = -6 \) is always true, which means the system of equations is **dependent**, implying there are **infinitely many solutions**.

### Step 4: Solve for \( y \)
Now, express \( y \) in terms of a free variable. From equation (1), we know:

\[
x = 6 + 3y
\]

So, solving for \( y \):

\[
y = \frac{x - 6}{3}
\]

Thus, the general solution must satisfy the equation:

\[
y = \frac{x - 6}{3}
\]

### Conclusion
The correct answer is that the system has infinitely many solutions, and \( y \) must satisfy the equation \( y = \frac{x - 6}{3} \).

---

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

Here are 5 related questions to consider:
1. What is a dependent system of equations?
2. How can you solve systems of linear equations by substitution?
3. What are the differences between consistent and inconsistent systems?
4. How can you graphically interpret a system with infinitely many solutions?
5. How do you identify whether a system has no solution, one solution, or infinitely many?

**Tip:** In a dependent system, the two equations are essentially the same line, meaning all points on the line satisfy both equations.

The system of equations labeled as 'System B': x = 6 + 3y, -x + 3y = -6. Analyze the system and determine whether it has no solution, a unique solution, or infinitely many solutions.

Analyze and solve the system of equations x = 6 + 3y and -x + 3y = -6. This problem demonstrates the properties of dependent systems and provides a detailed solution with infinitely many answers. Suitable for Grades 9-11.

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