https://file.aiquickdraw.com/mathbot/file/1726182541613djrcmzi6.jpg

Let's analyze the two systems of equations to determine the nature of their solutions.

### **System A:**
1. Equations:  
   \[
   \begin{cases}
   3x + y = 6 \\
   3x - y = 6
   \end{cases}
   \]

2. **Add the equations:**

   \[
   (3x + y) + (3x - y) = 6 + 6 \implies 6x = 12 \implies x = 2
   \]

3. **Substitute \(x = 2\) into one of the equations (e.g., \(3x + y = 6\)):**

   \[
   3(2) + y = 6 \implies 6 + y = 6 \implies y = 0
   \]

4. **Solution for System A:**  
   The system has a unique solution: \((x, y) = (2, 0)\).

### **System B:**
1. Equations:  
   \[
   \begin{cases}
   -4x - y + 4 = 0 \\
   4x + y = 4
   \end{cases}
   \]

2. **Simplify the equations:**  
   Rewrite both equations:

   - First equation:  
   \[
   -4x - y = -4 \implies 4x + y = 4
   \]
   - Second equation:  
   \[
   4x + y = 4
   \]

   Both equations are the same: \(4x + y = 4\).

3. **Conclusion for System B:**  
   Since the two equations are identical, there are infinitely many solutions. The solutions satisfy the equation:  
   \[
   y = -4x + 4
   \]

### **Final Answers:**
- **System A:** The system has a unique solution: \((x, y) = (2, 0)\).
- **System B:** The system has infinitely many solutions; they satisfy the equation \(y = -4x + 4\).

Would you like more details or have any questions?

### Related Questions:
1. How do you determine if a system of equations has no solution?
2. What methods can be used to solve systems of equations?
3. What does it mean for two lines to have infinitely many solutions?
4. Can there be a system of equations with exactly two solutions?
5. How do you graphically represent a system of equations?

**Tip:** When solving systems of equations, always check if the equations are dependent (identical or multiples of each other) or inconsistent (parallel but not identical).

Two systems of equations are given below. For each system, choose the best description of its solution. If applicable, give the solution.

Learn how to determine the nature of solutions for systems of linear equations. Explore examples where systems have unique solutions or infinitely many solutions, and understand the conditions for each case.

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