https://file.aiquickdraw.com/mathbot/file/1727361312843fqhyhfn3.jpg

Let's analyze both systems of equations from the image.

### System A:
\[
\begin{aligned}
-x + 3y - 9 &= 0 \quad \text{(Equation 1)} \\
x - 3y &= 9 \quad \text{(Equation 2)}
\end{aligned}
\]

**Step 1: Simplify both equations.**

- From Equation 1: \(-x + 3y = 9\).
- Equation 2 remains \(x - 3y = 9\).

**Step 2: Add the equations.**
\[
(-x + 3y) + (x - 3y) = 9 + 9
\]
This simplifies to:
\[
0 = 18
\]
Since this is a contradiction, it means **System A has no solution**.

### System B:
\[
\begin{aligned}
-x - 4y &= 8 \quad \text{(Equation 1)} \\
x - 4y &= 8 \quad \text{(Equation 2)}
\end{aligned}
\]

**Step 1: Add the two equations.**
\[
(-x - 4y) + (x - 4y) = 8 + 8
\]
This simplifies to:
\[
-8y = 16
\]
Solving for \(y\):
\[
y = -2
\]

**Step 2: Substitute \(y = -2\) into one of the original equations, say Equation 2:**
\[
x - 4(-2) = 8 \quad \Rightarrow \quad x + 8 = 8 \quad \Rightarrow \quad x = 0
\]

So, **System B has a unique solution**: \((x, y) = (0, -2)\).

### Summary:
- **System A** has no solution.
- **System B** has a unique solution: \((x, y) = (0, -2)\).

Let me know if you need any more details or explanations!

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Here are 5 follow-up questions based on this:

1. How can you tell if a system of equations has no solution by looking at the equations?
2. What does it mean when a system of equations has infinitely many solutions?
3. How can you use substitution or elimination to solve a system of equations?
4. What is the geometric interpretation of a system of linear equations having no solution?
5. Can two distinct linear equations ever have exactly two solutions?

**Tip:** When solving systems of equations, contradictions like \(0 = 18\) suggest no solution, while identities like \(0 = 0\) imply infinite solutions.

Determine the nature of solutions for the given systems of equations. Identify if the system has no solution, a unique solution, or infinitely many solutions, and find the specific values if applicable.

Explore how to determine if a system of linear equations has no solution, a unique solution, or infinitely many solutions. Using System A and System B as examples, this explanation covers algebraic simplification and the conditions for different types of solutions. Suitable for high school students (Grades 9-12).

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