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The system of equations presented appears to be:

\[
\begin{pmatrix}
3 & -1 & | & 2 \\
-3 & 3 + 8a & | & -6 - 4b
\end{pmatrix}
\]

The question asks for:

### Part a:
Given \( b \neq -1 \), for which value of \( a \) does the system have no solutions?

### Part b:
For which value of \( a \) does the system have a unique solution for all except one value of \( a \)?

### Part c:
For which values of \( a \) and \( b \) does this system have infinitely many solutions?

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Let's approach each part by analyzing the augmented matrix and checking conditions for no solutions, unique solutions, and infinitely many solutions based on the determinant and row reductions.

I'll solve this step by step.

Consider the system: Given that b ≠ -1, for which value of the parameter a does the system have no solutions? This system has a unique solution for all except one value of a. What is this value of a? For which values of a, b does this system have infinitely many solutions?

Learn how to determine when a system of linear equations has no solutions, a unique solution, or infinitely many solutions. This problem involves analyzing an augmented matrix and applying concepts such as row reduction and determinants. Suitable for advanced high school or undergraduate students studying linear algebra.

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