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Let's analyze the problem presented in the image and solve it step by step.

We are given a system of linear equations in the form of a matrix:

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

**Part a**: We need to determine for which value of \(a\) the system has no solutions, given that \(b \neq -1\).

### Condition for no solution:
A system of equations will have no solutions if the determinant of the coefficient matrix is zero, leading to inconsistent equations.

Let's compute the determinant of the matrix and solve for \(a\) under this condition.

Matrix:
\[
\begin{pmatrix}
3 & -1 \\
-3 & 3 + 8a \\
\end{pmatrix}
\]

### Determinant of this 2x2 matrix:
\[
\text{Det} = 3(3 + 8a) - (-1)(-3) = 9 + 24a - 3 = 6 + 24a
\]

For no solutions, the determinant must be zero:

\[
6 + 24a = 0 \implies a = -\frac{6}{24} = -\frac{1}{4}
\]

So, \(a = -\frac{1}{4}\) when the system has no solutions, provided that \(b \neq -1\).

**Part b**: The system has a unique solution for all values of \(a\) except one. This value of \(a\) corresponds to the case where the determinant is zero, which we already found in part (a). Hence, the value is:

\[
a = -\frac{1}{4}
\]

**Part c**: The system has infinitely many solutions when both the determinant is zero and the system is consistent. For consistency, the second equation must be a scalar multiple of the first.

We already know that for no solutions, \(a = -\frac{1}{4}\). Now, we need to check when \(b = -1\), since that would allow the system to have infinitely many solutions due to linear dependency.

Thus, for infinitely many solutions, we have:

\[
a = -\frac{1}{4}, b = -1
\]

### Final answers:
- **a)** \( a = -\frac{1}{4} \)
- **b)** \( a = -\frac{1}{4} \)
- **c)** \( a = -\frac{1}{4}, b = -1 \)

Let me know if you need further clarifications or have more questions! Here's a tip: always check for both determinant and consistency when solving linear systems.

#### Related Questions:
1. How do you calculate the determinant of a 2x2 matrix?
2. What does it mean for a system of equations to have no solution geometrically?
3. How do you find the condition for infinitely many solutions in a system of linear equations?
4. What role does the determinant play in determining the number of solutions?
5. How does changing a parameter in a matrix affect the solution of a system?

**Tip**: When solving linear systems, always try to simplify the matrix (if possible) using row operations to make the calculations easier.

Consider the system of linear equations given by the matrix: [3, -1, 2; -3, 3 + 8a, -6 - 4b]. Find: (a) the value of a for which the system has no solutions when b ≠ -1, (b) the value of a where the system has a unique solution for all except one value, (c) the values of a and b for which the system has infinitely many solutions.

Analyze a system of linear equations to find the values of parameters a and b that result in no solutions, a unique solution, or infinitely many solutions. This detailed problem covers linear algebra concepts like determinants, matrix theory, and theorems for systems of equations.

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