To analyze the consistency and solution type of the given systems, I'll work through each system provided in the image.

System a: $3x + 2y - 5z = 4, x + y + 2z = 1, 5x + 3y - 8z = 6$

1. Matrix Formulation:

   3 & 2 & -5 \\ 1 & 1 & 2 \\ 5 & 3 & -8 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 4 \\ 1 \\ 6 \end{bmatrix}$$

2. Augmented Matrix:

   3 & 2 & -5 & | & 4 \\ 1 & 1 & 2 & | & 1 \\ 5 & 3 & -8 & | & 6 \end{bmatrix}$$

3. Row Reduction:

   $R_2 \leftarrow R_2 - \frac{1}{3}R_1$ $R_3 \leftarrow R_3 - \frac{5}{3}R_1$

   $R_3 \leftarrow R_3 - k \times R_2$ (where $k$ makes the coefficient of $y$ or $z$ in $R_3$ zero)

4. Checking Consistency: After simplifying, check for contradictions such as $0 = \text{non-zero}$ in the last row.

System b: Augmented Matrix $\begin{bmatrix} 1 & -2 & 1 & 0 \\ 0 & 2 & 8 & -8 \\ -1 & 1 & -1 & -1 \end{bmatrix}$

1. Row Reduction:

   $R_3 \leftarrow R_3 + R_1$

   The row operations aim to achieve zeros in the lower left corner below the main diagonal.

2. Checking Consistency: Consistency is indicated by the lack of contradictory rows such as $0 = \text{non-zero}$.

3. Solution Type: A row of all zeros in the system matrix (not the augmented part) indicates a dependent system, potentially with infinitely many solutions.

System c: $x + 9y - z = -27, x - 8y + 16z = 10, 2x + y + 15z = 37$

1. Matrix Formulation: 1 & 9 & -1 \\ 1 & -8 & 16 \\ 2 & 1 & 15 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -27 \\ 10 \\ 37 \end{bmatrix}$$
2. Augmented Matrix: 1 & 9 & -1 & | & -27 \\ 1 & -8 & 16 & | & 10 \\ 2 & 1 & 15 & | & 37 \end{bmatrix}$$
3. Row Reduction: Perform similar operations as before to eliminate $x$ and $y$ from $R_2$ and $R_3$, checking the final form for contradictions.

I'll compute the row reduction steps to determine the consistency and type of solutions for each system.

Please wait a moment while I calculate.