determine a and b for which the system has a unique solution, no solution, infinite solution
3x -2 y + z = b,
-5x - 8y + 9z = 3,
2x + y + az = -1

We have a system of three linear equations with three unknowns \(x\), \(y\), and \(z\):

\[
\begin{aligned}
1. \quad 3x - 2y + z &= b \\
2. \quad -5x - 8y + 9z &= 3 \\
3. \quad 2x + y + az &= -1
\end{aligned}
\]

To analyze when the system has a unique solution, no solution, or infinitely many solutions, we use the concept of the determinant of the coefficient matrix. The system will have:

- **A unique solution** if the determinant of the coefficient matrix is non-zero.
- **No solution** if the system is inconsistent (e.g., rows lead to contradictory equations).
- **Infinite solutions** if the determinant of the coefficient matrix is zero but the system is consistent.

### Step 1: Coefficient Matrix
The coefficient matrix of this system is:

\[
A = \begin{pmatrix}
3 & -2 & 1 \\
-5 & -8 & 9 \\
2 & 1 & a
\end{pmatrix}
\]

### Step 2: Determinant of the Coefficient Matrix
We calculate the determinant of \(A\). For a \(3 \times 3\) matrix:

\[
\det(A) = 3 \begin{vmatrix} -8 & 9 \\ 1 & a \end{vmatrix} - (-2) \begin{vmatrix} -5 & 9 \\ 2 & a \end{vmatrix} + 1 \begin{vmatrix} -5 &

Learn how to determine values of a and b that result in a unique solution, no solution, or infinite solutions for a system of linear equations. This detailed step-by-step guide explores concepts like determinants and matrix theory, perfect for Grades 10-12.

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