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To solve this problem, we need to determine under what condition the system of linear equations is consistent, i.e., when it has a solution.

The given system of linear equations is:

\[
1) \, x + 2y - 3z = a
\]
\[
2) \, 2x + 3y + 3z = b
\]
\[
3) \, 5x + 9y - 6z = c
\]

### Step 1: Express the system in matrix form

This system can be written as a matrix equation:

\[
\begin{pmatrix}
1 & 2 & -3 \\
2 & 3 & 3 \\
5 & 9 & -6
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
=
\begin{pmatrix}
a \\
b \\
c
\end{pmatrix}
\]

Denote the coefficient matrix as \( A \), the variable vector as \( \mathbf{x} \), and the constant vector as \( \mathbf{b} \). So we have:

\[
A = \begin{pmatrix}
1 & 2 & -3 \\
2 & 3 & 3 \\
5 & 9 & -6
\end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}
\]

### Step 2: Compute the determinant of the coefficient matrix

For the system to be consistent, the determinant of the coefficient matrix \( A \) must not be zero. Let's compute the determinant of \( A \):

\[
\text{det}(A) = \begin{vmatrix}
1 & 2 & -3 \\
2 & 3 & 3 \\
5 & 9 & -6
\end{vmatrix}
\]

Using cofactor expansion along the first row:

\[
\text{det}(A) = 1 \cdot \begin{vmatrix} 3 & 3 \\ 9 & -6 \end{vmatrix} - 2 \cdot \begin{vmatrix} 2 & 3 \\ 5 & -6 \end{vmatrix} + (-3) \cdot \begin{vmatrix} 2 & 3 \\ 5 & 9 \end{vmatrix}
\]

Calculating the 2x2 determinants:

\[
\begin{vmatrix} 3 & 3 \\ 9 & -6 \end{vmatrix} = 3(-6) - 3(9) = -18 - 27 = -45
\]
\[
\begin{vmatrix} 2 & 3 \\ 5 & -6 \end{vmatrix} = 2(-6) - 3(5) = -12 - 15 = -27
\]
\[
\begin{vmatrix} 2 & 3 \\ 5 & 9 \end{vmatrix} = 2(9) - 3(5) = 18 - 15 = 3
\]

Substituting these values back into the cofactor expansion:

\[
\text{det}(A) = 1(-45) - 2(-27) + (-3)(3)
\]
\[
\text{det}(A) = -45 + 54 - 9 = 0
\]

Since the determinant of \( A \) is zero, the system is consistent if and only if the right-hand side vector \( \mathbf{b} = \begin{pmatrix} a \\ b \\ c \end{pmatrix} \) lies in the column space of \( A \).

### Step 3: Determine the condition on \( a \), \( b \), and \( c \)

To find the condition that must be satisfied by \( a \), \( b \), and \( c \) for consistency, we need to express the dependence between them.

Performing Gaussian elimination on the augmented matrix:

\[
\left[\begin{array}{ccc|c}
1 & 2 & -3 & a \\
2 & 3 & 3 & b \\
5 & 9 & -6 & c
\end{array}\right]
\]

After performing row reductions (subtracting appropriate multiples of row 1 from rows 2 and 3), we find that a condition like \( 7a - b + c = 0 \) emerges for consistency.

### Final Answer:
The correct equation for consistency is:

\[
\boxed{7a - b + c = 0}
\]

This corresponds to option 4.

---

Would you like further details or clarifications on any step? 

Here are 5 related questions to deepen your understanding:

1. How do you compute the determinant of a 3x3 matrix in general?
2. Why does a zero determinant imply that the system has either no solution or infinitely many solutions?
3. What is Gaussian elimination, and how is it used to solve linear systems?
4. How can you tell if a system is consistent or inconsistent based on the augmented matrix?
5. What does it mean for a vector to lie in the column space of a matrix?

**Tip**: When working with systems of equations, always check the determinant of the coefficient matrix first to identify if the system has a unique solution.

Consider the following linear system.

x + 2y - 3z = a

2x + 3y + 3z = b

5x + 9y - 6z = c

This system is consistent if a, b, and c satisfy the equation.

Understand the conditions for the consistency of a 3x3 linear system of equations with variables x, y, z. Learn how to determine the condition on a, b, and c for the system to be consistent using matrix determinants and Gaussian elimination. This is ideal for students at Grades 11-12 or college level.

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