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The system of linear equations provided is:

\[
\begin{aligned}
4x + y + 6z &= 7 \\
3x + 3y + 2z &= 17 \\
-x - y + z &= -9
\end{aligned}
\]

To determine the number of solutions, we can analyze the system by attempting to reduce it using methods such as substitution, elimination, or Gaussian elimination.

### Step 1: Write the system in matrix form
We can represent the system as an augmented matrix:

\[
\begin{pmatrix}
4 & 1 & 6 & | & 7 \\
3 & 3 & 2 & | & 17 \\
-1 & -1 & 1 & | & -9
\end{pmatrix}
\]

### Step 2: Row reduction (Gaussian elimination)
I will now perform row operations to reduce the matrix to its row echelon form and determine if the system has no solution, one solution, or infinitely many solutions. Let me calculate this for you.The matrix ranks of both the coefficient matrix \(A\) and the augmented matrix are equal and both are 3, meaning that the system is consistent and has a unique solution.

Thus, the system of equations has **exactly one solution**.

Would you like to see the detailed steps for solving this system or have any other questions? Here are 5 related questions to consider:

1. How do we use Gaussian elimination to solve a system of equations?
2. What does the rank of a matrix tell us about the number of solutions?
3. How do we interpret the difference between consistent and inconsistent systems?
4. Can this system be solved using other methods, such as substitution or elimination?
5. How does the determinant of a matrix help in determining the existence of solutions?

**Tip:** For any system of equations, if the number of variables equals the rank of the coefficient matrix and the augmented matrix, the system has a unique solution.

Identify the number of solutions of the system of linear equations:
4x + y + 6z = 7
3x + 3y + 2z = 17
-x - y + z = -9

Learn how to solve a system of linear equations using matrix methods like Gaussian elimination. This problem helps students understand matrix rank, consistency of systems, and the criteria for unique, no, or infinite solutions.

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