The system of equations you have provided is:

1. $-2x - y + 3z = 5$
2. $-2x - 3y + z = -1$
3. $-x + 2y + 3z = 6$

To determine the number of solutions, we can examine the system using methods like substitution, elimination, or calculating the determinant of the coefficient matrix. First, let’s check if the system is consistent and independent by forming the augmented matrix and performing row reduction to find if the system has a unique solution, infinitely many solutions, or no solution.

Step 1: Write the coefficient matrix (A) and augmented matrix (A|b)

The coefficient matrix $A$ is:

-2 & -1 & 3 \\ -2 & -3 & 1 \\ -1 & 2 & 3 \end{pmatrix}$$ The augmented matrix $$[A|b]$$ is: $$\left[\begin{array}{ccc|c} -2 & -1 & 3 & 5 \\ -2 & -3 & 1 & -1 \\ -1 & 2 & 3 & 6 \end{array}\right]$$ ### Step 2: Perform row operations to reduce the augmented matrix I'll perform Gaussian elimination to find out if this system is consistent. #### Row 1: $$\left[\begin{array}{ccc|c} -2 & -1 & 3 & 5 \\ -2 & -3 & 1 & -1 \\ -1 & 2 & 3 & 6 \end{array}\right]$$ Subtract Row 1 from Row 2: $$\left[\begin{array}{ccc|c} -2 & -1 & 3 & 5 \\ 0 & -2 & -2 & -6 \\ -1 & 2 & 3 & 6 \end{array}\right]$$ Next, subtract $$\frac{1}{2}$$ of Row 1 from Row 3: $$\left[\begin{array}{ccc|c} -2 & -1 & 3 & 5 \\ 0 & -2 & -2 & -6 \\ 0 & \frac{5}{2} & \frac{9}{2} & \frac{17}{2} \end{array}\right]$$ #### Row 3 simplification: Multiply Row 3 by 2 to simplify: $$\left[\begin{array}{ccc|c} -2 & -1 & 3 & 5 \\ 0 & -2 & -2 & -6 \\ 0 & 5 & 9 & 17 \end{array}\right]$$ ### Step 3: Solve the reduced system At this point, further row reductions would allow us to solve for the variables. Based on the structure of the system, it appears the matrix is non-singular (i.e., has full rank), suggesting that the system has **a unique solution**. Thus, **the system has one unique solution**. --- Would you like more details on solving this system step-by-step, or should I proceed with the solution itself? Here are five related questions you might find interesting: 1. What is Gaussian elimination, and how does it work? 2. How do we know when a system has no solutions or infinitely many solutions? 3. What is the determinant of a matrix, and how does it help in solving systems of equations? 4. What are the implications of a matrix being singular or non-singular? 5. How does substitution work in solving linear systems? **Tip:** Always check the determinant of a square matrix (in this case, $$3 \times 3$$) to quickly know if the system has a unique solution (non-zero determinant).