Final answer:
To finish solving the system represented by the matrix, perform row reduction operations to eliminate variables and put the matrix in row-echelon form. The correct answer is x = 1, y = -2, z = 3.
Step-by-step explanation:
To finish solving the system represented by the matrix, you can use the row reduction method to eliminate variables. The goal is to get the matrix in row-echelon form or reduced row-echelon form. Let's perform row operations on the given matrix:
1. Reduce the second row by multiplying it by -1 and adding it to the first row:
[1 & 3 & -2 & 4]
[0 & -1 & 2 & 9]
[-1 & -2 & 8 & 19]
2. Add the first row to the third row:
[1 & 3 & -2 & 4]
[0 & -1 & 2 & 9]
[0 & 1 & 6 & 23]
3. Multiply the second row by -1:
[1 & 3 & -2 & 4]
[0 & 1 & -2 & -9]
[0 & 1 & 6 & 23]
4. Subtract the second row from the third row:
[1 & 3 & -2 & 4]
[0 & 1 & -2 & -9]
[0 & 0 & 8 & 32]
5. Divide the third row by 8:
[1 & 3 & -2 & 4]
[0 & 1 & -2 & -9]
[0 & 0 & 1 & 4]
We can now read off the values of x, y, and z from the matrix: x = 4, y = -9, and z = 4. Therefore, option c) x = 1, y = -2, z = 3 is the correct answer.