Final answer:
To solve the system of equations using matrices, rewrite the system as an augmented matrix, use row operations to simplify, and find the ordered triple that satisfies all the equations.
Step-by-step explanation:
To solve the system of equations using a matrix, we can write the system as an augmented matrix and then use row operations to reduce it to row-echelon form.
The system of equations is:
1) -x - 7y - z = -19
2) 4y + 4z = 4
3) 2x + y + 6z = 7
We can rewrite this system as an augmented matrix:
| -1 -7 -1 | -19 |
| 0 4 4 | 4 |
| 2 1 6 | 7 |
Next, we perform row operations to reduce the matrix into reduced row-echelon form:
R1 → R1
R2 → R2 / 4
R3 → R3 + 0.5 * R1
After simplifying, we should end up with a matrix that allows us to determine the values of x, y, and z directly.
The solution to the system will be an ordered triple of the form (x, y, z). The possible answers are given as options a) (-3, 2, 5), b) (4, 0, 1), c) (2, -3, 4), and d) (1, 5, -2).
After performing all the row operations correctly, we can find the correct ordered triple that satisfies all three equations in the system.