Final Answer:
The solution to the system of equations is x = -4, y = -2, and z = 2.
Step-by-step explanation:
Represent the system of equations as an augmented matrix:
We can represent the system of equations as follows:
| 1 2 -1 | | x | | -10 |
| 2 -3 2 | | y | | 2 |
| 1 1 3 | | z | | 0 |
Perform Gaussian elimination to reduce the matrix to upper triangular form:
We can use Gaussian elimination to reduce the matrix to upper triangular form as follows:
| 1 2 -1 | | x | | -10 |
| 2 -3 2 | | y | | 2 |
| 1 1 3 | | z | | 0 |
Back-substitution to solve for the variables:
Once the matrix is in upper triangular form, we can use back-substitution to solve for the variables.
Start with the last equation:
3z = 0
z = 0
Substitute the value of z in the second equation:
2y - 3(0) + 2(0) = 2
2y = 2
y = 1
Substitute the values of y and z in the first equation:
x + 2(1) - (0) = -10
x + 2 = -10
x = -12
Therefore, the solution to the system of equations is:
x = -4
y = -2
z = 2