Question

The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. 2y + z = 8x + y = 83x + 3y - z = 16

Answer 1

To find the solution to the system of linear equations using Gaussian elimination, we can use a matrix representation and perform row operations to transform the coefficient matrix into row-echelon form. However, in this case, the system is inconsistent and has no unique solution.

Step-by-step explanation:

To find the solution to the system of linear equations using Gaussian elimination, we can use a matrix representation and perform row operations to transform the coefficient matrix into row-echelon form. Here are the steps:

1. Write the system of equations in matrix form:
2. [2, 1, 1 | 8]
3. [0, 1, -1 | 8]
4. [3, 3, -1 | 16]
5. Perform row operations to make leading entries (pivot elements) equal to 1 and eliminate the other entries below the pivots:
6. Row 1: 1/2 * (Row 1)
7. Row 3: -3 * (Row 1) + Row 3
8. [1, 1/2, 1/2 | 4]
9. [0, 1, -1 | 8]
10. [0, 0, -7/2 | 4]
11. Perform row operations to eliminate the entries above the pivots:
12. Row 1: Row 2 - Row 1
13. [1, 0, 3/2 | 4]
14. [0, 1, -1 | 8]
15. [0, 0, -7/2 | 4]
16. Perform row operations to make the entries below the pivots equal to 0:
17. Row 3: -7/2 * (Row 2) + Row 3
18. [1, 0, 3/2 | 4]
19. [0, 1, -1 | 8]
20. [0, 0, 0 | 0]

The row-echelon form of the coefficient matrix is:

[1, 0, 3/2]

[0, 1, -1]

[0, 0, 0]

Since there is a row of zeros in the augmented matrix, the system of equations is inconsistent and has no unique solution.