Answer: To determine the values of m and n for which the system of equations has no solution, a unique solution, or more than one solution, we can use the concept of determinants in linear algebra.
The given system of equations can be represented in matrix form as:
| 1 1 1 | | x | | 5 |
| 1 2 3 | * | y | = | 10 |
| 1 2 m | | z | | n |
The determinant of the coefficient matrix (the left-hand side) of the system is given by:
D = | 1 1 1 |
| 1 2 3 |
| 1 2 m |
The system has no solution when the determinant of the coefficient matrix is zero (D = 0). In this case, the equations represent parallel planes in three-dimensional space, and they do not intersect.
For a unique solution, the determinant of the coefficient matrix (D) should not be zero, and the determinant of the augmented matrix (which includes the constants on the right-hand side) should also not be zero.
The system has more than one solution when the determinant of the coefficient matrix (D) is zero, and the determinant of the augmented matrix is also zero.
Now, let's calculate the determinant of the coefficient matrix:
D = | 1 1 1 |
| 1 2 3 |
| 1 2 m |
D = (1)((2 * m) - (3 * 2)) - (1)((1 * m) - (3 * 1)) + (1)((1 * 2) - (2 * 1))
D = 2m - 6 - m + 3 + 2 - 2
D = m - 3
Now, we can analyze the values of m and n for each case:
No solution (D = 0):
This occurs when m - 3 = 0
Solving for m: m = 3
Unique solution (D ≠ 0):
This occurs when m - 3 ≠ 0
Solving for m: m ≠ 3
More than one solution (D = 0):
This occurs when m - 3 = 0 and n ≠ 5
Solving for m: m = 3
Solving for n: n ≠ 5
In summary:
- No solution: m = 3
- Unique solution: m ≠ 3
- More than one solution: m = 3 and n ≠ 5
Note: When m = 3 and n = 5, the system has no solution because the determinant is zero, but this case is already covered in the "No solution" category. For the "More than one solution" case, we consider all other values of n except n = 5.