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Which of the following are correct in respect of the system of equations x + y + z =8, x -y +2z = 6 and 3x -y +5z = k?
1. They have no solution, if k = 15.
2. They have infinitely many solutions, if k = 20.
3. They have unique solution, if k = 25
Q. Select the correct answer using the code given below:
- a)1 and 2 only
- b)2 and 3 only
- c)1 and 3 only
- d)1,2 and 3
Correct answer is option 'A'. Can you explain this answer?
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Which of the following are correct in respect of the system of equatio...
( ∵ system is inconsistent i.e., it has no solution)
( ∵ system has infinitely many solutions)
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Which of the following are correct in respect of the system of equatio...
Question Analysis:
We are given a system of equations and we need to determine the conditions under which the system has no solution, infinitely many solutions, or a unique solution. The system is given by:
x + y + z = 8 (Equation 1)
x - y + 2z = 6 (Equation 2)
3x - y + 5z = k (Equation 3)
Answer Explanation:
To determine the conditions under which the system has no solution, infinitely many solutions, or a unique solution, we can use the concept of matrix equations and determinants.
1. Determining No Solution:
If the system has no solution, it means that the equations are inconsistent and cannot be satisfied simultaneously. In matrix form, the system can be represented as:
[A] [X] = [B]
where [A] is the coefficient matrix, [X] is the variable matrix, and [B] is the constant matrix.
To determine if the system has no solution, we need to check if the determinant of the coefficient matrix [A] is zero. If the determinant is zero, then the system has no solution.
2. Determining Infinitely Many Solutions:
If the system has infinitely many solutions, it means that the equations are dependent and represent the same line or plane in space. In matrix form, the system can be represented as:
[A] [X] = [B]
To determine if the system has infinitely many solutions, we need to check if the determinant of the coefficient matrix [A] is zero and the determinant of the augmented matrix [A|B] is also zero. If both determinants are zero, then the system has infinitely many solutions.
3. Determining Unique Solution:
If the system has a unique solution, it means that the equations are consistent and represent distinct lines or planes in space. In matrix form, the system can be represented as:
[A] [X] = [B]
To determine if the system has a unique solution, we need to check if the determinant of the coefficient matrix [A] is non-zero. If the determinant is non-zero, then the system has a unique solution.
Applying the Method to the Given System:
Let's apply the above method to the given system of equations.
The coefficient matrix [A] can be written as:
| 1 1 1 |
| 1 -1 2 |
| 3 -1 5 |
The determinant of the coefficient matrix [A] is given by:
| 1 1 1 |
| 1 -1 2 |
| 3 -1 5 | = 1(-1)(5) + 1(2)(3) + 1(1)(-1) - 1(-1)(1) - 1(2)(3) - 1(1)(5) = -5 + 6 - 1 + 1 + 6 - 5 = 2
1. Determining No Solution:
To check if the system has no solution, we need to check if the determinant of the coefficient matrix [A] is zero. In this case, the determinant is 2, which is non-zero. Therefore, the system does not have no solution
We are given a system of equations and we need to determine the conditions under which the system has no solution, infinitely many solutions, or a unique solution. The system is given by:
x + y + z = 8 (Equation 1)
x - y + 2z = 6 (Equation 2)
3x - y + 5z = k (Equation 3)
Answer Explanation:
To determine the conditions under which the system has no solution, infinitely many solutions, or a unique solution, we can use the concept of matrix equations and determinants.
1. Determining No Solution:
If the system has no solution, it means that the equations are inconsistent and cannot be satisfied simultaneously. In matrix form, the system can be represented as:
[A] [X] = [B]
where [A] is the coefficient matrix, [X] is the variable matrix, and [B] is the constant matrix.
To determine if the system has no solution, we need to check if the determinant of the coefficient matrix [A] is zero. If the determinant is zero, then the system has no solution.
2. Determining Infinitely Many Solutions:
If the system has infinitely many solutions, it means that the equations are dependent and represent the same line or plane in space. In matrix form, the system can be represented as:
[A] [X] = [B]
To determine if the system has infinitely many solutions, we need to check if the determinant of the coefficient matrix [A] is zero and the determinant of the augmented matrix [A|B] is also zero. If both determinants are zero, then the system has infinitely many solutions.
3. Determining Unique Solution:
If the system has a unique solution, it means that the equations are consistent and represent distinct lines or planes in space. In matrix form, the system can be represented as:
[A] [X] = [B]
To determine if the system has a unique solution, we need to check if the determinant of the coefficient matrix [A] is non-zero. If the determinant is non-zero, then the system has a unique solution.
Applying the Method to the Given System:
Let's apply the above method to the given system of equations.
The coefficient matrix [A] can be written as:
| 1 1 1 |
| 1 -1 2 |
| 3 -1 5 |
The determinant of the coefficient matrix [A] is given by:
| 1 1 1 |
| 1 -1 2 |
| 3 -1 5 | = 1(-1)(5) + 1(2)(3) + 1(1)(-1) - 1(-1)(1) - 1(2)(3) - 1(1)(5) = -5 + 6 - 1 + 1 + 6 - 5 = 2
1. Determining No Solution:
To check if the system has no solution, we need to check if the determinant of the coefficient matrix [A] is zero. In this case, the determinant is 2, which is non-zero. Therefore, the system does not have no solution
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Which of the following are correct in respect of the system of equations x + y + z =8, x -y +2z = 6 and 3x -y +5z = k?1. They have no solution, if k = 15.2. They have infinitely many solutions, if k = 20.3. They have unique solution, if k = 25Q. Select the correct answer using the code given below:a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1,2 and 3Correct answer is option 'A'. Can you explain this answer?
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