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The system of equation kx + y + z = 1,
x + ky + z = k and x = y + kz = k2 has nosolution if k equals
x + ky + z = k and x = y + kz = k2 has nosolution if k equals
- a)0
- b)1
- c)-1
- d)-2
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The system of equation kx + y + z = 1,x + ky + z = k and x = y + kz = ...
⟹ k(k2 − 1) − (k − 1) + (1 − k) = 0
⟹ k(k+1)−1−1] = 0 ⟹ k2 +k−2 = 0
⟹ 1, −2
For k =1, first two equations will become same. ⟹ k = −2
Most Upvoted Answer
The system of equation kx + y + z = 1,x + ky + z = k and x = y + kz = ...
Understanding the System of Equations
The given system of equations is:
1. kx + y + z = 1
2. x + ky + z = k
3. x = y + kz = k²
We want to find the value of k for which this system has no solution.
Condition for No Solution
A system of linear equations has no solution if it is inconsistent. This typically occurs when the equations represent parallel lines or planes that do not intersect.
Analyzing the Equations
- From the third equation, we can express y and z in terms of x:
y = x - kz
z = (k² - x) / k.
- Substituting y and z into the first two equations helps us simplify and analyze the system.
Substituting Values
1. Substitute y and z into the first equation:
kx + (x - kz) + (k² - x) / k = 1.
2. Substitute y and z into the second equation:
x + k(x - kz) + (k² - x) / k = k.
These substitutions can be expanded and simplified to find conditions for k.
Finding the Value of k
Through the simplification process, you will arrive at a situation where the coefficients of x and the constants lead to a contradiction for a specific value of k.
Conclusion
After solving the equations, you will find that k = -2 causes the system to become inconsistent, leading to no solution. Thus, the correct answer is:
- d) -2
The given system of equations is:
1. kx + y + z = 1
2. x + ky + z = k
3. x = y + kz = k²
We want to find the value of k for which this system has no solution.
Condition for No Solution
A system of linear equations has no solution if it is inconsistent. This typically occurs when the equations represent parallel lines or planes that do not intersect.
Analyzing the Equations
- From the third equation, we can express y and z in terms of x:
y = x - kz
z = (k² - x) / k.
- Substituting y and z into the first two equations helps us simplify and analyze the system.
Substituting Values
1. Substitute y and z into the first equation:
kx + (x - kz) + (k² - x) / k = 1.
2. Substitute y and z into the second equation:
x + k(x - kz) + (k² - x) / k = k.
These substitutions can be expanded and simplified to find conditions for k.
Finding the Value of k
Through the simplification process, you will arrive at a situation where the coefficients of x and the constants lead to a contradiction for a specific value of k.
Conclusion
After solving the equations, you will find that k = -2 causes the system to become inconsistent, leading to no solution. Thus, the correct answer is:
- d) -2
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The system of equation kx + y + z = 1,x + ky + z = k and x = y + kz = k2 has nosolution if k equalsa)0b)1c)-1d)-2Correct answer is option 'D'. Can you explain this answer?
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Solutions for The system of equation kx + y + z = 1,x + ky + z = k and x = y + kz = k2 has nosolution if k equalsa)0b)1c)-1d)-2Correct answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for Defence.
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