ACT Exam > ACT Questions > The value of k, for which the following syste...
Start Learning for Free
The value of k, for which the following system of linear equations has a non-trivial solution.
x + 2y - 3z = 0
2x + y + z = 0
x - y + kz = 0
- a)4
- b)2
- c)3
- d)-4
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
The value of k, for which the following system of linear equations has...
Concept:
Consider the system of m linear equations
a11 x1 + a12 x2 + … + a1n xn = 0
a21 x1 + a22 x2 + … + a2n xn = 0
…
am1 x1 + am2 x2 + … + amn xn = 0
The above equations containing the n unknowns x1, x2, …, xn. To determine whether the above system of equations is consistent or not, we need to find the rank of the following matrix.
A is the coefficient matrix of the given system of equations.
- The system of homogeneous equations has a unique solution (trivial solution) if and only if the determinant of A is non-zero.
- The system of homogeneous equations has a Non - trivial solution if and only if the determinant of A is zero.
Calculation:
Given:
x + 2y – 3z = 0
2x + y + z = 0
x – y + kz = 0
For non-trivial solution the determinant should be zero
∴ 1(k + 1) – 2(2k - 1) – 3(-2 - 1) = 0
∴ k + 1 – 4k + 2 + 9 = 0
∴ 12 = 3k
∴ k = 4
Free Test
FREE
| Start Free Test |
Community Answer
The value of k, for which the following system of linear equations has...
To find the value of k for which the given system of linear equations has a non-trivial solution, we need to analyze the coefficients of the variables in each equation and determine the conditions for a non-trivial solution.
The given system of equations is:
1) x + 2y - 3z = 0
2) 2x + y + z = 0
3) x - y + kz = 0
In order for the system to have a non-trivial solution, there must be dependence between the equations, meaning that one equation can be obtained as a linear combination of the other two.
Let's analyze the coefficients of the variables in the system:
1) The coefficient of x in equation 1 is 1.
2) The coefficient of x in equation 2 is 2.
3) The coefficient of x in equation 3 is 1.
Since the coefficients of x in the three equations are different, there is no linear combination that can eliminate the variable x. Therefore, the value of k does not affect the existence of a non-trivial solution.
Next, let's analyze the coefficients of y in the system:
1) The coefficient of y in equation 1 is 2.
2) The coefficient of y in equation 2 is 1.
3) The coefficient of y in equation 3 is -1.
The coefficients of y in equations 1 and 2 are different, but we can eliminate the variable y by adding equation 1 and equation 3. This gives us:
(x + 2y - 3z) + (x - y + kz) = 0
2x + (y - y) + (k - 3)z = 0
2x + (k - 3)z = 0
Now, let's analyze the coefficients of z in the system:
1) The coefficient of z in equation 1 is -3.
2) The coefficient of z in equation 2 is 1.
3) The coefficient of z in equation 3 is k.
If we want to eliminate the variable z, we need the coefficients in equations 1 and 2 to be equal. Therefore, we have the condition:
-3 = 1
This condition is not satisfied, so we cannot eliminate the variable z.
Therefore, the value of k does not affect the existence of a non-trivial solution. Hence, the correct answer is option A) 4.
The given system of equations is:
1) x + 2y - 3z = 0
2) 2x + y + z = 0
3) x - y + kz = 0
In order for the system to have a non-trivial solution, there must be dependence between the equations, meaning that one equation can be obtained as a linear combination of the other two.
Let's analyze the coefficients of the variables in the system:
1) The coefficient of x in equation 1 is 1.
2) The coefficient of x in equation 2 is 2.
3) The coefficient of x in equation 3 is 1.
Since the coefficients of x in the three equations are different, there is no linear combination that can eliminate the variable x. Therefore, the value of k does not affect the existence of a non-trivial solution.
Next, let's analyze the coefficients of y in the system:
1) The coefficient of y in equation 1 is 2.
2) The coefficient of y in equation 2 is 1.
3) The coefficient of y in equation 3 is -1.
The coefficients of y in equations 1 and 2 are different, but we can eliminate the variable y by adding equation 1 and equation 3. This gives us:
(x + 2y - 3z) + (x - y + kz) = 0
2x + (y - y) + (k - 3)z = 0
2x + (k - 3)z = 0
Now, let's analyze the coefficients of z in the system:
1) The coefficient of z in equation 1 is -3.
2) The coefficient of z in equation 2 is 1.
3) The coefficient of z in equation 3 is k.
If we want to eliminate the variable z, we need the coefficients in equations 1 and 2 to be equal. Therefore, we have the condition:
-3 = 1
This condition is not satisfied, so we cannot eliminate the variable z.
Therefore, the value of k does not affect the existence of a non-trivial solution. Hence, the correct answer is option A) 4.
|
Explore Courses for ACT exam
|
|
Top Courses for ACTView all
The value of k, for which the following system of linear equations has a non-trivial solution.x + 2y - 3z = 02x + y + z = 0x - y + kz = 0a)4b)2c)3d)-4Correct answer is option 'A'. Can you explain this answer?
Question Description
The value of k, for which the following system of linear equations has a non-trivial solution.x + 2y - 3z = 02x + y + z = 0x - y + kz = 0a)4b)2c)3d)-4Correct answer is option 'A'. Can you explain this answer? for ACT 2025 is part of ACT preparation. The Question and answers have been prepared according to the ACT exam syllabus. Information about The value of k, for which the following system of linear equations has a non-trivial solution.x + 2y - 3z = 02x + y + z = 0x - y + kz = 0a)4b)2c)3d)-4Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for ACT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The value of k, for which the following system of linear equations has a non-trivial solution.x + 2y - 3z = 02x + y + z = 0x - y + kz = 0a)4b)2c)3d)-4Correct answer is option 'A'. Can you explain this answer?.
The value of k, for which the following system of linear equations has a non-trivial solution.x + 2y - 3z = 02x + y + z = 0x - y + kz = 0a)4b)2c)3d)-4Correct answer is option 'A'. Can you explain this answer? for ACT 2025 is part of ACT preparation. The Question and answers have been prepared according to the ACT exam syllabus. Information about The value of k, for which the following system of linear equations has a non-trivial solution.x + 2y - 3z = 02x + y + z = 0x - y + kz = 0a)4b)2c)3d)-4Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for ACT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The value of k, for which the following system of linear equations has a non-trivial solution.x + 2y - 3z = 02x + y + z = 0x - y + kz = 0a)4b)2c)3d)-4Correct answer is option 'A'. Can you explain this answer?.
Solutions for The value of k, for which the following system of linear equations has a non-trivial solution.x + 2y - 3z = 02x + y + z = 0x - y + kz = 0a)4b)2c)3d)-4Correct answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for ACT.
Download more important topics, notes, lectures and mock test series for ACT Exam by signing up for free.
Here you can find the meaning of The value of k, for which the following system of linear equations has a non-trivial solution.x + 2y - 3z = 02x + y + z = 0x - y + kz = 0a)4b)2c)3d)-4Correct answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The value of k, for which the following system of linear equations has a non-trivial solution.x + 2y - 3z = 02x + y + z = 0x - y + kz = 0a)4b)2c)3d)-4Correct answer is option 'A'. Can you explain this answer?, a detailed solution for The value of k, for which the following system of linear equations has a non-trivial solution.x + 2y - 3z = 02x + y + z = 0x - y + kz = 0a)4b)2c)3d)-4Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of The value of k, for which the following system of linear equations has a non-trivial solution.x + 2y - 3z = 02x + y + z = 0x - y + kz = 0a)4b)2c)3d)-4Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The value of k, for which the following system of linear equations has a non-trivial solution.x + 2y - 3z = 02x + y + z = 0x - y + kz = 0a)4b)2c)3d)-4Correct answer is option 'A'. Can you explain this answer? tests, examples and also practice ACT tests.
|
|
Explore Courses for ACT exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup