Mathematics Exam > Mathematics Questions > If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z =...
Start Learning for Free
If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y + 4z = 0 be a system of equations, then
- a)it is inconsistent
- b)it has only trivial solution
- c)it can be reduced to a single equation and so a solution does not exist
- d)the determinant of the matrix of coefficient is zero
Correct answer is option 'B'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z = 0 be a system of equatio...
Given:
The system of equations is:
1) 3x + 2y + z = 0
2) x + 4y + z = 0
3) 2x + y + 4z = 0
To find:
The nature of the system of equations.
Solution:
To determine the nature of the system of equations, we need to consider the coefficient matrix and solve it using the determinant method.
Determinant Method:
The system of equations can be written in matrix form as AX = 0, where A is the coefficient matrix and X is the column matrix of variables (x, y, z).
The coefficient matrix A is given by:
A = |3 2 1|
|1 4 1|
|2 1 4|
The determinant of the coefficient matrix is calculated as follows:
|A| = 3(4*4 - 1*1) - 2(1*4 - 2*1) + 1(1*1 - 2*4)
= 3(16 - 1) - 2(4 - 2) + 1(1 - 8)
= 3(15) - 2(2) + 1(-7)
= 45 - 4 - 7
= 34
Analysis:
If the determinant of the coefficient matrix (|A|) is zero, then the system of equations has either no solution or infinitely many solutions. If |A| is non-zero, then the system has a unique solution.
Conclusion:
Since the determinant of the coefficient matrix is non-zero (34 ≠ 0), the system of equations has a unique solution. Therefore, the correct answer is option 'B' - the system of equations has only a trivial solution.
The system of equations is:
1) 3x + 2y + z = 0
2) x + 4y + z = 0
3) 2x + y + 4z = 0
To find:
The nature of the system of equations.
Solution:
To determine the nature of the system of equations, we need to consider the coefficient matrix and solve it using the determinant method.
Determinant Method:
The system of equations can be written in matrix form as AX = 0, where A is the coefficient matrix and X is the column matrix of variables (x, y, z).
The coefficient matrix A is given by:
A = |3 2 1|
|1 4 1|
|2 1 4|
The determinant of the coefficient matrix is calculated as follows:
|A| = 3(4*4 - 1*1) - 2(1*4 - 2*1) + 1(1*1 - 2*4)
= 3(16 - 1) - 2(4 - 2) + 1(1 - 8)
= 3(15) - 2(2) + 1(-7)
= 45 - 4 - 7
= 34
Analysis:
If the determinant of the coefficient matrix (|A|) is zero, then the system of equations has either no solution or infinitely many solutions. If |A| is non-zero, then the system has a unique solution.
Conclusion:
Since the determinant of the coefficient matrix is non-zero (34 ≠ 0), the system of equations has a unique solution. Therefore, the correct answer is option 'B' - the system of equations has only a trivial solution.
|
Explore Courses for Mathematics exam
|
|
Similar Mathematics Doubts
If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z = 0 be a system of equations, thena)it is inconsistentb)it has only trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer?
Question Description
If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z = 0 be a system of equations, thena)it is inconsistentb)it has only trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z = 0 be a system of equations, thena)it is inconsistentb)it has only trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z = 0 be a system of equations, thena)it is inconsistentb)it has only trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer?.
If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z = 0 be a system of equations, thena)it is inconsistentb)it has only trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z = 0 be a system of equations, thena)it is inconsistentb)it has only trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z = 0 be a system of equations, thena)it is inconsistentb)it has only trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer?.
Solutions for If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z = 0 be a system of equations, thena)it is inconsistentb)it has only trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free.
Here you can find the meaning of If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z = 0 be a system of equations, thena)it is inconsistentb)it has only trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z = 0 be a system of equations, thena)it is inconsistentb)it has only trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z = 0 be a system of equations, thena)it is inconsistentb)it has only trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z = 0 be a system of equations, thena)it is inconsistentb)it has only trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z = 0 be a system of equations, thena)it is inconsistentb)it has only trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice Mathematics tests.
|
|
Explore Courses for Mathematics exam
|
|
Suggested Free Tests
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