GATE Exam > GATE Questions > x + 2y + z = 42x + y + 2z = 5x - y + z = 1The...
Start Learning for Free
x + 2y + z = 4
2x + y + 2z = 5
x - y + z = 1
The system of algebraic equations given above has
- a)a unique solution of x = 1, y = 1 and z = 1.
- b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).
- c)infinite number of solutions.
- d)no feasible solution.
Correct answer is option 'C'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equa...
Free Test
FREE
| Start Free Test |
Community Answer
x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equa...
Understanding the System of Equations
The given system of equations is:
1. \( x + 2y + z = 4 \)
2. \( 2x + y + 2z = 5 \)
3. \( 5x - y + z = 1 \)
To analyze the solutions, we can represent this system in matrix form and apply methods such as row reduction.
Row Reduction Process
We can write the augmented matrix as follows:
\[
\begin{bmatrix}
1 & 2 & 1 & | & 4 \\
2 & 1 & 2 & | & 5 \\
5 & -1 & 1 & | & 1
\end{bmatrix}
\]
Performing row operations to simplify:
- Multiply row 1 by 2 and subtract from row 2.
- Multiply row 1 by 5 and subtract from row 3.
After performing these operations, we may find contradictions or dependencies among the equations.
Identifying Solutions
After simplifying:
- The equations may reduce to a form indicating that one equation is a linear combination of the others.
- This indicates that the system has infinitely many solutions or is consistent.
Conclusion
Given the dependencies among the equations, we conclude that:
- The system does not restrict \(x\), \(y\), and \(z\) to unique values.
- Thus, the system has **infinitely many solutions**.
Hence, the correct answer is option **'C'**: infinite number of solutions.
The given system of equations is:
1. \( x + 2y + z = 4 \)
2. \( 2x + y + 2z = 5 \)
3. \( 5x - y + z = 1 \)
To analyze the solutions, we can represent this system in matrix form and apply methods such as row reduction.
Row Reduction Process
We can write the augmented matrix as follows:
\[
\begin{bmatrix}
1 & 2 & 1 & | & 4 \\
2 & 1 & 2 & | & 5 \\
5 & -1 & 1 & | & 1
\end{bmatrix}
\]
Performing row operations to simplify:
- Multiply row 1 by 2 and subtract from row 2.
- Multiply row 1 by 5 and subtract from row 3.
After performing these operations, we may find contradictions or dependencies among the equations.
Identifying Solutions
After simplifying:
- The equations may reduce to a form indicating that one equation is a linear combination of the others.
- This indicates that the system has infinitely many solutions or is consistent.
Conclusion
Given the dependencies among the equations, we conclude that:
- The system does not restrict \(x\), \(y\), and \(z\) to unique values.
- Thus, the system has **infinitely many solutions**.
Hence, the correct answer is option **'C'**: infinite number of solutions.
|
Explore Courses for GATE exam
|
|
Similar GATE Doubts
x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equations given above hasa)a unique solution of x = 1, y = 1 and z = 1.b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).c)infinite number of solutions.d)no feasible solution.Correct answer is option 'C'. Can you explain this answer?
Question Description
x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equations given above hasa)a unique solution of x = 1, y = 1 and z = 1.b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).c)infinite number of solutions.d)no feasible solution.Correct answer is option 'C'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equations given above hasa)a unique solution of x = 1, y = 1 and z = 1.b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).c)infinite number of solutions.d)no feasible solution.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equations given above hasa)a unique solution of x = 1, y = 1 and z = 1.b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).c)infinite number of solutions.d)no feasible solution.Correct answer is option 'C'. Can you explain this answer?.
x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equations given above hasa)a unique solution of x = 1, y = 1 and z = 1.b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).c)infinite number of solutions.d)no feasible solution.Correct answer is option 'C'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equations given above hasa)a unique solution of x = 1, y = 1 and z = 1.b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).c)infinite number of solutions.d)no feasible solution.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equations given above hasa)a unique solution of x = 1, y = 1 and z = 1.b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).c)infinite number of solutions.d)no feasible solution.Correct answer is option 'C'. Can you explain this answer?.
Solutions for x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equations given above hasa)a unique solution of x = 1, y = 1 and z = 1.b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).c)infinite number of solutions.d)no feasible solution.Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for GATE.
Download more important topics, notes, lectures and mock test series for GATE Exam by signing up for free.
Here you can find the meaning of x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equations given above hasa)a unique solution of x = 1, y = 1 and z = 1.b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).c)infinite number of solutions.d)no feasible solution.Correct answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equations given above hasa)a unique solution of x = 1, y = 1 and z = 1.b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).c)infinite number of solutions.d)no feasible solution.Correct answer is option 'C'. Can you explain this answer?, a detailed solution for x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equations given above hasa)a unique solution of x = 1, y = 1 and z = 1.b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).c)infinite number of solutions.d)no feasible solution.Correct answer is option 'C'. Can you explain this answer? has been provided alongside types of x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equations given above hasa)a unique solution of x = 1, y = 1 and z = 1.b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).c)infinite number of solutions.d)no feasible solution.Correct answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equations given above hasa)a unique solution of x = 1, y = 1 and z = 1.b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).c)infinite number of solutions.d)no feasible solution.Correct answer is option 'C'. Can you explain this answer? tests, examples and also practice GATE tests.
|
|
Explore Courses for GATE 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