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Consider the following linear system.
x + 2y - 3z = a
2x + 3y + 3z = b
5x + 9y - 6z = c
This system is consistent if a,b and c satisfy the equation
- a)7a - b - c = 0
- b)3a + b - c = 0
- c)3a - b + c = 0
- d)7a - b + c = 0
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Consider the following linear system.x + 2y - 3z = a2x + 3y + 3z = b5x...
(AB) = (1 2 -3 a 2 3 3 b 5 9 - 6 c)
(R2 - 2R1) ; (R3-5R1)
≈ (1 2 - 3 a 0 -1 9 b - 2a 0 -1 9 c - 5a)
(R3 - R2)
≈ (1 2 - 3 a 0 - 1 9 b - 2a 0 0 0 (c - b - 3a))
(c - b - 3a)
3a + b - c = 0
Most Upvoted Answer
Consider the following linear system.x + 2y - 3z = a2x + 3y + 3z = b5x...
Explanation:
Consistent System:
- A linear system is consistent if it has at least one solution.
- In other words, the system has a solution when the number of equations is equal to or less than the number of unknown variables.
Given System:
- x + 2y - 3z = a
- 2x + 3y + 3z = b
- 5x + 9y - 6z = c
Rewriting the System:
- We can rewrite the given system in matrix form as: AX = B, where:
A = [[1, 2, -3], [2, 3, 3], [5, 9, -6]], X = [[x], [y], [z]], and B = [[a], [b], [c]].
Consistency Condition:
- For the system to be consistent, the determinant of matrix A must be equal to the determinant of the augmented matrix [A|B].
- The system is consistent if det(A) = det([A|B]).
Determinant Calculation:
- det(A) = -3(3*9 - 3*9) - 2(2*9 - 3*5) + 5(2*3 - 3*3) = -27 + 6 + 15 = -6
- det([A|B]) = -3(3(c) - 3(b)) - 2(2(c) - 3(a)) + 5(2(b) - 3(a))
= -3(3c - 3b) - 2(2c - 3a) + 5(2b - 3a)
= -9c + 9b - 4c + 6a + 10b - 15a
= 6a + 19b - 9c
Consistency Condition (cont.):
- For the system to be consistent, det(A) = det([A|B]) must hold true.
- Therefore, -6 = 6a + 19b - 9c
- Simplifying, we get 6a + 19b - 9c = -6
- Comparing with the given options, we find that 3a + b - c = 0, which matches the consistency condition.
Therefore, the correct answer is option B) 3a + b - c = 0.
Consistent System:
- A linear system is consistent if it has at least one solution.
- In other words, the system has a solution when the number of equations is equal to or less than the number of unknown variables.
Given System:
- x + 2y - 3z = a
- 2x + 3y + 3z = b
- 5x + 9y - 6z = c
Rewriting the System:
- We can rewrite the given system in matrix form as: AX = B, where:
A = [[1, 2, -3], [2, 3, 3], [5, 9, -6]], X = [[x], [y], [z]], and B = [[a], [b], [c]].
Consistency Condition:
- For the system to be consistent, the determinant of matrix A must be equal to the determinant of the augmented matrix [A|B].
- The system is consistent if det(A) = det([A|B]).
Determinant Calculation:
- det(A) = -3(3*9 - 3*9) - 2(2*9 - 3*5) + 5(2*3 - 3*3) = -27 + 6 + 15 = -6
- det([A|B]) = -3(3(c) - 3(b)) - 2(2(c) - 3(a)) + 5(2(b) - 3(a))
= -3(3c - 3b) - 2(2c - 3a) + 5(2b - 3a)
= -9c + 9b - 4c + 6a + 10b - 15a
= 6a + 19b - 9c
Consistency Condition (cont.):
- For the system to be consistent, det(A) = det([A|B]) must hold true.
- Therefore, -6 = 6a + 19b - 9c
- Simplifying, we get 6a + 19b - 9c = -6
- Comparing with the given options, we find that 3a + b - c = 0, which matches the consistency condition.
Therefore, the correct answer is option B) 3a + b - c = 0.
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Consider the following linear system.x + 2y - 3z = a2x + 3y + 3z = b5x + 9y - 6z = cThis system is consistent if a,b and c satisfy the equationa)7a - b - c = 0b)3a + b - c = 0c)3a - b + c = 0d)7a - b + c = 0Correct answer is option 'B'. Can you explain this answer?
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Consider the following linear system.x + 2y - 3z = a2x + 3y + 3z = b5x + 9y - 6z = cThis system is consistent if a,b and c satisfy the equationa)7a - b - c = 0b)3a + b - c = 0c)3a - b + c = 0d)7a - b + c = 0Correct answer is option 'B'. 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 Consider the following linear system.x + 2y - 3z = a2x + 3y + 3z = b5x + 9y - 6z = cThis system is consistent if a,b and c satisfy the equationa)7a - b - c = 0b)3a + b - c = 0c)3a - b + c = 0d)7a - b + c = 0Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following linear system.x + 2y - 3z = a2x + 3y + 3z = b5x + 9y - 6z = cThis system is consistent if a,b and c satisfy the equationa)7a - b - c = 0b)3a + b - c = 0c)3a - b + c = 0d)7a - b + c = 0Correct answer is option 'B'. Can you explain this answer?.
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