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If the system
2x – y + 3z = 2
x + y + 2z = 2
5x – y + az = b
Has infinitely many solutions, then the values of a and b, respectively, are
- a)– 8 and 6
- b)8 and 6
- c)– 8 and –6
- d)8 and –6
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If the system2x – y + 3z = 2x + y + 2z = 25x – y + az = bH...
Concept:
Consider the system of m linear equations
a11 x1 + a12 x2 + … + a1n xn = b1
a21 x1 + a22 x2 + … + a2n xn = b2
…
am1 x1 + am2 x2 + … + amn xn = bm
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 following matrices.
A is the coefficient matrix and [A|B] is called as augmented matrix of the given system of equations.
We can find the consistency of the given system of equations as follows:
- If the rank of matrix A is equal to the rank of augmented matrix and it is equal to the number of unknowns, then the system is consistent and there is a unique solution, i.e.
Rank of A = Rank of augmented matrix = n - If the rank of matrix A is equal to the rank of augmented matrix and it is less than the number of unknowns, then the system is consistent and there are an infinite number of solutions.
Rank of A = Rank of augmented matrix < n - If the rank of matrix A is not equal to the rank of the augmented matrix, then the system is inconsistent, and it has no solution.
Rank of A ≠ Rank of augmented matrix
Calculation:
Given linear system is
2x – y + 3z = 2
x + y + 2z = 2
5x – y + az = b
Then augmented matrix form is written below;
For rank (A) < n = 3
‘a’ must be = 8
For rank [A|B] < 3, b = 6
Therefore a = 8 & b = 6
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If the system2x – y + 3z = 2x + y + 2z = 25x – y + az = bH...
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If the system2x – y + 3z = 2x + y + 2z = 25x – y + az = bHas infinitely many solutions, then the values of a and b, respectively, area)– 8 and 6b)8 and 6c)– 8 and –6d)8 and –6Correct answer is option 'B'. Can you explain this answer?
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