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The simultaneous equations
2x + ay + z = 20
x + 3y + 4z = b
x + 2y + 3z = c
has unique solution then what is the value of a, b and c respectively?
- a)a ≠ ± 1, b and c can be any value
- b)a ≠ ± 1, b ≠ 0, c ≠ 1
- c)a ≠ -1, b and c can be any value
- d)a, b and c can be any value
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The simultaneous equations2x + ay + z = 20x + 3y + 4z = bx + 2y + 3z =...
For given system of equations to have unique solution, the determinant of the coefficient matrix should be not equal to 0
2 (9−8) − a(3−4) + (2−3) ≠ 0
2 + a − 1 ≠ 0
a ≠ −1
Since Δ of coefficient matrix is not equal to 0 then b and c can take any value
Note:
If coefficient matrix’s Δ ≠ 0 then rank of coefficient matrix is always equal rank of augmented matrix so right-hand side of given equation can take any value
Most Upvoted Answer
The simultaneous equations2x + ay + z = 20x + 3y + 4z = bx + 2y + 3z =...
The given system of equations is:
2x + ay + z = 20 (equation 1)
x + 3y + 4z = b (equation 2)
x + 2y + 3z = c (equation 3)
To determine the value of a, b, and c, we need to use the fact that the system has a unique solution.
One way to solve the system is by using the method of elimination. We can start by eliminating x from equations 2 and 3:
Multiply equation 2 by 2 and equation 3 by -1:
2(x + 3y + 4z) = 2b
-1(x + 2y + 3z) = -c
Simplifying:
2x + 6y + 8z = 2b (equation 4)
-x - 2y - 3z = -c (equation 5)
Now, add equation 1 and equation 5:
2x + ay + z + (-x - 2y - 3z) = 20 + (-c)
Simplifying:
x + (ay - 2y) + (z - 3z) = 20 - c
Combining like terms:
x + (ay - 2y) - 2z = 20 - c (equation 6)
Now, we have the following system of equations:
2x + 6y + 8z = 2b (equation 4)
x + (ay - 2y) - 2z = 20 - c (equation 6)
To have a unique solution, the determinant of the coefficient matrix must be non-zero.
The determinant of the coefficient matrix is given by:
|2 6 8|
|1 a-2 -2|
|0 1 -3|
Expanding along the first row, we have:
2[(a-2)(-3) - (-2)(1)] - 6[(-2)(-3) - 1(0)] + 8[1(1) - (a-2)(0)]
Simplifying:
6a - 12 + 12 - 6 + 8 = 6a - 6
For a unique solution, the determinant must be non-zero:
6a - 6 ≠ 0
Solving for a:
6a ≠ 6
a ≠ 1
Therefore, for the system to have a unique solution, a cannot be 1.
So, the value of a is not determined.
The values of b and c are also not determined from the given information.
2x + ay + z = 20 (equation 1)
x + 3y + 4z = b (equation 2)
x + 2y + 3z = c (equation 3)
To determine the value of a, b, and c, we need to use the fact that the system has a unique solution.
One way to solve the system is by using the method of elimination. We can start by eliminating x from equations 2 and 3:
Multiply equation 2 by 2 and equation 3 by -1:
2(x + 3y + 4z) = 2b
-1(x + 2y + 3z) = -c
Simplifying:
2x + 6y + 8z = 2b (equation 4)
-x - 2y - 3z = -c (equation 5)
Now, add equation 1 and equation 5:
2x + ay + z + (-x - 2y - 3z) = 20 + (-c)
Simplifying:
x + (ay - 2y) + (z - 3z) = 20 - c
Combining like terms:
x + (ay - 2y) - 2z = 20 - c (equation 6)
Now, we have the following system of equations:
2x + 6y + 8z = 2b (equation 4)
x + (ay - 2y) - 2z = 20 - c (equation 6)
To have a unique solution, the determinant of the coefficient matrix must be non-zero.
The determinant of the coefficient matrix is given by:
|2 6 8|
|1 a-2 -2|
|0 1 -3|
Expanding along the first row, we have:
2[(a-2)(-3) - (-2)(1)] - 6[(-2)(-3) - 1(0)] + 8[1(1) - (a-2)(0)]
Simplifying:
6a - 12 + 12 - 6 + 8 = 6a - 6
For a unique solution, the determinant must be non-zero:
6a - 6 ≠ 0
Solving for a:
6a ≠ 6
a ≠ 1
Therefore, for the system to have a unique solution, a cannot be 1.
So, the value of a is not determined.
The values of b and c are also not determined from the given information.
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The simultaneous equations2x + ay + z = 20x + 3y + 4z = bx + 2y + 3z = chas unique solution then what is the value of a, b and c respectively?a)a ≠ ± 1, b and c can be any valueb)a ≠ ± 1, b ≠ 0, c ≠ 1c)a ≠ -1, b and c can be any valued)a, b and c can be any valueCorrect answer is option 'C'. Can you explain this answer?
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The simultaneous equations2x + ay + z = 20x + 3y + 4z = bx + 2y + 3z = chas unique solution then what is the value of a, b and c respectively?a)a ≠ ± 1, b and c can be any valueb)a ≠ ± 1, b ≠ 0, c ≠ 1c)a ≠ -1, b and c can be any valued)a, b and c can be any valueCorrect answer is option 'C'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about The simultaneous equations2x + ay + z = 20x + 3y + 4z = bx + 2y + 3z = chas unique solution then what is the value of a, b and c respectively?a)a ≠ ± 1, b and c can be any valueb)a ≠ ± 1, b ≠ 0, c ≠ 1c)a ≠ -1, b and c can be any valued)a, b and c can be any valueCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The simultaneous equations2x + ay + z = 20x + 3y + 4z = bx + 2y + 3z = chas unique solution then what is the value of a, b and c respectively?a)a ≠ ± 1, b and c can be any valueb)a ≠ ± 1, b ≠ 0, c ≠ 1c)a ≠ -1, b and c can be any valued)a, b and c can be any valueCorrect answer is option 'C'. Can you explain this answer?.
The simultaneous equations2x + ay + z = 20x + 3y + 4z = bx + 2y + 3z = chas unique solution then what is the value of a, b and c respectively?a)a ≠ ± 1, b and c can be any valueb)a ≠ ± 1, b ≠ 0, c ≠ 1c)a ≠ -1, b and c can be any valued)a, b and c can be any valueCorrect answer is option 'C'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about The simultaneous equations2x + ay + z = 20x + 3y + 4z = bx + 2y + 3z = chas unique solution then what is the value of a, b and c respectively?a)a ≠ ± 1, b and c can be any valueb)a ≠ ± 1, b ≠ 0, c ≠ 1c)a ≠ -1, b and c can be any valued)a, b and c can be any valueCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The simultaneous equations2x + ay + z = 20x + 3y + 4z = bx + 2y + 3z = chas unique solution then what is the value of a, b and c respectively?a)a ≠ ± 1, b and c can be any valueb)a ≠ ± 1, b ≠ 0, c ≠ 1c)a ≠ -1, b and c can be any valued)a, b and c can be any valueCorrect answer is option 'C'. Can you explain this answer?.
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