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Find the value of k for which the following simultaneous equations
x + y + z = 3; x + 2y + 3z = 4; x + 4y + kz = 6 will not have a unique solution.
x + y + z = 3; x + 2y + 3z = 4; x + 4y + kz = 6 will not have a unique solution.
- a)0
- b)5
- c)6
- d)7
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Find the value of k for which the following simultaneous equationsx + ...
We need to find the value of k for which the following simultaneous equations
x + y + z= 3
x + 2y + 3z = 4
x + 4y + kz = 6
will not have a unique solution. This system of equation may be written as,
Reduce this system of equation to echelon form using the operations
"R2 → R2 - R1" and R3 → R3 - R1.
These operations yield -
and also, R3 → R3 - 3R2 gives,
also this system of equation has not unique solution if,
k-7 = 0
k = 7
x + y + z= 3
x + 2y + 3z = 4
x + 4y + kz = 6
will not have a unique solution. This system of equation may be written as,
Reduce this system of equation to echelon form using the operations
"R2 → R2 - R1" and R3 → R3 - R1.
These operations yield -
and also, R3 → R3 - 3R2 gives,
also this system of equation has not unique solution if,
k-7 = 0
k = 7
Most Upvoted Answer
Find the value of k for which the following simultaneous equationsx + ...
We need to find the value of k for which the following simultaneous equations
x + y + z= 3
x + 2y + 3z = 4
x + 4y + kz = 6
will not have a unique solution. This system of equation may be written as,
Reduce this system of equation to echelon form using the operations
"R2 → R2 - R1" and R3 → R3 - R1.
These operations yield -
and also, R3 → R3 - 3R2 gives,
also this system of equation has not unique solution if,
k-7 = 0
k = 7
x + y + z= 3
x + 2y + 3z = 4
x + 4y + kz = 6
will not have a unique solution. This system of equation may be written as,
Reduce this system of equation to echelon form using the operations
"R2 → R2 - R1" and R3 → R3 - R1.
These operations yield -
and also, R3 → R3 - 3R2 gives,
also this system of equation has not unique solution if,
k-7 = 0
k = 7
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Community Answer
Find the value of k for which the following simultaneous equationsx + ...
Understanding the Problem
To determine the value of \( k \) for which the given system of equations does not have a unique solution, we need to analyze the conditions under which a system of linear equations fails to have a unique solution. This occurs when the determinant of the coefficient matrix is zero.
Given Equations
The equations provided are:
1. \( x + y + z = 3 \)
2. \( x + 2y + 3z = 4 \)
3. \( x + 4y + kz = 6 \)
We can represent this system in matrix form as \( AX = B \), where:
\[ A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & k \end{pmatrix} \]
\[ X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 3 \\ 4 \\ 6 \end{pmatrix} \]
Calculating the Determinant
The determinant of matrix \( A \) must be calculated:
\[
\text{det}(A) = 1 \begin{vmatrix} 2 & 3 \\ 4 & k \end{vmatrix} - 1 \begin{vmatrix} 1 & 3 \\ 1 & k \end{vmatrix} + 1 \begin{vmatrix} 1 & 2 \\ 1 & 4 \end{vmatrix}
\]
Calculating the minors:
1. \( \begin{vmatrix} 2 & 3 \\ 4 & k \end{vmatrix} = 2k - 12 \)
2. \( \begin{vmatrix} 1 & 3 \\ 1 & k \end{vmatrix} = k - 3 \)
3. \( \begin{vmatrix} 1 & 2 \\ 1 & 4 \end{vmatrix} = 4 - 2 = 2 \)
Combining these, we have:
\[
\text{det}(A) = 1(2k - 12) - 1(k - 3) + 1(2) = 2k - 12 - k + 3 + 2 = k - 7
\]
Setting the Determinant to Zero
For the system not to have a unique solution, set the determinant to zero:
\[
k - 7 = 0 \implies k = 7
\]
Conclusion
Thus, the value of \( k \) for which the given system of equations does not have a unique solution is:
k = 7 (Option D)
To determine the value of \( k \) for which the given system of equations does not have a unique solution, we need to analyze the conditions under which a system of linear equations fails to have a unique solution. This occurs when the determinant of the coefficient matrix is zero.
Given Equations
The equations provided are:
1. \( x + y + z = 3 \)
2. \( x + 2y + 3z = 4 \)
3. \( x + 4y + kz = 6 \)
We can represent this system in matrix form as \( AX = B \), where:
\[ A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & k \end{pmatrix} \]
\[ X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 3 \\ 4 \\ 6 \end{pmatrix} \]
Calculating the Determinant
The determinant of matrix \( A \) must be calculated:
\[
\text{det}(A) = 1 \begin{vmatrix} 2 & 3 \\ 4 & k \end{vmatrix} - 1 \begin{vmatrix} 1 & 3 \\ 1 & k \end{vmatrix} + 1 \begin{vmatrix} 1 & 2 \\ 1 & 4 \end{vmatrix}
\]
Calculating the minors:
1. \( \begin{vmatrix} 2 & 3 \\ 4 & k \end{vmatrix} = 2k - 12 \)
2. \( \begin{vmatrix} 1 & 3 \\ 1 & k \end{vmatrix} = k - 3 \)
3. \( \begin{vmatrix} 1 & 2 \\ 1 & 4 \end{vmatrix} = 4 - 2 = 2 \)
Combining these, we have:
\[
\text{det}(A) = 1(2k - 12) - 1(k - 3) + 1(2) = 2k - 12 - k + 3 + 2 = k - 7
\]
Setting the Determinant to Zero
For the system not to have a unique solution, set the determinant to zero:
\[
k - 7 = 0 \implies k = 7
\]
Conclusion
Thus, the value of \( k \) for which the given system of equations does not have a unique solution is:
k = 7 (Option D)
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Find the value of k for which the following simultaneous equationsx + y + z = 3; x + 2y + 3z = 4; x + 4y + kz = 6will not have a unique solution.a)0b)5c)6d)7Correct answer is option 'D'. Can you explain this answer? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Find the value of k for which the following simultaneous equationsx + y + z = 3; x + 2y + 3z = 4; x + 4y + kz = 6will not have a unique solution.a)0b)5c)6d)7Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the value of k for which the following simultaneous equationsx + y + z = 3; x + 2y + 3z = 4; x + 4y + kz = 6will not have a unique solution.a)0b)5c)6d)7Correct answer is option 'D'. Can you explain this answer?.
Solutions for Find the value of k for which the following simultaneous equationsx + y + z = 3; x + 2y + 3z = 4; x + 4y + kz = 6will not have a unique solution.a)0b)5c)6d)7Correct answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
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