Let the system of linear equations $ x+y+ kz =2$ $ 2 x+3 y-z=1 $ $ 3 x+4 y +2 z= k$ have infinitely many solutions Then the system $ ( k +1) x+(2 k -1) y=7$ $ (2 k +1) x+( k +5) y=10$ has:
- infinitely many solutions
- unique solution satisfying $x-y=1$
- no solution
- unique solution satisfying $x+y=1$
The Correct Option is D
Approach Solution - 1
We are given the system of equations:
\[ x + y + kz = 2, \quad 2x + 3y - z = 1, \quad 3x + 4y + 2z = k. \]
To find the value of \(k\) for which the system has infinitely many solutions, we first find the determinant of the coefficient matrix.
Step 1: Coefficient Matrix
The coefficient matrix is:
\[ A = \begin{bmatrix} 1 & 1 & k \\ 2 & 3 & -1 \\ 3 & 4 & 2 \end{bmatrix}. \]
Step 2: Determinant of the Matrix
The determinant of \(A\) is calculated as:
\[ \text{Det}(A) = \begin{vmatrix} 1 & 1 & k \\ 2 & 3 & -1 \\ 3 & 4 & 2 \end{vmatrix}. \]
Expanding the determinant along the first row:
\[ \text{Det}(A) = 1 \cdot \begin{vmatrix} 3 & -1 \\ 4 & 2 \end{vmatrix} - 1 \cdot \begin{vmatrix} 2 & -1 \\ 3 & 2 \end{vmatrix} + k \cdot \begin{vmatrix} 2 & 3 \\ 3 & 4 \end{vmatrix}. \]
Step 3: Calculate the 2x2 Determinants
Compute each 2x2 determinant:
- \[ \begin{vmatrix} 3 & -1 \\ 4 & 2 \end{vmatrix} = (3)(2) - (4)(-1) = 6 + 4 = 10. \]
- \[ \begin{vmatrix} 2 & -1 \\ 3 & 2 \end{vmatrix} = (2)(2) - (3)(-1) = 4 + 3 = 7. \]
- \[ \begin{vmatrix} 2 & 3 \\ 3 & 4 \end{vmatrix} = (2)(4) - (3)(3) = 8 - 9 = -1. \]
Step 4: Substitute Back
Substitute these values into the determinant expression:
\[ \text{Det}(A) = 1(10) - 1(7) + k(-1) = 10 - 7 - k = 3 - k. \]
Step 5: Infinitely Many Solutions
For the system to have infinitely many solutions, the determinant must be zero:
\[ 3 - k = 0 \implies k = 3. \]
Step 6: Reduced System
For \(k = 3\), the system becomes:
\[ 4x + 5y = 7 \quad \text{(1)}, \]
\[ 7x + 8y = 10 \quad \text{(2)}. \]
Subtract equation (1) from equation (2):
\[ (7x + 8y) - (4x + 5y) = 10 - 7. \]
This simplifies to:
\[ 3x + 3y = 3 \implies x + y = 1. \]
Conclusion
The value of \(k\) for which the system has infinitely many solutions is:
\[ \boxed{3}. \]
Furthermore, the system satisfies \(x + y = 1\).
Approach Solution -2
For system is
(1)
and (2)
Clearly, they have a unique solution
(2)
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
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- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.