$x + y + z = 6$,
$x + 2y + 5z = 10$,
$2x + 3y + \lambda z = \mu$
has infinitely many solutions, then the value of $\lambda + \mu$ equals :
Analyzing Linear Equations for Infinitely Many Solutions
We are given a system of three linear equations:
- $x + y + z = 6$
- $x + 2y + 5z = 10$
- $2x + 3y + \lambda z = \mu$
For this system to have infinitely many solutions, two conditions must be met:
- The determinant of the coefficient matrix must be zero.
- The system must be consistent (the ranks of the coefficient matrix and the augmented matrix must be equal).
Calculating the Determinant
The coefficient matrix is:
$ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 5 \\ 2 & 3 & \lambda \end{bmatrix} $The determinant is calculated as:
$ \det(A) = 1 \begin{vmatrix} 2 & 5 \\ 3 & \lambda \end{vmatrix} - 1 \begin{vmatrix} 1 & 5 \\ 2 & \lambda \end{vmatrix} + 1 \begin{vmatrix} 1 & 2 \\ 2 & 3 \end{vmatrix} $ $ \det(A) = 1(2\lambda - 15) - 1(\lambda - 10) + 1(3 - 4) $ $ \det(A) = 2\lambda - 15 - \lambda + 10 - 1 $ $ \det(A) = \lambda - 6 $For infinitely many solutions, $det(A) = 0$.
$ \lambda - 6 = 0 \implies \lambda = 6 $Checking Consistency with Gaussian Elimination
Substitute $\lambda = 6$ into the augmented matrix and perform row operations:
$ \begin{bmatrix} 1 & 1 & 1 & | & 6 \\ 1 & 2 & 5 & | & 10 \\ 2 & 3 & 6 & | & \mu \end{bmatrix} $Apply $R_2 \rightarrow R_2 - R_1$ and $R_3 \rightarrow R_3 - 2R_1$:
$ \begin{bmatrix} 1 & 1 & 1 & | & 6 \\ 0 & 1 & 4 & | & 4 \\ 0 & 1 & 4 & | & \mu - 12 \end{bmatrix} $Apply $R_3 \rightarrow R_3 - R_2$:
$ \begin{bmatrix} 1 & 1 & 1 & | & 6 \\ 0 & 1 & 4 & | & 4 \\ 0 & 0 & 0 & | & (\mu - 12) - 4 \end{bmatrix} $ $ \begin{bmatrix} 1 & 1 & 1 & | & 6 \\ 0 & 1 & 4 & | & 4 \\ 0 & 0 & 0 & | & \mu - 16 \end{bmatrix} $For infinitely many solutions, the last row must be entirely zeros:
$ \mu - 16 = 0 \implies \mu = 16 $Final Calculation
We found $\lambda = 6$ and $\mu = 16$.
Therefore, the value of $\lambda + \mu$ is:
$ \lambda + \mu = 6 + 16 = 22 $Similar Questions
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Important Questions from Algebra
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $P = [p_{ij}]$ and $Q = [q_{ij}]$ be two square matrices of order 3 such that $q_{ij} = 2^{(i + j - 1)} p_{ij}$ and $\det(Q) = 2^{10}$. Then the value of $\det(\text{adj}(\text{adj } P))$ is:
