Consider the system of linear equations: $x + y + z = 6$, $x + 2y + 3z = 10$, and $x + 2y + kz = \mu$. If the system has infinitely many solutions, the value of $k$ must be equal to ________ (answer in integer).
Answer Key:
3
Step-by-Step Solution & Explanation
Form the augmented matrix $[A|b]$:
$\begin{pmatrix} 1 & 1 & 1 & 6 \\ 1 & 2 & 3 & 10 \\ 1 & 2 & k & \mu \end{pmatrix}$
Applying elementary row operations $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_1$:
$\begin{pmatrix} 1 & 1 & 1 & 6 \\ 0 & 1 & 2 & 4 \\ 0 & 1 & k-1 & \mu-6 \end{pmatrix}$
Now apply $R_3 \to R_3 - R_2$:
$\begin{pmatrix} 1 & 1 & 1 & 6 \\ 0 & 1 & 2 & 4 \\ 0 & 0 & k-3 & \mu-10 \end{pmatrix}$
For the system to have infinitely many solutions, we must have $\operatorname{rank}(A) = \operatorname{rank}([A|b]) < 3$.
This requires the entire last row to be all zeros:
$k - 3 = 0 \implies k = 3$
$\mu - 10 = 0 \implies \mu = 10$
Hence, the required value of $k$ is $3$.