For the system of linear equations:
$\begin{aligned} x + y + z &= 6 \\ x + 2y + 3z &= 10 \\ x + 2y + k z &= \mu \end{aligned}$
Calculate the value of $k$ for which the system has infinitely many solutions (given $\mu = 10$).
Answer Key:
3
Step-by-Step Solution & Explanation
Using row reduction on the augmented matrix $[A | B]$:
$\begin{pmatrix} 1 & 1 & 1 & | & 6 \\ 1 & 2 & 3 & | & 10 \\ 1 & 2 & k & | & \mu \end{pmatrix}$
Row operations $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_2$:
$\begin{pmatrix} 1 & 1 & 1 & | & 6 \\ 0 & 1 & 2 & | & 4 \\ 0 & 0 & k - 3 & | & \mu - 10 \end{pmatrix}$
For infinitely many solutions, the rank of $A$ must equal the rank of $[A|B] < 3$:
$k - 3 = 0 \implies k = 3$
and $\mu - 10 = 0 \implies \mu = 10$.