For what value of $k$ do the equations $3x - y + 8 = 0$ and $6x - ky = -16$ represent coincident lines?
$\dfrac{1}{2}$
$-\dfrac{1}{2}$
$2$
$-2$
Step-by-Step Solution
Key Concept: For coincident lines, $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$.
Stepwise Solution:
Rewrite second equation: $6x - ky + 16 = 0$. Here $a_1=3, b_1=-1, c_1=8$ and $a_2=6, b_2=-k, c_2=16$. [0.5 Mark]
$\dfrac{3}{6} = \dfrac{-1}{-k} = \dfrac{8}{16} \Rightarrow \dfrac{1}{2} = \dfrac{1}{k} \Rightarrow k = 2$. [0.5 Mark]
Marking Scheme:
• Setting up ratio condition for coincident lines: 0.5 Mark
• Solving for $k = 2$: 0.5 Mark
Correct Answer: $2$