Step 1: Write both equations in standard form:
$$\frac{1}{2}x+\frac{1}{3}y-5=0\quad\text{and}\quad 2x+ky-7=0$$
Step 2: For an inconsistent system (no solution): $\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\ne\dfrac{c_1}{c_2}$.
Step 3: $$\frac{1/2}{2}=\frac{1/3}{k}\ \Rightarrow\ \frac{1}{4}=\frac{1}{3k}$$
Step 4: $3k=4\Rightarrow k=\dfrac{4}{3}$.
Step 5: Check: $\dfrac{c_1}{c_2}=\dfrac{-5}{-7}=\dfrac{5}{7}\ne\dfrac14$, so the system is indeed inconsistent.
Answer: (B) $\dfrac{4}{3}$