Answer: B. $\dfrac{1}{2}$

Explanation:1. Rewrite the equation by factoring out x: $(-3+6r)x = 42$

2. Understand the condition for a linear equation $Ax=B$ to have no solution: A linear equation $Ax=B$ has no solution if and only if $A=0$ and $B \neq 0$. In our equation, $A = (-3+6r)$ and $B = 42$.

3. Apply the condition for no solution: For no solution, we need: 1. $-3+6r = 0$ 2. $42 \neq 0$ (This condition is already true)

4. Solve for r using the first condition: $-3+6r = 0$ $6r = 3$ $r = \dfrac{3}{6}$ $r = \dfrac{1}{2}$

5. Verify that if $r=\dfrac{1}{2}$, the equation form is indeed $0x = \text{non-zero}$: If $r=\dfrac{1}{2}$, then $-3+6(\dfrac{1}{2}) = -3+3 = 0$. The equation becomes $0x = 42$, which simplifies to $0 = 42$. This is a false statement, indicating there is no value of x that can satisfy it. Thus, the equation has no solution.

6. Check the options: Option [B] is $\dfrac{1}{2}$, which matches our result.

7. Conclusion: For the equation to have no solution, the value of r must be $\dfrac{1}{2}$. Option [B] is correct.