Answer: $\dfrac{5}{12}$

Explanation:

1. Understand the condition for an equation to have no solution: A linear equation of the form $Ax + B = Cx + D$ has no solution if and only if the coefficients of $x$ are equal ($A = C$) and the constant terms are not equal ($B \neq D$). This corresponds to parallel lines with different $y$-intercepts.

2. Identify the coefficients of $x$ on both sides: The given equation is $-\dfrac{3}{14}rx + \dfrac{s}{9} = 10 - \dfrac{5}{56}x$. The coefficient of $x$ on the left is $-\dfrac{3}{14}r$; on the right it is $-\dfrac{5}{56}$.

3. Set the coefficients equal: $-\dfrac{3}{14}r = -\dfrac{5}{56}$.

4. Solve for $r$: Multiplying both sides by $-1$ gives $\dfrac{3}{14}r = \dfrac{5}{56}$. Multiplying both sides by $\dfrac{14}{3}$ isolates $r$: $r = \dfrac{5}{56} \times \dfrac{14}{3} = \dfrac{5 \times 14}{56 \times 3}$. Since $56 = 4 \times 14$, the $14$ cancels, leaving $r = \dfrac{5}{4 \times 3} = \dfrac{5}{12}$.

5. Check the constant terms: For no solution the constants must be unequal, so $\dfrac{s}{9} \neq 10$, meaning $s \neq 90$. This condition constrains $s$ but does not affect $r$.

6. Conclusion: The value of $r$ is $\dfrac{5}{12}$.

Note that $5/12$, $.4166$, $.4167$, $0.416$, and $0.417$ are examples of ways to enter a correct answer.