Let's analyze the given system of equations:

1. $48x - 72y = 30y + 24$
2. $ry = \frac{1}{6} - 16x$

We need to find the value of $r$ for which the system has no solution.

Step-by-Step Solution:

1. Simplify the first equation: $48x - 72y = 30y + 24$ Combine like terms: $48x - 102y = 24$ Dividing the entire equation by 6: $8x - 17y = 4 \quad \text{(Equation 1)}$

2. Rearrange the second equation: $ry = \frac{1}{6} - 16x$ Multiply through by 6 to clear the fraction: $6ry = 1 - 96x$ Rearrange to solve for $y$: $y = \frac{1 - 96x}{6r} \quad \text{(Equation 2)}$

3. Substitute $y$ from Equation 2 into Equation 1: $8x - 17 \left( \frac{1 - 96x}{6r} \right) = 4$ Simplify: $8x - \frac{17(1 - 96x)}{6r} = 4$ Multiply through by $6r$ to eliminate the denominator: $48xr - 17(1 - 96x) = 24r$ Expand: $48xr - 17 + 1632x = 24r$ Combine like terms: $(48r + 1632)x = 24r + 17$ $x = \frac{24r + 17}{48r + 1632}$

4. Determine the condition for no solution: For the system to have no solution, the denominator must be zero, since it would lead to a division by zero, making $x$ undefined. Set the denominator equal to zero:

   $48r + 1632 = 0$ $48r = -1632$ $r = -34$

Therefore, the value of $r$ for which the system has no solution is $\boxed{-34}$.

Would you like more details on any step or have any questions?

Here are some related questions to explore:

1. What happens if $r = -34$ in the system of equations?
2. How would you solve this system if $r$ had a different value?
3. Can you find the solution for the system if $r = 1$?
4. What geometric interpretation corresponds to a system with no solution?
5. What is the significance of parallel lines in the context of this problem?

Tip: When solving systems of equations, always consider simplifying one equation before substitution or elimination to reduce calculation complexity.