The problem you've uploaded presents a system of equations:

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

You are asked to find the value of $r$ if the system has no solution.

Step 1: Simplify the first equation

We start by simplifying the first equation:

$48x - 72y = 30y + 24$

Move all terms involving $y$ to one side:

$48x - 72y - 30y = 24$

$48x - 102y = 24$

This is the simplified form of the first equation.

Step 2: Rearrange the second equation

The second equation is:

$ry = \frac{1}{6} - 16x$

We can leave this equation as is for now.

Step 3: Analyze the conditions for no solution

For the system to have no solution, the two lines represented by these equations must be parallel. Parallel lines have the same slope but different intercepts. Therefore, we need to express both equations in slope-intercept form (i.e., $y = mx + b$) and compare their slopes.

First equation in slope-intercept form:

$48x - 102y = 24$

Solving for $y$:

$-102y = -48x + 24$ $y = \frac{48}{102}x - \frac{24}{102}$ $y = \frac{24}{51}x - \frac{12}{51}$

The slope of the first equation is $\frac{24}{51}$.

Second equation in slope-intercept form:

From the second equation, we solve for $y$:

$ry = \frac{1}{6} - 16x$ $y = \frac{\frac{1}{6} - 16x}{r}$ $y = \frac{1}{6r} - \frac{16x}{r}$

The slope of this equation is $-\frac{16}{r}$.

Step 4: Set slopes equal for parallel lines

For the system to have no solution, the slopes must be equal:

$\frac{24}{51} = -\frac{16}{r}$

Now, solve for $r$:

$r = -\frac{16 \times 51}{24}$ $r = -\frac{816}{24}$ $r = -34$

Thus, the value of $r$ is $\boxed{-34}$.

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How do you determine when two lines are parallel?
2. What would happen if the system had exactly one solution?
3. Can the value of $r$ change if different methods are used to solve the system?
4. How do you interpret the meaning of a system of equations having no solution geometrically?
5. What would the solution be if the system had an infinite number of solutions?

Tip: When solving systems of equations, always check for special cases such as parallel lines or identical lines to determine if the system has no solution or infinite solutions.