To determine which table represents a proportional relationship between $x$ and $y$, we need to check whether the ratio $\frac{y}{x}$ is constant for all $(x, y)$ pairs in the table. Here’s how to evaluate each option:

A:

$\text{Ratios: } \frac{7}{-6}, \frac{9}{-3}, \frac{3}{6}, \frac{-3}{15}$

• $\frac{7}{-6} = -\frac{7}{6}$
• $\frac{9}{-3} = -3$
• $\frac{3}{6} = \frac{1}{2}$
• $\frac{-3}{15} = -\frac{1}{5}$

The ratios are not constant. Not proportional.

B:

$\text{Ratios: } \frac{4}{-3}, \frac{6}{-1}, \frac{7}{1}, \frac{10}{3}$

• $\frac{4}{-3} = -\frac{4}{3}$
• $\frac{6}{-1} = -6$
• $\frac{7}{1} = 7$
• $\frac{10}{3} = \frac{10}{3}$

The ratios are not constant. Not proportional.

C:

$\text{Ratios: } \frac{-12}{-3}, \frac{-8}{-2}, \frac{0}{0}, \frac{16}{4}$

• $\frac{-12}{-3} = 4$
• $\frac{-8}{-2} = 4$
• $\frac{0}{0}$ is undefined, but we can ignore it for proportionality.
• $\frac{16}{4} = 4$

The ratios are constant ($4$). This table represents a proportional relationship.

D:

$\text{Ratios: } \frac{-5}{-6}, \frac{1}{-2}, \frac{4}{0}, \frac{16}{8}$

• $\frac{-5}{-6} = \frac{5}{6}$
• $\frac{1}{-2} = -\frac{1}{2}$
• $\frac{4}{0}$ is undefined.
• $\frac{16}{8} = 2$

The ratios are not constant. Not proportional.

Conclusion:

Table C represents a proportional relationship between $x$ and $y$.

Would you like further details or clarifications?

5 Questions to Explore:

1. How do we handle undefined ratios in proportionality checks?
2. Can proportionality exist with non-zero $y$-intercepts?
3. What real-world examples illustrate proportional relationships?
4. How do unit rates relate to proportional relationships?
5. How can graphing $y$ versus $x$ confirm proportionality?

Tip: When checking for proportionality, always compute and compare the ratios $\frac{y}{x}$ (or $\frac{x}{y}$) across all pairs.