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Is the following relationship proportional? Explain.

To determine if the relationship is proportional, we need to check if the ratio between the total cost of tickets (\( y \)) and the number of movie tickets (\( x \)) is constant. 

Here is the data from the table:

\[
\begin{array}{c|c}
\text{Number of Movie Tickets } (x) & \text{Total Cost of Tickets } (y) \\
\hline
1 & -6 \\
2 & -12 \\
3 & -18 \\
4 & -24 \\
\end{array}
\]

To check if the relationship is proportional, we can calculate the ratio \(\frac{y}{x}\) for each pair:

\[
\frac{y}{x} = \frac{-6}{1} = -6
\]
\[
\frac{y}{x} = \frac{-12}{2} = -6
\]
\[
\frac{y}{x} = \frac{-18}{3} = -6
\]
\[
\frac{y}{x} = \frac{-24}{4} = -6
\]

Since the ratio \(\frac{y}{x}\) is \(-6\) for each pair, the relationship is indeed proportional. The constant of proportionality here is \(-6\).

**Explanation**: A relationship is proportional if the ratio between the two variables is constant. In this case, the total cost \( y \) is always \(-6\) times the number of tickets \( x \), making the relationship proportional.

Would you like more details or have any questions?

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Here are five related questions:
1. What does the constant of proportionality represent in this context?
2. How would the relationship change if the ratio was not constant?
3. Can a proportional relationship have a positive constant of proportionality?
4. What are some examples of proportional relationships in real life?
5. How would you graph the relationship between \( x \) and \( y \) on a coordinate plane?

**Tip**: Always verify the constancy of ratios when determining proportionality in a table of values.

Explore whether the relationship between the number of movie tickets and total cost of tickets is proportional. Detailed analysis includes calculating ratios and understanding the constant of proportionality. Suitable for grades 6-8.

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