https://file.aiquickdraw.com/mathbot/file/17270178821777moappfw.jpg

The image shows a table with two rows labeled \(x\) and \(y\), and various numbers in the columns. The table seems to represent a relationship between \(x\) and \(y\), possibly a proportional or arithmetic one. The incomplete table looks like this:

\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 8 & \_ & 56 & 72 \\
\hline
y & 2 & 10 & \_ & \_ \\
\hline
\end{array}
\]

To solve the missing entries, we need to identify the relationship between \(x\) and \(y\).

### Step 1: Analyze the existing pairs.
- For \(x = 8\), \(y = 2\).
- For \(x = 56\), \(y = \_ \).
- For \(x = 72\), \(y = \_ \).

If we assume a direct proportionality (i.e., \(x = k \cdot y\), where \(k\) is a constant), we can find \(k\) using the first pair:

\[
k = \frac{x}{y} = \frac{8}{2} = 4.
\]

### Step 2: Calculate the missing values using \(k = 4\).
- For \(x = 56\), \(y = \frac{56}{4} = 14\).
- For \(x = 72\), \(y = \frac{72}{4} = 18\).
  
### Step 3: Find the missing \(x\) when \(y = 10\).
Since \(x = 4 \cdot y\), for \(y = 10\):
\[
x = 4 \cdot 10 = 40.
\]

Thus, the completed table is:

\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 8 & 40 & 56 & 72 \\
\hline
y & 2 & 10 & 14 & 18 \\
\hline
\end{array}
\]

Would you like more details on this, or do you have any questions?

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Here are 5 related questions to expand on this:
1. How can you verify whether a relationship between variables is proportional?
2. What happens if the relationship is not proportional but linear instead?
3. How do you find the constant \(k\) in a directly proportional relationship?
4. Can this table be modeled using a different mathematical function, such as quadratic?
5. What is the significance of the constant ratio in real-world applications?

**Tip:** Always check for patterns in known values when dealing with sequences or tables to determine relationships like proportionality or other functions.

What are the missing values in the table? The table shows two rows labeled x and y with values like (x, y) = (8, 2), (x, y) = (56, ?), (x, y) = (72, ?).

Learn how to find the missing values in a table that shows a proportional relationship between x and y. This problem involves identifying the constant ratio between variables and applying algebraic methods to complete the table. Suitable for students in Grades 6-8.

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