find the constant of variation k and the undetermined value in the table if y is directly proportional to x. 

To find the constant of variation \( k \) and the undetermined value in a table when \( y \) is directly proportional to \( x \), we can use the direct variation relationship formula:

\[
y = kx
\]

Here, \( k \) is the constant of variation, and \( y \) varies directly with \( x \).

### Steps to Solve the Problem:

1. **Identify the Relationship**: Since \( y \) is directly proportional to \( x \), we can write:
   \[
   y = kx
   \]
   
2. **Find \( k \)**: To find \( k \), we need at least one pair of \( x \) and \( y \) values from the table.

3. **Calculate the Undetermined Value**: Once we find \( k \), we can use it to calculate the missing \( y \) or \( x \) value.

### Example Problem

Let’s say we have the following table with some values missing:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
2 & 10 \\
4 & y \\
x & 30 \\
\hline
\end{array}
\]

#### 1. Find the constant \( k \)

Using the pair \((x, y) = (2, 10)\), we can substitute into the direct variation formula:

\[
y = kx \Rightarrow 10 = k \times 2
\]

\[
k = \frac{10}{2} = 5
\]

So, the constant of variation \( k \) is 5.

#### 2. Calculate the undetermined value \( y \) when \( x = 4 \)

Using the direct variation formula with \( k = 5 \):

\[
y = kx \Rightarrow y = 5 \times 4 = 20
\]

Thus, \( y = 20 \) when \( x = 4 \).

#### 3. Calculate the undetermined value \( x \) when \( y = 30 \)

Again, using the formula \( y = kx \) with \( k = 5 \):

\[
30 = 5x
\]

\[
x = \frac{30}{5} = 6
\]

Thus, \( x = 6 \) when \( y = 30 \).

### Conclusion

- The constant of variation \( k \) is 5.
- When \( x = 4 \), \( y = 20 \).
- When \( y = 30 \), \( x = 6 \).

---

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

### Related Questions:

1. How would the calculation change if \( y \) were inversely proportional to \( x \)?
2. What are some examples of real-life situations where direct variation is applicable?
3. How do we handle cases where the relationship is not directly proportional but instead involves a more complex equation?
4. What if the constant of variation \( k \) was a fraction? How would that affect the calculations?
5. How can we graph the relationship between \( x \) and \( y \) for a direct variation?

**Tip:** Direct variation problems are easier to solve when you identify pairs of known values for \( x \) and \( y \) because they allow you to determine the constant \( k \), which you can then use to find other missing values.

Learn how to find the constant of variation (k) and the undetermined value in a table when y is directly proportional to x. Includes step-by-step solution and example problems. Suitable for Grades 7-9.

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