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The problem states that \( y \) varies directly as \( x \), meaning that the relationship between \( y \) and \( x \) can be written as:

\[
y = kx
\]

where \( k \) is the constant of variation (also called the constant of proportionality).

Let's solve each problem step by step.

### 1. If \( y = 12 \) when \( x = 4 \), find \( y \) when \( x = 12 \).

First, find the constant \( k \):
\[
12 = k \times 4 \quad \Rightarrow \quad k = \frac{12}{4} = 3
\]
Now, use \( k = 3 \) to find \( y \) when \( x = 12 \):
\[
y = 3 \times 12 = 36
\]

### 2. If \( y = 25 \) when \( x = 5 \), find \( x \) when \( y = 100 \).

Find the constant \( k \):
\[
25 = k \times 5 \quad \Rightarrow \quad k = \frac{25}{5} = 5
\]
Now, use \( k = 5 \) to find \( x \) when \( y = 100 \):
\[
100 = 5 \times x \quad \Rightarrow \quad x = \frac{100}{5} = 20
\]

### 3. If \( y = -8 \) when \( x = 2 \), find \( x \) when \( y = 24 \).

Find the constant \( k \):
\[
-8 = k \times 2 \quad \Rightarrow \quad k = \frac{-8}{2} = -4
\]
Now, use \( k = -4 \) to find \( x \) when \( y = 24 \):
\[
24 = -4 \times x \quad \Rightarrow \quad x = \frac{24}{-4} = -6
\]

### 4. If \( y = 14 \) when \( x = 7 \), find \( y \) when \( x = 21 \).

Find the constant \( k \):
\[
14 = k \times 7 \quad \Rightarrow \quad k = \frac{14}{7} = 2
\]
Now, use \( k = 2 \) to find \( y \) when \( x = 21 \):
\[
y = 2 \times 21 = 42
\]

### 5. If \( y = -18 \) when \( x = -6 \), find \( x \) when \( y = 90 \).

Find the constant \( k \):
\[
-18 = k \times -6 \quad \Rightarrow \quad k = \frac{-18}{-6} = 3
\]
Now, use \( k = 3 \) to find \( x \) when \( y = 90 \):
\[
90 = 3 \times x \quad \Rightarrow \quad x = \frac{90}{3} = 30
\]

### Summary of Answers:
1. \( y = 36 \)
2. \( x = 20 \)
3. \( x = -6 \)
4. \( y = 42 \)
5. \( x = 30 \)

Would you like more details or have any questions?

Here are five related questions for further practice:
1. What happens to \( y \) if \( x \) is halved, assuming direct variation?
2. If \( y \) varies directly as \( x \) and \( y = 18 \) when \( x = 6 \), what is \( y \) when \( x = 15 \)?
3. How do you determine the constant of proportionality in a direct variation problem?
4. If the constant \( k \) is negative, what effect does this have on the relationship between \( y \) and \( x \)?
5. If \( y = 50 \) when \( x = 10 \), what is the formula for the relationship between \( y \) and \( x \)?

**Tip:** In problems of direct variation, once you find the constant \( k \), solving for any other values becomes a straightforward application of the formula \( y = kx \).

Test III. In each of the following, y varies directly as x. Find the value as indicated.

Learn how to solve direct variation problems involving y and x. This step-by-step guide covers key concepts of direct proportionality and algebra, suitable for students in Grades 7-9.

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