https://file.aiquickdraw.com/mathbot/file/17268584917873yqvklf3.jpg

Let's solve the problem step by step:

The problem states that \( y \) varies directly with \( x \). This means that \( y = kx \), where \( k \) is the constant of variation.

### Step 1: Find the constant of variation (\( k \)).

We are given the ordered pair \( (12, -4) \). Plugging this into the direct variation equation:

\[
-4 = k \cdot 12
\]

To solve for \( k \):

\[
k = \frac{-4}{12} = -\frac{1}{3}
\]

So, the constant of variation is \( k = -\frac{1}{3} \).

### Step 2: Find the value of \( x \) when \( y = -6 \).

Using the equation \( y = kx \), substitute \( y = -6 \) and \( k = -\frac{1}{3} \):

\[
-6 = -\frac{1}{3} \cdot x
\]

Multiply both sides by \( -3 \) to solve for \( x \):

\[
x = 18
\]

### Final answers:
- The constant of variation is \( -\frac{1}{3} \) (or -⅓).
- The value of \( x \) when \( y = -6 \) is \( 18 \).

Now, using the available choices in the image:
- The constant of variation is \( -\frac{1}{3} \).
- The value of \( x \) is \( 18 \).

Would you like more details on this? Here are five related questions to expand your understanding:

1. How do you determine if a relationship shows direct variation?
2. What would change if \( y \) varied inversely with \( x \) instead of directly?
3. Can a constant of variation ever be zero in direct variation?
4. How do we interpret the graph of a direct variation relationship?
5. How would you solve for \( k \) if multiple ordered pairs are given?

**Tip:** In a direct variation, as one variable increases or decreases, the other does so proportionally. The constant \( k \) helps establish this relationship.

Mrs. Jones gave her students the ordered pair (12, -4) and told them y varies directly with x. What is the constant of variation when y is a function of x, and what is the value of x when y = -6?

Learn how to solve a direct variation problem where y varies directly with x, given the ordered pair (12, -4). Calculate the constant of variation and find the value of x when y = -6 using algebraic formulas. Suitable for students in Grades 8-9.

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