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.