In a direct variation, the relationship between \( x \) and \( y \) can be expressed as:
\[ y = kx \]
where \( k \) is the constant of variation.
From the equation \( 4x = -6y \), we can rearrange it to find the relationship between \( x \) and \( y \):
\[ -6y = 4x \]
Dividing both sides by -6 gives:
\[ y = -\frac{4}{6}x \]
Simplifying \(-\frac{4}{6}\):
\[ y = -\frac{2}{3}x \]
Now, comparing this to the direct variation formula \( y = kx \), we can see that the constant of variation \( k \) is:
\[ k = -\frac{2}{3} \]
Thus, the constant of variation \( k \) is \(-\frac{2}{3}\).