In a direct variation, the relationship between two variables can be expressed in the form \( y = kx \), where \( k \) is the constant of variation.
Given the equation \( 4x = -6y \), we can rearrange it to find \( k \).
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First, solve for \( y \): \[ -6y = 4x \] \[ y = -\frac{4}{6}x \] \[ y = -\frac{2}{3}x \]
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Here, we can see that the constant of variation \( k \) is \(-\frac{2}{3}\).
Therefore, we conclude: \[ k = -\frac{2}{3} \]