To determine the constant of variation, [tex]\( k \)[/tex], for the direct variation equation [tex]\( y = kx \)[/tex] through the point [tex]\((-3, 2)\)[/tex], we can follow these steps:
1. Identify the given point:
- We are given the coordinates [tex]\((x, y) = (-3, 2)\)[/tex].
2. Write the direct variation equation:
- The equation for direct variation is [tex]\( y = kx \)[/tex].
3. Substitute the given point into the equation:
- Substitute [tex]\( x = -3 \)[/tex] and [tex]\( y = 2 \)[/tex] into the equation [tex]\( y = kx \)[/tex]:
[tex]\[
2 = k(-3)
\][/tex]
4. Solve for [tex]\( k \)[/tex]:
- Isolate [tex]\( k \)[/tex] by dividing both sides of the equation by [tex]\(-3\)[/tex]:
[tex]\[
k = \frac{2}{-3}
\][/tex]
5. Simplify the expression:
- [tex]\(\frac{2}{-3}\)[/tex] simplifies to [tex]\(-\frac{2}{3}\)[/tex].
Therefore, the constant of variation, [tex]\( k \)[/tex], is [tex]\(-\frac{2}{3}\)[/tex]. The correct answer is not listed among the choices given as they seem to have positive constants, but based on the steps we have followed, the actual constant is [tex]\(-\frac{2}{3}\)[/tex].
