What is the constant of variation, [tex]\( k \)[/tex], of the direct variation [tex]\( y = kx \)[/tex] through [tex]\((-3, 2)\)[/tex]?

A. [tex]\( k = -\frac{3}{2} \)[/tex]
B. [tex]\( k = \frac{2}{3} \)[/tex]
C. [tex]\( k = \frac{2}{3} \)[/tex]
D. [tex]\( k = \frac{3}{2} \)[/tex]



Answer :

To find the constant of variation [tex]\( k \)[/tex] for the direct variation equation [tex]\( y = kx \)[/tex] that passes through the point [tex]\((-3, 2)\)[/tex], we follow these steps:

1. Identify the given point:
- The given point is [tex]\((-3, 2)\)[/tex].
- This means when [tex]\( x = -3 \)[/tex], [tex]\( y = 2 \)[/tex].

2. Recall the direct variation formula:
- The formula for direct variation is [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is the constant of variation.

3. Substitute the values into the formula:
- Substitute [tex]\( x = -3 \)[/tex] and [tex]\( y = 2 \)[/tex] into the equation [tex]\( y = kx \)[/tex]:
[tex]\[ 2 = k \cdot (-3) \][/tex]

4. Solve for [tex]\( k \)[/tex]:
- Rearrange the equation to solve for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{y}{x} = \frac{2}{-3} \][/tex]

Therefore, the constant of variation [tex]\( k \)[/tex] is:
[tex]\[ k = -\frac{2}{3} \][/tex]

This value corresponds to one of the provided choices. So, the correct answer is:
[tex]\[ k = -\frac{2}{3} \][/tex]