To determine the constant of variation, [tex]\( k \)[/tex], for the line [tex]\( y = kx \)[/tex] that passes through the points [tex]\((3, 18)\)[/tex] and [tex]\((5, 30)\)[/tex], follow these steps:
1. Identify the given points:
- [tex]\((x_1, y_1) = (3, 18)\)[/tex]
- [tex]\((x_2, y_2) = (5, 30)\)[/tex]
2. Use the formula for the constant of variation [tex]\( k \)[/tex] for a line passing through two points:
[tex]\[
k = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
3. Substitute the coordinates of the points into the formula:
[tex]\[
k = \frac{30 - 18}{5 - 3}
\][/tex]
4. Calculate the numerator and the denominator:
- Numerator: [tex]\( 30 - 18 = 12 \)[/tex]
- Denominator: [tex]\( 5 - 3 = 2 \)[/tex]
5. Divide the results to find [tex]\( k \)[/tex]:
[tex]\[
k = \frac{12}{2} = 6.0
\][/tex]
Thus, the constant of variation [tex]\( k \)[/tex] of the line [tex]\( y = kx \)[/tex] through the points [tex]\((3, 18)\)[/tex] and [tex]\((5, 30)\)[/tex] is [tex]\( 6 \)[/tex].
Therefore, the answer is:
[tex]\[
\boxed{6}
\][/tex]
