To determine which points lie on the line of direct variation that goes through the point (5, 12), we first need to find the equation of the line. Since it’s a direct variation, the line can be described by the equation [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is the constant of variation (slope of the line).
1. Find the slope [tex]\( k \)[/tex] of the direct variation line:
Given the point (5, 12),
[tex]\[
k = \frac{y}{x} = \frac{12}{5} = 2.4
\][/tex]
Therefore, the line's equation is:
[tex]\[
y = 2.4x
\][/tex]
2. Check each given point to see if it satisfies the equation [tex]\( y = 2.4x \)[/tex]:
- [tex]\((0, 0)\)[/tex]:
[tex]\[
y = 2.4 \cdot 0 = 0
\][/tex]
The point (0, 0) lies on the line.
- [tex]\((2.5, 6)\)[/tex]:
[tex]\[
y = 2.4 \cdot 2.5 = 6
\][/tex]
The point (2.5, 6) lies on the line.
- [tex]\((3, 10)\)[/tex]:
[tex]\[
y = 2.4 \cdot 3 = 7.2
\][/tex]
The point (3, 10) does not lie on the line.
- [tex]\((7.5, 18)\)[/tex]:
[tex]\[
y = 2.4 \cdot 7.5 = 18
\][/tex]
The point (7.5, 18) lies on the line.
- [tex]\((12.5, 24)\)[/tex]:
[tex]\[
y = 2.4 \cdot 12.5 = 30
\][/tex]
The point (12.5, 24) does not lie on the line.
- [tex]\((15, 36)\)[/tex]:
[tex]\[
y = 2.4 \cdot 15 = 36
\][/tex]
The point (15, 36) lies on the line.
3. Summary of points that lie on the line:
- [tex]\((0, 0)\)[/tex]
- [tex]\((2.5, 6)\)[/tex]
- [tex]\((7.5, 18)\)[/tex]
- [tex]\((15, 36)\)[/tex]
Therefore, the points that lie on the line of direct variation through (5, 12) are:
[tex]\[
(0, 0), (2.5, 6), (7.5, 18), (15, 36)
\][/tex]
