To find the equation of the line passing through the points [tex]\((6,9)\)[/tex] and [tex]\((0,0)\)[/tex], we should follow these steps:
1. Determine the slope (m) of the line:
The slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be calculated using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates of the given points [tex]\((6, 9)\)[/tex] and [tex]\((0, 0)\)[/tex]:
[tex]\[
m = \frac{0 - 9}{0 - 6} = \frac{-9}{-6} = \frac{3}{2}
\][/tex]
2. Use the point-slope form of the equation of a line:
The point-slope form is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Since the line passes through the origin [tex]\((0,0)\)[/tex], the y-intercept (b) is 0. Thus, the equation simplifies to:
[tex]\[
y = mx
\][/tex]
Substituting the slope [tex]\(m = \frac{3}{2}\)[/tex]:
[tex]\[
y = \frac{3}{2} x
\][/tex]
3. Verify the equation fits given options:
Among the provided options:
a. [tex]\(y = x - 3\)[/tex] does not match.
b. [tex]\(y = x + 3\)[/tex] does not match.
c. [tex]\(y = \frac{2}{3} x\)[/tex] does not match.
d. [tex]\(y = \frac{3}{2} x\)[/tex] matches the equation derived.
So the correct answer is:
d. [tex]\(y = \frac{3}{2} x\)[/tex]
