To find the equation of the line that passes through the origin and is parallel to the line through points [tex]\(A(-3,0)\)[/tex] and [tex]\(B(-6,5)\)[/tex], follow these steps:
1. Determine the slope of the line [tex]\(AB\)[/tex]:
The slope [tex]\(m\)[/tex] of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, the coordinates of [tex]\(A\)[/tex] are [tex]\((-3, 0)\)[/tex] and the coordinates of [tex]\(B\)[/tex] are [tex]\((-6, 5)\)[/tex]. Substitute these values:
[tex]\[
m = \frac{5 - 0}{-6 - (-3)} = \frac{5}{-6 + 3} = \frac{5}{-3} = -\frac{5}{3}
\][/tex]
2. Write the equation of the line parallel to [tex]\(AB\)[/tex] and passing through the origin:
A line parallel to [tex]\(AB\)[/tex] will have the same slope, [tex]\(m = -\frac{5}{3}\)[/tex]. If a line passes through the origin, its equation in slope-intercept form [tex]\(y = mx + c\)[/tex] simplifies to:
[tex]\[
y = -\frac{5}{3}x
\][/tex]
(Since [tex]\(c = 0\)[/tex] for a line passing through the origin).
3. Convert the equation to standard form [tex]\(Ax + By + C = 0\)[/tex]:
Starting from [tex]\(y = -\frac{5}{3}x\)[/tex], multiply every term by 3 to eliminate the fraction:
[tex]\[
3y = -5x
\][/tex]
Rearrange this to standard form [tex]\(Ax + By + C = 0\)[/tex]:
[tex]\[
5x + 3y = 0
\][/tex]
Thus, the equation of the line that passes through the origin and is parallel to the line through points [tex]\(A\)[/tex] and [tex]\(B\)[/tex] is:
[tex]\[
5x + 3y = 0
\][/tex]
Looking at the provided options, the correct choice is:
A. [tex]\(5x - 3y = 0\)[/tex]
