To find the equation of the line that passes through the origin (0, 0) and is parallel to the line that passes through points [tex]\( A(-3, 0) \)[/tex] and [tex]\( B(-6, 5) \)[/tex], follow these steps:
### Step 1: Find the Slope of Line [tex]\(AB\)[/tex]
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1,y_1)\)[/tex] and [tex]\((x_2,y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
m = \frac{5 - 0}{-6 - (-3)}
\][/tex]
[tex]\[
m = \frac{5}{-6 + 3}
\][/tex]
[tex]\[
m = \frac{5}{-3} = -\frac{5}{3}
\][/tex]
### Step 2: Determine the Equation of the Line through the Origin
The slope of the line passing through the origin and parallel to line [tex]\(AB\)[/tex] will be the same as the slope of line [tex]\(AB\)[/tex].
The slope-intercept form of a line is:
[tex]\[
y = mx + c
\][/tex]
Since the line passes through the origin (0, 0), the y-intercept [tex]\( c \)[/tex] is 0. Thus, the equation is:
[tex]\[
y = -\frac{5}{3}x
\][/tex]
### Step 3: Convert to Standard Form
To convert [tex]\( y = -\frac{5}{3}x \)[/tex] to standard form [tex]\( Ax + By = C \)[/tex]:
1. Multiply both sides by 3 to eliminate the fraction:
[tex]\[
3y = -5x
\][/tex]
2. Rearrange all terms to one side to get the equation in standard form:
[tex]\[
5x + 3y = 0
\][/tex]
### Conclusion
The correct equation of the line that passes through the origin and is parallel to the line passing through points [tex]\( A(-3, 0) \)[/tex] and [tex]\( B(-6, 5) \)[/tex] is:
[tex]\[
5x + 3y = 0
\][/tex]
This corresponds to option [tex]\( \boxed{A} \)[/tex].
