To find the equation of the line that passes through the origin and is parallel to line [tex]\( AB \)[/tex] which passes through points [tex]\( A(-3,0) \)[/tex] and [tex]\( B(-6,5) \)[/tex], we follow these steps:
1. Calculate the slope of line [tex]\( AB \)[/tex]:
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is:
[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]\[
m = \frac{5 - 0}{-6 - (-3)} = \frac{5}{-3} = -\frac{5}{3}
\][/tex]
2. Find the equation of the line passing through the origin and parallel to [tex]\( AB \)[/tex]:
Since parallel lines have the same slope, the slope of the line we are looking for is also [tex]\( -\frac{5}{3} \)[/tex].
Using the point-slope form of the equation of a line [tex]\( y = mx + b \)[/tex], where [tex]\( b \)[/tex] is the y-intercept, and since this line passes through the origin, [tex]\( (0,0) \)[/tex]:
[tex]\[
y = -\frac{5}{3}x + 0
\][/tex]
Simplifying, we get:
[tex]\[
y = -\frac{5}{3}x
\][/tex]
3. Convert [tex]\( y = -\frac{5}{3}x \)[/tex] to the standard form [tex]\( Ax + By = 0 \)[/tex]:
Multiply every term by 3 to clear the fraction:
[tex]\[
3y = -5x
\][/tex]
Rearrange to the form [tex]\( Ax + By = 0 \)[/tex]:
[tex]\[
5x + 3y = 0
\][/tex]
Therefore, the equation of the line that passes through the origin and is parallel to line [tex]\( AB \)[/tex] is:
[tex]\[
5x + 3y = 0
\][/tex]
4. Match the solution with the given options:
The options are:
- [tex]\( 5x - 3y = 0 \)[/tex]
- [tex]\( -x + 3y = 0 \)[/tex]
- [tex]\( -5x - 3y = 0 \)[/tex]
- [tex]\( 3x + 5y = 0 \)[/tex]
- [tex]\( -3x + 5y = 0 \)[/tex]
The correct equation [tex]\( 5x + 3y = 0 \)[/tex] matches none of the given options.
Hence, based on the given options, there is no correct match. The answer derived did not fit into any of the provided choices. Therefore, further verification shows that none of the options is a correct representation of the derived equation [tex]\( 5x + 3y = 0 \)[/tex].
