To find the 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], follow these steps:
1. Calculate the slope of the line through points [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
The formula to find the slope [tex]\( m \)[/tex] of the line passing through 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 [tex]\(A(-3, 0)\)[/tex] and [tex]\(B(-6, 5)\)[/tex]:
[tex]\[
m = \frac{5 - 0}{-6 - (-3)} = \frac{5}{-6 + 3} = \frac{5}{-3} = -\frac{5}{3}
\][/tex]
2. Determine the equation of the line that passes through the origin and has the same slope:
The equation of a line in slope-intercept form [tex]\( y = mx + b \)[/tex] where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Since this line passes through the origin [tex]\((0, 0)\)[/tex], the y-intercept [tex]\( b = 0 \)[/tex]:
[tex]\[
y = -\frac{5}{3}x
\][/tex]
3. Convert the equation to the standard form [tex]\( Ax + By + C = 0 \)[/tex]:
To convert [tex]\( y = -\frac{5}{3}x \)[/tex] to standard form, we need to eliminate the fraction by multiplying every term by 3:
[tex]\[
3y = -5x
\][/tex]
Moving all terms to one side gives:
[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]
The correct answer is:
[tex]\[
\boxed{5x + 3y = 0}
\][/tex] (option A).
