Let's solve the problem step-by-step:
1. Identify Points and Calculate the Slope of Line [tex]\( AB \)[/tex]:
We have two points:
[tex]\[
A(-3, 0) \quad \text{and} \quad B(-6, 5)
\][/tex]
The formula for 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:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
m = \frac{5 - 0}{-6 - (-3)} = \frac{5}{-6 + 3} = \frac{5}{-3} = -\frac{5}{3}
\][/tex]
Thus, the slope of line [tex]\( AB \)[/tex] is [tex]\( -\frac{5}{3} \)[/tex].
2. Determine the Equation of the Line Passing Through the Origin and Parallel to Line [tex]\( AB \)[/tex]:
If a line passes through the origin [tex]\((0, 0)\)[/tex] and is parallel to line [tex]\( AB \)[/tex], it will have the same slope as line [tex]\( AB \)[/tex].
The general form of the equation of a line with slope [tex]\( m \)[/tex] passing through the origin is:
[tex]\[
y = mx
\][/tex]
Since the slope [tex]\( m \)[/tex] is [tex]\( -\frac{5}{3} \)[/tex], the equation of the line is:
[tex]\[
y = -\frac{5}{3}x
\][/tex]
3. Rewrite the Equation in Standard Form:
To match the given answer choices, we need to transform [tex]\( y = -\frac{5}{3}x \)[/tex] into the standard linear form [tex]\( Ax + By = 0 \)[/tex].
Multiply both sides by 3 to eliminate the fraction:
[tex]\[
3y = -5x
\][/tex]
Rearrange the equation to get it into the form [tex]\( Ax + By = 0 \)[/tex]:
[tex]\[
5x + 3y = 0
\][/tex]
Therefore, the equation of the line passing through the origin and parallel to line [tex]\( AB \)[/tex] is:
[tex]\[
5x + 3y = 0
\][/tex]
4. Match the Equation with the Given Answer Choices:
- A. [tex]\(5x - 3y = 0\)[/tex]
- B. [tex]\(-x + 3y = 0\)[/tex]
- C. [tex]\(-5x - 3y = 0\)[/tex]
- D. [tex]\(3x + 5y = 0\)[/tex]
- E. [tex]\(-3x + 5y = 0\)[/tex]
The accurate match is:
[tex]\[
\boxed{5x + 3y = 0}
\][/tex]
