To determine the equation of the line that passes through the origin and is parallel to line [tex]\( AB \)[/tex] which passes through the points [tex]\( A(-3, 0) \)[/tex] and [tex]\( B(-6, 5) \)[/tex], we can follow these steps:
### Step 1: Calculate the slope of line [tex]\( AB \)[/tex]
The formula to calculate the slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Applying the formula to our points:
[tex]\[
\text{slope of } AB = \frac{5 - 0}{-6 - (-3)} = \frac{5 - 0}{-6 + 3} = \frac{5}{-3} = -\frac{5}{3}
\][/tex]
So, the slope of line [tex]\( AB \)[/tex] is [tex]\(-\frac{5}{3}\)[/tex].
### Step 2: Determine the equation of the line through the origin with the same slope
A line that is parallel to [tex]\( AB \)[/tex] and passes through the origin will have the same slope, [tex]\(-\frac{5}{3}\)[/tex]. The general form for the equation of a line in slope-intercept form [tex]\( y = mx + b \)[/tex] passing through the origin [tex]\((0, 0)\)[/tex] simplifies to:
[tex]\[
y = mx \Rightarrow y = -\frac{5}{3}x
\][/tex]
### Step 3: Convert the equation to standard form
To convert [tex]\( y = -\frac{5}{3}x \)[/tex] to standard form [tex]\( Ax + By = C \)[/tex], we can multiply every term by [tex]\( 3 \)[/tex] to clear the fraction:
[tex]\[
y = -\frac{5}{3}x
\][/tex]
Multiply by 3:
[tex]\[
3y = -5x
\][/tex]
Rearrange terms to match the format [tex]\( Ax + By = C \)[/tex]:
[tex]\[
5x + 3y = 0 \Rightarrow 5x - (-3)y = 0
\][/tex]
Comparing with the options given:
- 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]
We see that the correct equation in standard form is:
### Final Answer
[tex]\[
\boxed{5x - 3y = 0}
\][/tex]
