Question

Line AB passes through A(-3, 0) and B(-6, 5). What is the equation of the line that passes through the origin and is parallel to line AB?
A. 5x − 3y = 0
B. -x + 3y = 0
C. -5x − 3y = 0
D. 3x + 5y = 0
E. -3x + 5y = 0

Answer 1

please choose one of the answers provided
A. 5x − 3y = 0
B. -x + 3y = 0
C. -5x − 3y = 0
D. 3x + 5y = 0
E. -3x + 5y = 0

Answer 2

To find the equation of the line that is parallel to line AB and passes through the origin, we first need to determine the slope of line AB.

1. **Calculate the slope of line AB**:

The coordinates of points A and B are A(-3, 0) and B(-6, 5), respectively. The slope \( m \) of the line passing through these points is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 0}{-6 - (-3)} = \frac{5}{-6 + 3} = \frac{5}{-3} = -\frac{5}{3}
\]

2. **Form the equation of the line parallel to AB and passing through the origin**:

Since the line we seek is parallel to line AB, it will have the same slope, \( m = -\frac{5}{3} \). The general form of a line's equation with slope \( m \) passing through the origin \((0, 0)\) is:
\[
y = mx
\]
Substituting \( m = -\frac{5}{3} \), we get:
\[
y = -\frac{5}{3}x
\]

3. **Convert to standard form**:

Multiply both sides by 3 to eliminate the fraction:
\[
3y = -5x
\]

Rearrange to standard form \( Ax + By = C \):
\[
5x + 3y = 0
\]

However, to match the standard form given in the options, we reverse the signs:
\[
-5x - 3y = 0
\]

Thus, the equation of the line that passes through the origin and is parallel to line AB is:
\[
\boxed{-5x - 3y = 0}
\]

Therefore, the correct option is not directly available but to match standard forms:

If rewritten closely similarly:
None of the given options are correct in simple multiplication check directly, assuming typographic errors of textbooks standard:

Correct form could try matching, but none of above matches exactly likely errors exist!

Answer 3

Let's re-examine our previous work and then find the closest match to the given options.

From our previous calculation:
- We determined that the slope \(m\) of line AB is \(-\frac{5}{3}\).
- The equation of the line that is parallel to AB and passes through the origin is \( y = -\frac{5}{3}x \).
- Converting this to standard form, we ended up with \( 5x + 3y = 0 \), but reversed it to match: \(-5x -3 y= 0\), not initially found.

Yet, ensuring the slope matches again if revised.
Initially parallel checking unchanged, signs still in correct form -5x - 3y=0 form:

Moving to options match by signs flip near:
Returning back correctly match correct carefully next aligned match not non-matching if allows.

From options matching:

1. A. \( 5x - 3y = 0\): Rearranging differently.
2. B. \( -x + 3y = 0\): different slopes not same!
3. C. \( -5x - 3y = 0\): Given properly closely upon corrections matched!
4. D. \( 3x + 5y = 0\): Again differently!
5. E. \( -3x + 5y =0

Final Correct noticed:

The correct matching then clearly option reread confirms:
Option:
C. \(-5x - 3y = 0)\)

Thus correct answering therefore:
\(\boxed{C)}).