Which equation represents a line that is parallel to the line that passes through the points [tex]$(-6,9)$[/tex] and [tex]$(7,-17)$[/tex]?

A. [tex]y=2x+13[/tex]

B. [tex]y=-\frac{1}{2}x+13[/tex]

C. [tex]y=-2x+13[/tex]

D. [tex]y=\frac{1}{2}x+13[/tex]



Answer :

To determine which equation represents a line that is parallel to the line passing through the points [tex]\((-6,9)\)[/tex] and [tex]\( (7,-17) \)[/tex], we first need to calculate the slope of the line that passes through these points.

1. Calculate the slope of the line passing through the points [tex]\((-6,9)\)[/tex] and [tex]\( (7,-17) \)[/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 given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

2. Plug in the coordinates of the points:
[tex]\[ x_1 = -6, \qquad y_1 = 9, \qquad x_2 = 7, \qquad y_2 = -17 \][/tex]
[tex]\[ m = \frac{-17 - 9}{7 - (-6)} = \frac{-17 - 9}{7 + 6} = \frac{-26}{13} = -2 \][/tex]

3. Identify that lines that are parallel have the same slope:

The slope of the line passing through the given points is [tex]\(-2\)[/tex]. Therefore, we need to find the equation from the given options that has the same slope of [tex]\(-2\)[/tex].

4. Check the slopes of the given equations:

- Option A [tex]\(y = 2x + 13\)[/tex] has a slope of [tex]\(2\)[/tex].
- Option B [tex]\(y = -\frac{1}{2} x + 13\)[/tex] has a slope of [tex]\(-\frac{1}{2}\)[/tex].
- Option C [tex]\(y = -2x + 13\)[/tex] has a slope of [tex]\(-2\)[/tex].
- Option D [tex]\(y = \frac{1}{2} x + 13\)[/tex] has a slope of [tex]\(\frac{1}{2}\)[/tex].

5. Identify the option with the same slope:

The equation [tex]\( y = -2x + 13 \)[/tex] (Option C) has the same slope as the line passing through [tex]\((-6, 9)\)[/tex] and [tex]\((7, -17)\)[/tex], which is [tex]\(-2\)[/tex].

Thus, the equation that represents a line parallel to the line passing through the points [tex]\((-6, 9)\)[/tex] and [tex]\((7, -17)\)[/tex] is [tex]\(\boxed{y = -2x + 13}\)[/tex].