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].
