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 must first understand the properties of parallel lines. Parallel lines have the same slope.
### Step-by-Step Solution
1. Identify the coordinates:
- The coordinates of the first point are [tex]\((-6, 9)\)[/tex].
- The coordinates of the second point are [tex]\((7, -17)\)[/tex].
2. Calculate the slope ([tex]\(m\)[/tex]) of the line passing through the given points:
- 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:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
- Substitute the coordinates:
[tex]\[
m = \frac{-17 - 9}{7 - (-6)} = \frac{-17 - 9}{7 + 6} = \frac{-26}{13} = -2
\][/tex]
3. Understand the implication for parallel lines:
- Since parallel lines have the same slope, any line parallel to the one passing through the points [tex]\((-6, 9)\)[/tex] and [tex]\((7, -17)\)[/tex] must also have a slope of [tex]\(-2\)[/tex].
4. Match the slope with the given options:
- Option A: [tex]\( y = -\frac{1}{2}x + 13 \)[/tex] has slope [tex]\(-\frac{1}{2}\)[/tex].
- Option B: [tex]\( y = \frac{1}{2}x + 13 \)[/tex] has slope [tex]\(\frac{1}{2}\)[/tex].
- Option C: [tex]\( y = -2x + 13 \)[/tex] has slope [tex]\(-2\)[/tex].
- Option D: [tex]\( y = 2x + 13 \)[/tex] has slope [tex]\(2\)[/tex].
5. Select the correct option:
- The slope of the original line is [tex]\(-2\)[/tex]. Hence, the equation that represents a line parallel to the given line must also have a slope of [tex]\(-2\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{y = -2x + 13} \][/tex]
