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 should follow these steps:
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:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
- Substituting the given points [tex]\((-6, 9)\)[/tex] and [tex]\((7, -17)\)[/tex]:
[tex]\[
m = \frac{-17 - 9}{7 + 6} = \frac{-26}{13} = -2
\][/tex]
So, the slope of the line passing through the points [tex]\((-6, 9)\)[/tex] and [tex]\((7, -17)\)[/tex] is [tex]\(-2\)[/tex].
2. Identify which of the given equations has the same slope (i.e., [tex]\(-2\)[/tex]):
- The slope-intercept form of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
- For each option, the slope [tex]\(m\)[/tex] is:
- A. [tex]\(y = 2x + 13\)[/tex] ⇒ slope [tex]\(m = 2\)[/tex]
- B. [tex]\(y = -2x + 13\)[/tex] ⇒ slope [tex]\(m = -2\)[/tex]
- C. [tex]\(y = \frac{1}{2}x + 13\)[/tex] ⇒ slope [tex]\(m = \frac{1}{2}\)[/tex]
- D. [tex]\(y = -\frac{1}{2}x + 13\)[/tex] ⇒ slope [tex]\(m = -\frac{1}{2}\)[/tex]
3. Determine the correct equation:
Since the slope of the line passing through the points [tex]\((-6, 9)\)[/tex] and [tex]\((7, -17)\)[/tex] is [tex]\(-2\)[/tex], the equation that represents a line that is parallel to this line must also have a slope of [tex]\(-2\)[/tex].
Hence, 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]\[ \boxed{B. \ y = -2x + 13} \][/tex]
