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 find the slope of this line.
The formula for the slope [tex]\( m \)[/tex] of a line passing through 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]
Given the points [tex]\((-6, 9)\)[/tex] and [tex]\((7, -17)\)[/tex], we can substitute these coordinates into the formula:
[tex]\[ m = \frac{-17 - 9}{7 - (-6)} \][/tex]
First, simplify the numerator and the denominator:
1. Calculate [tex]\(-17 - 9\)[/tex]:
[tex]\[ -17 - 9 = -26 \][/tex]
2. Calculate [tex]\(7 - (-6)\)[/tex]:
[tex]\[ 7 - (-6) = 7 + 6 = 13 \][/tex]
Now, substitute these values back into the slope formula:
[tex]\[ m = \frac{-26}{13} = -2 \][/tex]
So, the slope of the line passing through the points [tex]\((-6, 9)\)[/tex] and [tex]\((7, -17)\)[/tex] is [tex]\(-2\)[/tex].
Next, we need to find the equation among the given options that has the same slope, [tex]\(-2\)[/tex]. Here are the equations with their corresponding slopes:
A. [tex]\( y = \frac{1}{2} x + 13 \)[/tex] has a slope of [tex]\(\frac{1}{2}\)[/tex]
B. [tex]\( y = -2 x + 13 \)[/tex] has a slope of [tex]\(-2\)[/tex]
C. [tex]\( y = 2 x + 13 \)[/tex] has a slope of [tex]\(2\)[/tex]
D. [tex]\( y = -\frac{1}{2} x + 13 \)[/tex] has a slope of [tex]\(-\frac{1}{2}\)[/tex]
The line with the same slope of [tex]\(-2\)[/tex] is:
[tex]\[ \boxed{B. \, y = -2x + 13} \][/tex]
This equation represents a line that is parallel to the line passing through the points [tex]\((-6, 9)\)[/tex] and [tex]\((7, -17)\)[/tex].
