To determine which equation represents a line parallel to the line passing through the points [tex]\((-6, 9)\)[/tex] and [tex]\((7, -17)\)[/tex], follow these steps:
### Step 1: Calculate the Slope of the Line
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 given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the points [tex]\((-6, 9)\)[/tex] and [tex]\((7, -17)\)[/tex]:
[tex]\[ m = \frac{-17 - 9}{7 - (-6)} \][/tex]
[tex]\[ m = \frac{-17 - 9}{7 + 6} \][/tex]
[tex]\[ m = \frac{-26}{13} \][/tex]
[tex]\[ m = -2 \][/tex]
So, the slope of the line passing through the points [tex]\((-6, 9)\)[/tex] and [tex]\((7, -17)\)[/tex] is [tex]\(-2\)[/tex].
### Step 2: Identify the Slope of the Parallel Line
Lines that are parallel have identical slopes. Therefore, the slope of the line parallel to the one passing through our points should also be [tex]\(-2\)[/tex].
### Step 3: Match the Slope with Given Equations
Now, we check the slopes of the given options:
A. [tex]\( y = 2x + 13 \)[/tex] [tex]\( \implies \)[/tex] Slope is [tex]\(2\)[/tex]
B. [tex]\( y = -2x + 13 \)[/tex] [tex]\( \implies \)[/tex] Slope is [tex]\(-2\)[/tex]
C. [tex]\( y = -\frac{1}{2}x + 13 \)[/tex] [tex]\( \implies \)[/tex] Slope is [tex]\(-\frac{1}{2}\)[/tex]
D. [tex]\( y = \frac{1}{2}x + 13 \)[/tex] [tex]\( \implies \)[/tex] Slope is [tex]\(\frac{1}{2}\)[/tex]
### Step 4: Select the Correct Option
The only equation with the same slope [tex]\(-2\)[/tex] as our original line is:
[tex]\[ \boxed{B \; y = -2x + 13} \][/tex]
Therefore, 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]\( y = -2x + 13 \)[/tex].
