To find the slope of the line passing through the points [tex]\( G(3, -1) \)[/tex] and [tex]\( H(-9, 7) \)[/tex], we will follow these steps:
1. Identify the coordinates of points [tex]\( G \)[/tex] and [tex]\( H \)[/tex].
2. Calculate the change in [tex]\( y \)[/tex]-coordinates ([tex]\(\Delta y\)[/tex]).
3. Calculate the change in [tex]\( x \)[/tex]-coordinates ([tex]\(\Delta x\)[/tex]).
4. Use the formula for the slope of a line, [tex]\( m \)[/tex], which is given by:
[tex]\[
m = \frac{\Delta y}{\Delta x}
\][/tex]
Let's go through each step in detail:
### Step 1: Identifying the coordinates
- Point [tex]\( G \)[/tex] has coordinates [tex]\( (3, -1) \)[/tex]:
- [tex]\( x_1 = 3 \)[/tex]
- [tex]\( y_1 = -1 \)[/tex]
- Point [tex]\( H \)[/tex] has coordinates [tex]\( (-9, 7) \)[/tex]:
- [tex]\( x_2 = -9 \)[/tex]
- [tex]\( y_2 = 7 \)[/tex]
### Step 2: Calculate the change in [tex]\( y \)[/tex]-coordinates ([tex]\(\Delta y\)[/tex])
The change in [tex]\( y \)[/tex]-coordinates ([tex]\(\Delta y\)[/tex]) is found by subtracting the [tex]\( y \)[/tex]-coordinate of point [tex]\( G \)[/tex] from the [tex]\( y \)[/tex]-coordinate of point [tex]\( H \)[/tex]:
[tex]\[
\Delta y = y_2 - y_1 = 7 - (-1) = 7 + 1 = 8
\][/tex]
### Step 3: Calculate the change in [tex]\( x \)[/tex]-coordinates ([tex]\(\Delta x\)[/tex])
The change in [tex]\( x \)[/tex]-coordinates ([tex]\(\Delta x\)[/tex]) is found by subtracting the [tex]\( x \)[/tex]-coordinate of point [tex]\( G \)[/tex] from the [tex]\( x \)[/tex]-coordinate of point [tex]\( H \)[/tex]:
[tex]\[
\Delta x = x_2 - x_1 = -9 - 3 = -12
\][/tex]
### Step 4: Calculate the slope ([tex]\( m \)[/tex])
Now, we will use the slope formula:
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{8}{-12}
\][/tex]
Simplify the fraction:
[tex]\[
m = \frac{8}{-12} = -\frac{2}{3}
\][/tex]
Therefore, the slope of the line passing through points [tex]\( G(3, -1) \)[/tex] and [tex]\( H(-9, 7) \)[/tex] is [tex]\( -\frac{2}{3} \)[/tex] or approximately [tex]\(-0.6667\)[/tex].
To sum up, the slope [tex]\( m \)[/tex] of the line [tex]\( GH \)[/tex] is [tex]\(-\frac{2}{3}\)[/tex] or [tex]\(-0.6667\)[/tex].
