Certainly! Let's solve the problem together.
### 1. Finding the Slope
To find the slope [tex]\(m\)[/tex] of the line that passes through the points [tex]\((-12, 8)\)[/tex] and [tex]\((6, 2)\)[/tex], we use the slope formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points [tex]\((x_1, y_1) = (-12, 8)\)[/tex] and [tex]\((x_2, y_2) = (6, 2)\)[/tex]:
[tex]\[
m = \frac{2 - 8}{6 + 12} = \frac{-6}{18} = -\frac{1}{3}
\][/tex]
So, the slope [tex]\(m\)[/tex] is [tex]\(-\frac{1}{3}\)[/tex].
### 2. Writing the Point-Slope Form
The point-slope form of the equation of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
We will use the point [tex]\((-12, 8)\)[/tex]:
[tex]\[
y - 8 = -\frac{1}{3}(x - (-12))
\][/tex]
Simplifying the equation, we get:
[tex]\[
y - 8 = -\frac{1}{3}(x + 12)
\][/tex]
This is the point-slope form of the equation of the line.
### 3. Converting to Slope-Intercept Form
To convert the point-slope form to the slope-intercept form, [tex]\(y = mx + b\)[/tex], let's start from:
[tex]\[
y - 8 = -\frac{1}{3}(x + 12)
\][/tex]
Distribute the slope [tex]\(-\frac{1}{3}\)[/tex]:
[tex]\[
y - 8 = -\frac{1}{3}x - 4
\][/tex]
Add 8 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[
y = -\frac{1}{3}x - 4 + 8
\][/tex]
Simplify:
[tex]\[
y = -\frac{1}{3}x + 4
\][/tex]
So, the slope-intercept form of the equation of the line is:
[tex]\[
y = -\frac{1}{3}x + 4
\][/tex]
### Summary
- Point-Slope Form: [tex]\( y - 8 = -\frac{1}{3}(x + 12) \)[/tex]
- Slope-Intercept Form: [tex]\( y = -\frac{1}{3}x + 4 \)[/tex]
