To find the slope and the [tex]\( y \)[/tex]-intercept of the line given by the equation [tex]\( x - 8y - 24 = 8 \)[/tex], we will start by rewriting the equation in slope-intercept form, which is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
### Step-by-Step Solution:
1. Start with the given equation:
[tex]\[
x - 8y - 24 = 8
\][/tex]
2. Isolate the [tex]\( y \)[/tex]-term:
[tex]\[
x - 24 = 8y + 8
\][/tex]
To achieve this, we first move the constant term on the right side of the equation to the left:
[tex]\[
x - 8y - 24 = 8 \ \Rightarrow \ x - 24 = 8 + 8y
\][/tex]
Simplify:
[tex]\[
x - 24 - 8 = 8y
\][/tex]
3. Solve for [tex]\( y \)[/tex] by isolating [tex]\( y \)[/tex]:
[tex]\[
8y = x - 32
\][/tex]
Divide every term by the coefficient of [tex]\( y \)[/tex] (which is 8):
[tex]\[
y = \frac{x - 32}{8}
\][/tex]
4. Simplify the right-hand side:
[tex]\[
y = \frac{x}{8} - \frac{32}{8}
\][/tex]
Simplifying further, we get:
[tex]\[
y = \frac{x}{8} - 4
\][/tex]
Now we have the equation in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
### Identifying the Slope and [tex]\( y \)[/tex]-intercept:
- The slope [tex]\( m \)[/tex] is the coefficient of [tex]\( x \)[/tex], which is:
[tex]\[
m = \frac{1}{8} \approx 0.125
\][/tex]
- The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is the constant term, which is:
[tex]\[
b = -4
\][/tex]
### Final Results:
- The slope of the line is:
[tex]\[
m = 0.125
\][/tex]
- The [tex]\( y \)[/tex]-intercept of the line is:
[tex]\[
b = -4
\][/tex]
Therefore, the slope of the line is [tex]\( 0.125 \)[/tex] and the [tex]\( y \)[/tex]-intercept is [tex]\( -4 \)[/tex].
