To address the problem systematically, let's break it down step-by-step for both perpendicular and parallel lines.
### Equation of the Perpendicular Line
1. Identify the slope of the original line:
The given line is [tex]\( y = -\frac{9}{8}x + 5 \)[/tex].
From this, we can see that the slope [tex]\( m \)[/tex] is [tex]\( -\frac{9}{8} \)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the slope of the original line. So, for a slope [tex]\( -\frac{9}{8} \)[/tex], the negative reciprocal is:
[tex]\[
\text{slope perpendicular} = -\left(\frac{1}{-\frac{9}{8}}\right) = \frac{8}{9}
\][/tex]
3. Use the point-slope form to find equation:
The point-slope form of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, the point given is [tex]\( (8, -3) \)[/tex] and the slope [tex]\( m \)[/tex] is [tex]\( \frac{8}{9} \)[/tex]. Plugging these values in:
[tex]\[
y - (-3) = \frac{8}{9}(x - 8)
\][/tex]
Simplifying:
[tex]\[
y + 3 = \frac{8}{9}(x - 8)
\][/tex]
So the equation of the perpendicular line is:
[tex]\[
\boxed{y + 3 = \frac{8}{9}(x - 8)}
\][/tex]
### Equation of the Parallel Line
1. Identify the slope of the parallel line:
The slope of a parallel line will be the same as the original line. Therefore, it remains:
[tex]\[
\text{slope parallel} = -\frac{9}{8}
\][/tex]
2. Use the point-slope form to find equation:
Again using the point-slope formula with the given point [tex]\( (8, -3) \)[/tex] and the slope [tex]\( -\frac{9}{8} \)[/tex]:
[tex]\[
y - (-3) = -\frac{9}{8}(x - 8)
\][/tex]
Simplifying:
[tex]\[
y + 3 = -\frac{9}{8}(x - 8)
\][/tex]
Therefore, the equation of the parallel line is:
[tex]\[
\boxed{y + 3 = -\frac{9}{8}(x - 8)}
\][/tex]
In summary:
- The equation of the perpendicular line is:
[tex]\[
\boxed{y + 3 = \frac{8}{9}(x - 8)}
\][/tex]
- The equation of the parallel line is:
[tex]\[
\boxed{y + 3 = -\frac{9}{8}(x - 8)}
\][/tex]
