To solve the problem step-by-step, let's start by understanding the given line equation and the properties of parallel and perpendicular lines.
The given line equation is:
[tex]\[ y = \frac{8}{7}x - 7 \][/tex]
### Part (a) - Slope of a Line Parallel to the Given Line
Firstly, let's recall that parallel lines have the same slope. The slope-intercept form of a line is given by:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] represents the slope and [tex]\( b \)[/tex] represents the y-intercept.
In the given line equation [tex]\( y = \frac{8}{7}x - 7 \)[/tex], comparing it to the slope-intercept form, we see that the slope [tex]\( m \)[/tex] of the given line is:
[tex]\[ \frac{8}{7} \][/tex]
Therefore, the slope of any line parallel to this given line will have the same slope. Hence, the slope of a line parallel to the given line is:
[tex]\[ \boxed{1.1428571428571428} \][/tex]
### Part (b) - Slope of a Line Perpendicular to the Given Line
Next, let's find the slope of a line that is perpendicular to the given line. Perpendicular lines have slopes that are negative reciprocals of each other. If the slope of one line is [tex]\( m \)[/tex], then the slope of a line perpendicular to it is [tex]\( -\frac{1}{m} \)[/tex].
For the given line with slope [tex]\( m = \frac{8}{7} \)[/tex], the slope of a perpendicular line will be:
[tex]\[ -\frac{1}{\left(\frac{8}{7}\right)} \][/tex]
[tex]\[ = -\frac{7}{8} \][/tex]
Therefore, the slope of a line perpendicular to the given line is:
[tex]\[ \boxed{-0.875} \][/tex]
In summary:
(a) The slope of a line parallel to the given line is [tex]\( \boxed{1.1428571428571428} \)[/tex].
(b) The slope of a line perpendicular to the given line is [tex]\( \boxed{-0.875} \)[/tex].
