Answer :
To solve this problem, we need to identify the slope of the given line, determine the slope of a line that is perpendicular to it, and also determine the slope of a line that is parallel to it.
1. Identify the slope of the given line.
The given line equation is in the form:
[tex]\[ y = \frac{3}{4} - \frac{2}{3} x \][/tex]
In the slope-intercept form \(y = mx + b\), where \(m\) is the slope, the equation can be rewritten as:
[tex]\[ y = -\frac{2}{3} x + \frac{3}{4} \][/tex]
From this, we can see that the slope \(m\) of the given line is \(-\frac{2}{3}\).
2. Determine the slope of a line perpendicular to the given line.
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. The negative reciprocal of \(-\frac{2}{3}\) is:
[tex]\[ -\left(\frac{1}{-\frac{2}{3}}\right) = \frac{3}{2} \][/tex]
3. Determine the slope of a line parallel to the given line.
The slope of a line that is parallel to another line is the same as the slope of the original line. Therefore, the slope of a line parallel to the given line is:
[tex]\[ -\frac{2}{3} \][/tex]
Thus, the solutions are:
- The slope of a line perpendicular to the given line: \(\boxed{1.5}\)
- The slope of a line parallel to the given line: [tex]\(\boxed{-0.6666666666666666}\)[/tex]
1. Identify the slope of the given line.
The given line equation is in the form:
[tex]\[ y = \frac{3}{4} - \frac{2}{3} x \][/tex]
In the slope-intercept form \(y = mx + b\), where \(m\) is the slope, the equation can be rewritten as:
[tex]\[ y = -\frac{2}{3} x + \frac{3}{4} \][/tex]
From this, we can see that the slope \(m\) of the given line is \(-\frac{2}{3}\).
2. Determine the slope of a line perpendicular to the given line.
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. The negative reciprocal of \(-\frac{2}{3}\) is:
[tex]\[ -\left(\frac{1}{-\frac{2}{3}}\right) = \frac{3}{2} \][/tex]
3. Determine the slope of a line parallel to the given line.
The slope of a line that is parallel to another line is the same as the slope of the original line. Therefore, the slope of a line parallel to the given line is:
[tex]\[ -\frac{2}{3} \][/tex]
Thus, the solutions are:
- The slope of a line perpendicular to the given line: \(\boxed{1.5}\)
- The slope of a line parallel to the given line: [tex]\(\boxed{-0.6666666666666666}\)[/tex]
