To find slopes of lines that are parallel and perpendicular to the given line [tex]\( y = \frac{1}{4} x + 7 \)[/tex], follow these steps:
### Step-by-Step Solution:
1. Identify the slope of the given line:
- The equation of the given line is [tex]\( y = \frac{1}{4} x + 7 \)[/tex].
- This is in the slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
- So, the slope [tex]\( m \)[/tex] of the given line is [tex]\( \frac{1}{4} \)[/tex].
2. Determine the slope of lines parallel to the given line:
- Lines that are parallel to each other have the same slope.
- Hence, the slope of lines parallel to the given line [tex]\( y = \frac{1}{4} x + 7 \)[/tex] will also be [tex]\( \frac{1}{4} \)[/tex].
3. Determine the slope of lines perpendicular to the given line:
- The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line.
- The negative reciprocal of [tex]\( \frac{1}{4} \)[/tex] is calculated as follows:
- Take the reciprocal of [tex]\( \frac{1}{4} \)[/tex], which is [tex]\( 4 \)[/tex] (since reciprocal means [tex]\( \frac{1}{a} \)[/tex] turns into [tex]\( \frac{a}{1} \)[/tex]).
- Change the sign to the opposite (negative in this case), giving us [tex]\( -4 \)[/tex].
### Summary:
- The slope of lines parallel to the given line [tex]\( y = \frac{1}{4} x + 7 \)[/tex] is [tex]\( 0.25 \)[/tex].
- The slope of lines perpendicular to the given line [tex]\( y = \frac{1}{4} x + 7 \)[/tex] is [tex]\( -4 \)[/tex].
Thus, the slopes are:
- Parallel slope: [tex]\( 0.25 \)[/tex]
- Perpendicular slope: [tex]\( -4 \)[/tex]
