To solve for the slopes of lines parallel and perpendicular to the given line [tex]\( 4x + 8y = 3 \)[/tex], we start by converting the equation of the line to the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
Step-by-Step Solution:
1. Start with the given equation of the line:
[tex]\( 4x + 8y = 3 \)[/tex]
2. Solve for [tex]\( y \)[/tex] to get the equation into the form [tex]\( y = mx + b \)[/tex]:
First, isolate the term involving [tex]\( y \)[/tex]:
[tex]\[ 8y = -4x + 3 \][/tex]
3. Divide every term by 8 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{4}{8}x + \frac{3}{8} \][/tex]
4. Simplify the fractions:
[tex]\[ y = -\frac{1}{2}x + \frac{3}{8} \][/tex]
From this equation, it is clear that the slope [tex]\( m \)[/tex] of the line is [tex]\( -\frac{1}{2} \)[/tex].
5. Identify the slope of a line parallel to the given line:
Lines that are parallel have the same slope. Therefore, the slope of a line parallel to the given line is:
[tex]\[ \text{Slope of a parallel line} = -\frac{1}{2} \][/tex]
6. Determine the slope of a line perpendicular to the given line:
To find the slope of a line perpendicular to another, take the negative reciprocal of the original slope. The negative reciprocal of [tex]\( -\frac{1}{2} \)[/tex] is:
[tex]\[ \text{Slope of a perpendicular line} = -\frac{1}{-\frac{1}{2}} = 2 \][/tex]
Therefore:
- The slope of a line parallel to the given line is [tex]\( -\frac{1}{2} \)[/tex].
- The slope of a line perpendicular to the given line is [tex]\( 2 \)[/tex].
So, we fill in the squares as follows:
- Slope of a perpendicular line: [tex]\( 2 \)[/tex]
- Slope of a parallel line: [tex]\( -\frac{1}{2} \)[/tex]
