Answer :
Let's solve the problem step-by-step.
### Given Data:
1. The equation of the line: \( y = \frac{1}{2}x - 4 \)
2. A point: \((-4, 2)\)
### Step 1: Finding the slope of the line and parallel line
The given line \( y = \frac{1}{2}x - 4 \) has a slope \( m = \frac{1}{2} \).
#### Slope of a Line Parallel to the Given Line
The slope of any line parallel to the given line is the same as the slope of the given line. Therefore, the slope of the parallel line is:
[tex]\[ \text{Parallel Line Slope} = \frac{1}{2} \][/tex]
### Step 2: Finding the equation of the parallel line
We need to find the y-intercept (b) of the parallel line that passes through the point \((-4, 2)\).
Using the point-slope form of the line equation:
[tex]\[ y = mx + b \][/tex]
Substitute \((x, y)\) with \((-4, 2)\) and \(m\) with \(\frac{1}{2}\):
[tex]\[ 2 = \frac{1}{2}(-4) + b \][/tex]
[tex]\[ 2 = -2 + b \][/tex]
[tex]\[ b = 4 \][/tex]
So, the equation of the line parallel to \( y = \frac{1}{2}x - 4 \) passing through \((-4, 2)\) is:
[tex]\[ y = \frac{1}{2}x + 4 \][/tex]
### Step 3: Finding the slope of a line 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. Therefore, the slope of the perpendicular line is:
[tex]\[ \text{Perpendicular Line Slope} = -\frac{1}{(\frac{1}{2})} = -2 \][/tex]
### Step 4: Finding the equation of the perpendicular line
We need to find the y-intercept (b) of the perpendicular line that passes through the point \((-4, 2)\).
Using the point-slope form of the line equation:
[tex]\[ y = mx + b \][/tex]
Substitute \((x, y)\) with \((-4, 2)\) and \(m\) with \(-2\):
[tex]\[ 2 = -2(-4) + b \][/tex]
[tex]\[ 2 = 8 + b \][/tex]
[tex]\[ b = -6 \][/tex]
So, the equation of the line perpendicular to \( y = \frac{1}{2}x - 4 \) passing through \((-4, 2)\) is:
[tex]\[ y = -2x - 6 \][/tex]
To summarize:
- Slope of the line parallel to the given line: \(\frac{1}{2}\)
- Equation of the parallel line passing through \((-4, 2)\): \( y = \frac{1}{2}x + 4 \)
- Slope of the line perpendicular to the given line: \(-2\)
- Equation of the perpendicular line passing through [tex]\((-4, 2)\)[/tex]: [tex]\( y = -2x - 6 \)[/tex]
### Given Data:
1. The equation of the line: \( y = \frac{1}{2}x - 4 \)
2. A point: \((-4, 2)\)
### Step 1: Finding the slope of the line and parallel line
The given line \( y = \frac{1}{2}x - 4 \) has a slope \( m = \frac{1}{2} \).
#### Slope of a Line Parallel to the Given Line
The slope of any line parallel to the given line is the same as the slope of the given line. Therefore, the slope of the parallel line is:
[tex]\[ \text{Parallel Line Slope} = \frac{1}{2} \][/tex]
### Step 2: Finding the equation of the parallel line
We need to find the y-intercept (b) of the parallel line that passes through the point \((-4, 2)\).
Using the point-slope form of the line equation:
[tex]\[ y = mx + b \][/tex]
Substitute \((x, y)\) with \((-4, 2)\) and \(m\) with \(\frac{1}{2}\):
[tex]\[ 2 = \frac{1}{2}(-4) + b \][/tex]
[tex]\[ 2 = -2 + b \][/tex]
[tex]\[ b = 4 \][/tex]
So, the equation of the line parallel to \( y = \frac{1}{2}x - 4 \) passing through \((-4, 2)\) is:
[tex]\[ y = \frac{1}{2}x + 4 \][/tex]
### Step 3: Finding the slope of a line 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. Therefore, the slope of the perpendicular line is:
[tex]\[ \text{Perpendicular Line Slope} = -\frac{1}{(\frac{1}{2})} = -2 \][/tex]
### Step 4: Finding the equation of the perpendicular line
We need to find the y-intercept (b) of the perpendicular line that passes through the point \((-4, 2)\).
Using the point-slope form of the line equation:
[tex]\[ y = mx + b \][/tex]
Substitute \((x, y)\) with \((-4, 2)\) and \(m\) with \(-2\):
[tex]\[ 2 = -2(-4) + b \][/tex]
[tex]\[ 2 = 8 + b \][/tex]
[tex]\[ b = -6 \][/tex]
So, the equation of the line perpendicular to \( y = \frac{1}{2}x - 4 \) passing through \((-4, 2)\) is:
[tex]\[ y = -2x - 6 \][/tex]
To summarize:
- Slope of the line parallel to the given line: \(\frac{1}{2}\)
- Equation of the parallel line passing through \((-4, 2)\): \( y = \frac{1}{2}x + 4 \)
- Slope of the line perpendicular to the given line: \(-2\)
- Equation of the perpendicular line passing through [tex]\((-4, 2)\)[/tex]: [tex]\( y = -2x - 6 \)[/tex]
