Question
Write an equation of the line that is perpendicular to the given line and un point. through the given pol \( \begin{array}{ll}\text { 7. } y=-4 x+2 ;(0,2) & \text { 8. } y=\frac{2}{3} x+6 ;(-6,2)\end{array} \)
Ask by Martinez Turnbull. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The equations of the perpendicular lines are:
1. \( y = \frac{1}{4}x + 2 \) (for Problem 7)
2. \( y = -\frac{3}{2}x - 7 \) (for Problem 8)
Solution
Sure! Let's find the equations of the lines that are **perpendicular** to the given lines and pass through the specified points.
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### **Problem 7**
**Given:**
- **Original Line:** \( y = -4x + 2 \)
- **Point:** \( (0, 2) \)
**Steps to Find the Perpendicular Line:**
1. **Determine the slope of the original line.**
- The given line is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope.
- **Slope (\( m_1 \))** of the original line: \( -4 \).
2. **Find the slope of the perpendicular line.**
- The slope of a line perpendicular to another is the **negative reciprocal** of the original slope.
- **Slope (\( m_2 \))** of the perpendicular line: \( m_2 = -\frac{1}{m_1} = -\frac{1}{-4} = \frac{1}{4} \).
3. **Use the point-slope form to write the equation of the perpendicular line.**
- **Point-Slope Form:** \( y - y_1 = m(x - x_1) \)
- Plugging in \( m = \frac{1}{4} \) and the point \( (0, 2) \):
\[
y - 2 = \frac{1}{4}(x - 0)
\]
- Simplifying:
\[
y = \frac{1}{4}x + 2
\]
**Answer:**
\[
\boxed{y = \frac{1}{4}x + 2}
\]
---
### **Problem 8**
**Given:**
- **Original Line:** \( y = \frac{2}{3}x + 6 \)
- **Point:** \( (-6, 2) \)
**Steps to Find the Perpendicular Line:**
1. **Determine the slope of the original line.**
- The given line is in the slope-intercept form \( y = mx + b \).
- **Slope (\( m_1 \))** of the original line: \( \frac{2}{3} \).
2. **Find the slope of the perpendicular line.**
- The slope of a line perpendicular to another is the **negative reciprocal** of the original slope.
- **Slope (\( m_2 \))** of the perpendicular line: \( m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{2}{3}} = -\frac{3}{2} \).
3. **Use the point-slope form to write the equation of the perpendicular line.**
- **Point-Slope Form:** \( y - y_1 = m(x - x_1) \)
- Plugging in \( m = -\frac{3}{2} \) and the point \( (-6, 2) \):
\[
y - 2 = -\frac{3}{2}(x - (-6)) \\
y - 2 = -\frac{3}{2}(x + 6)
\]
- Simplifying:
\[
y - 2 = -\frac{3}{2}x - 9 \\
y = -\frac{3}{2}x - 7
\]
**Answer:**
\[
\boxed{y = -\frac{3}{2}x - 7}
\]
---
These are the equations of the lines perpendicular to the given lines and passing through the specified points.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To find the equation of a line perpendicular to the given lines, we need to first determine the slope of the perpendicular lines. For the equation \( y = -4x + 2 \), the slope is -4, so the slope of the perpendicular line will be \( \frac{1}{4} \). Using the point (0,2), we apply the point-slope form: \[ y - 2 = \frac{1}{4}(x - 0) \implies y = \frac{1}{4}x + 2 \] For the second equation \( y = \frac{2}{3}x + 6 \), the slope is \( \frac{2}{3} \), meaning the slope of the perpendicular line is \( -\frac{3}{2} \). Using the point (-6,2): \[ y - 2 = -\frac{3}{2}(x + 6) \implies y = -\frac{3}{2}x - 7 \]
