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Question 14 Write an equation for a line perpendicular to \( y=-2 x-2 \) and passing through the point \( (4,1) \) \( y=\square \) Question Help:Masmit Message instructor Substion
Ask by Cox Carlson. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The equation of the line perpendicular to \( y = -2x - 2 \) and passing through the point \( (4, 1) \) is \( y = \frac{1}{2}x - 1 \).
Solution
To find the equation of a line perpendicular to \( y = -2x - 2 \) and passing through the point \( (4, 1) \), follow these steps:
### 1. **Determine the Slope of the Given Line**
The given equation is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope.
\[ y = -2x - 2 \]
So, the slope (\( m \)) of the given line is **-2**.
### 2. **Find the Slope of the Perpendicular Line**
Perpendicular lines have slopes that are negative reciprocals of each other.
\[ m_{\text{perpendicular}} = -\frac{1}{m} = -\frac{1}{-2} = \frac{1}{2} \]
So, the slope of the perpendicular line is **\( \frac{1}{2} \)**.
### 3. **Use the Point-Slope Form to Find the Equation**
The point-slope form of a line is:
\[ y - y_1 = m(x - x_1) \]
Where:
- \( m \) is the slope
- \( (x_1, y_1) \) is the point the line passes through
Plugging in the values:
\[ y - 1 = \frac{1}{2}(x - 4) \]
### 4. **Simplify to Slope-Intercept Form**
Expand and simplify the equation:
\[
\begin{align*}
y - 1 &= \frac{1}{2}x - 2 \\
y &= \frac{1}{2}x - 2 + 1 \\
y &= \frac{1}{2}x - 1
\end{align*}
\]
### **Final Equation**
\[ y = \frac{1}{2}x - 1 \]
**Answer:**
\[ y = \frac{1}{2}x - 1 \]
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Simplify this solution The Deep Dive
To find the equation of a line that is perpendicular to \( y = -2x - 2 \), we need the slope of the new line. The slope of the original line is -2, so the slope of the perpendicular line is the negative reciprocal, which is \( \frac{1}{2} \). Now, using the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (4, 1) \) and \( m = \frac{1}{2} \), we have: \[ y - 1 = \frac{1}{2}(x - 4) \] Simplifying this gives: \[ y - 1 = \frac{1}{2}x - 2 \] \[ y = \frac{1}{2}x - 1 \] So, the final answer is: \( y = \frac{1}{2}x - 1 \)
