Question
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Answer
The equation of the line perpendicular to \( y = -x - 4 \) and passing through \( (-1, 0) \) is \( y = x + 1 \).
Solution
To find the equation of the line that is **perpendicular** to \( y = -x - 4 \) and **passes through** the point \( (-1, 0) \), 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**,
- \( b \) is the **y-intercept**.
For \( y = -x - 4 \):
\[
m_1 = -1
\]
### 2. Find the Slope of the Perpendicular Line
Lines that are perpendicular to each other have slopes that are **negative reciprocals**.
\[
m_2 = -\frac{1}{m_1} = -\frac{1}{-1} = 1
\]
So, the slope of the perpendicular line is \( m_2 = 1 \).
### 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:
- \( (x_1, y_1) \) is a point on the line,
- \( m \) is the slope.
Given the point \( (-1, 0) \) and slope \( m_2 = 1 \):
\[
y - 0 = 1(x - (-1))
\]
\[
y = 1(x + 1)
\]
\[
y = x + 1
\]
### **Final Answer**
The equation of the line **perpendicular** to \( y = -x - 4 \) and **passing through** the point \( (-1, 0) \) is:
\[
\boxed{y = x + 1}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To find a line that passes through the point \((-1, 0)\) and is perpendicular to the line described by the equation \(y = -x - 4\), we first need to determine the slope of the given line. The slope-intercept form of the equation reveals that the slope is \(-1\). The slope of the line that is perpendicular to this will be the negative reciprocal, which is \(1\). Using the point-slope formula, the equation of the new line that passes through \((-1, 0)\) with a slope of \(1\) can be written as: \[ y - 0 = 1(x + 1) \] Simplifying, we get: \[ y = x + 1 \] So the line that passes through \((-1, 0)\) and is perpendicular to \(y = -x - 4\) is given by \(y = x + 1\).
