line v is y=-5x+6. line w includes point (-1,-2) and is perpendicular to v. what’s equation for w

To find the equation of line \( w \), which is perpendicular to line \( v \) and passes through the point \((-1, -2)\), we can follow these steps:

### 1. Determine the slope of line \( v \):
The equation of line \( v \) is given as \( y = -5x + 6 \). The slope-intercept form of a line is \( y = mx + b \), where \( m \) represents the slope. From the equation, the slope of line \( v \) is \( m_v = -5 \).

### 2. Find the slope of line \( w \):
Since line \( w \) is perpendicular to line \( v \), the slope of line \( w \) will be the negative reciprocal of the slope of line \( v \). The negative reciprocal of \( -5 \) is:

\[
m_w = \frac{1}{5}
\]

### 3. Use the point-slope form to find the equation of line \( w \):
Now that we know the slope of line \( w \) is \( \frac{1}{5} \), and that line \( w \) passes through the point \((-1, -2)\), we can use the point-slope form of the equation of a line:

\[
y - y_1 = m(x - x_1)
\]

Substitute the point \((-1, -2)\) and the slope \( m_w = \frac{1}{5} \) into the equation:

\[
y - (-2) = \frac{1}{5}(x - (-1))
\]

Simplify:

\[
y + 2 = \frac{1}{5}(x + 1)
\]

### 4. Solve for \( y \) to get the equation in slope-intercept form:
Now distribute the \( \frac{1}{5} \) on the right-hand side:

\[
y + 2 = \frac{1}{5}x + \frac{1}{5}
\]

Subtract 2 from both sides to isolate \( y \):

\[
y = \frac{1}{5}x + \frac{1}{5} - 2
\]

\[
y = \frac{1}{5}x - \frac{9}{5}
\]

So, the equation of line \( w \) is:

\[
y = \frac{1}{5}x - \frac{9}{5}
\]

### Final Answer:
The equation of line \( w \) is \( y = \frac{1}{5}x - \frac{9}{5} \).

---

Do you want more details or have any questions about this solution?

Here are 5 related questions you could explore:
1. How do you find the slope of a line given two points?
2. What is the general process to find the equation of a line parallel to a given line?
3. How can you verify if two lines are perpendicular by comparing their slopes?
4. What is the difference between point-slope form and slope-intercept form?
5. Can a horizontal line ever be perpendicular to another line?

**Tip:** The product of the slopes of two perpendicular lines is always \(-1\).

Find the equation of a line perpendicular to y=-5x+6 that passes through the point (-1,-2). This solution explains the concepts of slopes, perpendicular lines, and uses point-slope form to determine the final equation, suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation