Consider the line y = -9/7x+6.
Find the equation of the line that is perpendicular to this line and passes through the point (7, - 6).
Find the equation of the line that is parallel to this line and passes through the point (7, - 6).

To find the equations of the lines that are perpendicular and parallel to the given line \( y = -\frac{9}{7}x + 6 \), and that pass through the point \( (7, -6) \), we follow these steps:

### 1. Equation of the Perpendicular Line
- **Step 1**: Determine the slope of the given line.
  The slope \( m \) of the given line is \( -\frac{9}{7} \).

- **Step 2**: Determine the slope of the perpendicular line.
  The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Therefore, the slope of the perpendicular line is:
  \[
  m_{\text{perpendicular}} = \frac{7}{9}
  \]

- **Step 3**: Use the point-slope form to find the equation of the perpendicular line.
  The point-slope form of a line is:
  \[
  y - y_1 = m(x - x_1)
  \]
  where \( m \) is the slope and \( (x_1, y_1) \) is the point the line passes through.

  Substituting \( m = \frac{7}{9} \), \( x_1 = 7 \), and \( y_1 = -6 \):
  \[
  y - (-6) = \frac{7}{9}(x - 7)
  \]
  Simplifying:
  \[
  y + 6 = \frac{7}{9}(x - 7)
  \]
  \[
  y + 6 = \frac{7}{9}x - \frac{49}{9}
  \]
  \[
  y = \frac{7}{9}x - \frac{49}{9} - 6
  \]
  \[
  y = \frac{7}{9}x - \frac{49}{9} - \frac{54}{9}
  \]
  \[
  y = \frac{7}{9}x - \frac{103}{9}
  \]

  So, the equation of the perpendicular line is:
  \[
  y = \frac{7}{9}x - \frac{103}{9}
  \]

### 2. Equation of the Parallel Line
- **Step 1**: Use the same slope as the original line because parallel lines have identical slopes.
  Therefore, the slope \( m_{\text{parallel}} = -\frac{9}{7} \).

- **Step 2**: Use the point-slope form to find the equation of the parallel line.
  Substituting \( m = -\frac{9}{7} \), \( x_1 = 7 \), and \( y_1 = -6 \):
  \[
  y - (-6) = -\frac{9}{7}(x - 7)
  \]
  Simplifying:
  \[
  y + 6 = -\frac{9}{7}(x - 7)
  \]
  \[
  y + 6 = -\frac{9}{7}x + \frac{63}{7}
  \]
  \[
  y + 6 = -\frac{9}{7}x + 9
  \]
  \[
  y = -\frac{9}{7}x + 9 - 6
  \]
  \[
  y = -\frac{9}{7}x + 3
  \]

  So, the equation of the parallel line is:
  \[
  y = -\frac{9}{7}x + 3
  \]

### Summary:
- The equation of the line that is **perpendicular** to \( y = -\frac{9}{7}x + 6 \) and passes through \( (7, -6) \) is \( y = \frac{7}{9}x - \frac{103}{9} \).
- The equation of the line that is **parallel** to \( y = -\frac{9}{7}x + 6 \) and passes through \( (7, -6) \) is \( y = -\frac{9}{7}x + 3 \).

Do you want more details or have any questions? Here are some related questions you might find interesting:

1. How can you verify if a point lies on a given line?
2. What is the significance of the y-intercept in a line's equation?
3. How do you convert a line equation from point-slope form to slope-intercept form?
4. What is the general form of a linear equation?
5. How do you find the distance between a point and a line?

**Tip:** When finding perpendicular slopes, always remember to take the negative reciprocal of the original slope. This ensures the new line will form a right angle with the original line.

Learn how to find the equations of perpendicular and parallel lines to a given line y = -9/7x + 6 passing through the point (7, -6). Understand the concepts of slope, point-slope form, and perpendicular slopes. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation