Consider the line y = -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 \( y = -7x + 6 \), 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 given line is in slope-intercept form \( y = mx + b \), where \( m \) is the slope. Here, the slope \( m \) of the given line is \( -7 \).

- **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:
  \[
  m_{\text{perpendicular}} = \frac{1}{7}
  \]

- **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 \( m = \frac{1}{7} \), \( x_1 = 7 \), and \( y_1 = -6 \):
  \[
  y - (-6) = \frac{1}{7}(x - 7)
  \]
  Simplifying:
  \[
  y + 6 = \frac{1}{7}(x - 7)
  \]
  \[
  y + 6 = \frac{1}{7}x - 1
  \]
  \[
  y = \frac{1}{7}x - 1 - 6
  \]
  \[
  y = \frac{1}{7}x - 7
  \]

  So, the equation of the perpendicular line is:
  \[
  y = \frac{1}{7}x - 7
  \]

### 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 \( m_{\text{parallel}} = -7 \).

- **Step 2**: Use the point-slope form to find the equation of the parallel line.
  Substituting \( m = -7 \), \( x_1 = 7 \), and \( y_1 = -6 \):
  \[
  y - (-6) = -7(x - 7)
  \]
  Simplifying:
  \[
  y + 6 = -7x + 49
  \]
  \[
  y = -7x + 49 - 6
  \]
  \[
  y = -7x + 43
  \]

  So, the equation of the parallel line is:
  \[
  y = -7x + 43
  \]

### Summary:
- The equation of the line that is **perpendicular** to \( y = -7x + 6 \) and passes through \( (7, -6) \) is \( y = \frac{1}{7}x - 7 \).
- The equation of the line that is **parallel** to \( y = -7x + 6 \) and passes through \( (7, -6) \) is \( y = -7x + 43 \).

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

1. How do you derive the equation of a line given two points?
2. What is the geometric interpretation of the slope of a line?
3. How can you determine if two lines are parallel or perpendicular just by looking at their equations?
4. What happens to the slope when a line is rotated 90 degrees?
5. How do you find the intersection point of two lines?

**Tip:** Remember, the slope of a line is a measure of its steepness and direction. Positive slopes rise from left to right, while negative slopes fall from left to right.

Learn how to find the equations of perpendicular and parallel lines to y = -7x + 6 that pass through the point (7, -6). This problem covers the concepts of slope, perpendicular lines, and parallel lines in mathematics. Suitable for students in Grades 9-12.

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