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.