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.