Let's analyze the problem.

The line given is:

$y = -\frac{7}{3}x + 6$

You are tasked with finding:

1. The equation of a line that is parallel to this line and passes through the point $(-7, -6)$.
2. The equation of a line that is perpendicular to this line and passes through the same point $(-7, -6)$.

1. Parallel Line:

• A line parallel to the given line will have the same slope, which is $-\frac{7}{3}$.

• The point-slope form of a line is given by: $y - y_1 = m(x - x_1)$ where $m$ is the slope and $(x_1, y_1)$ is a point on the line.

  Using the point $(-7, -6)$ and slope $-\frac{7}{3}$: $y + 6 = -\frac{7}{3}(x + 7)$ Expanding this: $y + 6 = -\frac{7}{3}x - \frac{49}{3}$ Subtracting 6 from both sides: $y = -\frac{7}{3}x - \frac{49}{3} - 6$ Converting 6 to thirds: $y = -\frac{7}{3}x - \frac{49}{3} - \frac{18}{3}$ So the equation of the parallel line is: $y = -\frac{7}{3}x - \frac{67}{3}$

2. Perpendicular Line:

• The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the original line. The slope of the given line is $-\frac{7}{3}$, so the slope of the perpendicular line is $\frac{3}{7}$.
• Using the point-slope form again with point $(-7, -6)$ and slope $\frac{3}{7}$: $y + 6 = \frac{3}{7}(x + 7)$ Expanding this: $y + 6 = \frac{3}{7}x + 3$ Subtracting 6 from both sides: $y = \frac{3}{7}x + 3 - 6$ Simplifying: $y = \frac{3}{7}x - 3$

Final Results:

• Equation of the parallel line: $y = -\frac{7}{3}x - \frac{67}{3}$
• Equation of the perpendicular line: $y = \frac{3}{7}x - 3$

Would you like further details or explanations?

Here are some additional questions to consider:

1. What is the significance of the slopes of parallel lines?
2. How do we calculate the slope of a perpendicular line?
3. How would the equations change if the given point were different?
4. Can you graph both the parallel and perpendicular lines to visualize them?
5. How would this process change if the original equation were in a different form?

Tip: When working with fractions in linear equations, it’s helpful to convert whole numbers to fractions with the same denominator to make calculations smoother.