In this problem, we are given the line equation:

$y = \frac{7}{9}x + 1$

We are tasked with finding two lines:

1. The equation of a line parallel to the given line and passing through the point $(-7, 4)$.
2. The equation of a line perpendicular to the given line and passing through the point $(-7, 4)$.

Step 1: Line parallel to the given line

Parallel lines have the same slope. The slope of the given line is $\frac{7}{9}$. So, the slope of the parallel line will also be $\frac{7}{9}$.

To find the equation of the parallel line that passes through $(-7, 4)$, we use the point-slope form of the line equation:

$y - y_1 = m(x - x_1)$

where $m$ is the slope and $(x_1, y_1)$ is the point through which the line passes. Plugging in the values:

$m = \frac{7}{9}, \quad (x_1, y_1) = (-7, 4)$

$y - 4 = \frac{7}{9}(x + 7)$

Now simplify this equation:

$y - 4 = \frac{7}{9}x + \frac{7}{9} \times 7$ $y - 4 = \frac{7}{9}x + \frac{49}{9}$

Adding 4 to both sides:

$y = \frac{7}{9}x + \frac{49}{9} + 4$

Convert 4 to a fraction with a denominator of 9:

$y = \frac{7}{9}x + \frac{49}{9} + \frac{36}{9}$

$y = \frac{7}{9}x + \frac{85}{9}$

So the equation of the parallel line is:

$y = \frac{7}{9}x + \frac{85}{9}$

Step 2: Line perpendicular to the given line

The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Since the slope of the given line is $\frac{7}{9}$, the slope of the perpendicular line will be $-\frac{9}{7}$.

To find the equation of the perpendicular line passing through $(-7, 4)$, again using the point-slope form:

$y - 4 = -\frac{9}{7}(x + 7)$

Now simplify this equation:

$y - 4 = -\frac{9}{7}x - \frac{9}{7} \times 7$ $y - 4 = -\frac{9}{7}x - 9$

Add 4 to both sides:

$y = -\frac{9}{7}x - 9 + 4$

$y = -\frac{9}{7}x - 5$

Thus, the equation of the perpendicular line is:

$y = -\frac{9}{7}x - 5$

Summary:

• The equation of the parallel line is: $y = \frac{7}{9}x + \frac{85}{9}$
• The equation of the perpendicular line is: $y = -\frac{9}{7}x - 5$

Do you need any further clarification or details on these steps? Here are some related questions for you to explore:

1. How can you convert the point-slope form of a line into the slope-intercept form?
2. What happens if the two lines are parallel but have different y-intercepts?
3. How do you find the intersection of two perpendicular lines?
4. How does the slope of a line affect its steepness?
5. How can you apply the concept of parallel and perpendicular lines in geometry?

Tip: Always remember that parallel lines have equal slopes, and perpendicular lines have slopes that are negative reciprocals of each other!