To solve this problem, let’s break it down into two parts: finding the equation of the line parallel and perpendicular to the given line, both passing through the point $(2, -4)$.

Step 1: Analyze the Given Line

The equation of the given line is: $-7x + 9y = 4$ To find parallel and perpendicular lines, we need to express this line in slope-intercept form, $y = mx + b$, where $m$ is the slope.

1. Rewrite the equation by isolating $y$: $9y = 7x + 4$ $y = \frac{7}{9}x + \frac{4}{9}$ Now, we can see that the slope of this line ($m$) is $\frac{7}{9}$.

Step 2: Equation of the Parallel Line

A line parallel to the given line will have the same slope, $\frac{7}{9}$, and will pass through the point $(2, -4)$.

1. Use the point-slope form of a line: $y - y_1 = m(x - x_1)$, where $m = \frac{7}{9}$ and $(x_1, y_1) = (2, -4)$: $y + 4 = \frac{7}{9}(x - 2)$
2. Simplify this equation: $y + 4 = \frac{7}{9}x - \frac{14}{9}$ $y = \frac{7}{9}x - \frac{14}{9} - \frac{36}{9}$ $y = \frac{7}{9}x - \frac{50}{9}$ So, the equation of the parallel line is: $y = \frac{7}{9}x - \frac{50}{9}$

Step 3: Equation of the Perpendicular Line

A line perpendicular to the given line will have a slope that is the negative reciprocal of $\frac{7}{9}$, which is $-\frac{9}{7}$.

1. Again, use the point-slope form of a line with the new slope $-\frac{9}{7}$ and the point $(2, -4)$: $y + 4 = -\frac{9}{7}(x - 2)$
2. Simplify this equation: $y + 4 = -\frac{9}{7}x + \frac{18}{7}$ $y = -\frac{9}{7}x + \frac{18}{7} - \frac{28}{7}$ $y = -\frac{9}{7}x - \frac{10}{7}$ Thus, the equation of the perpendicular line is: $y = -\frac{9}{7}x - \frac{10}{7}$

Final Answer

• Equation of the parallel line: $y = \frac{7}{9}x - \frac{50}{9}$
• Equation of the perpendicular line: $y = -\frac{9}{7}x - \frac{10}{7}$

Would you like further details on any of the steps, or have any questions about this solution?

Here are five related questions to deepen your understanding:

1. How would you find the distance between the two parallel lines in this problem?
2. What would the equations look like if the given line were written in standard form without solving for $y$?
3. How do you determine the slope of a line given two points on the line?
4. How can you verify that two lines are perpendicular using only their slopes?
5. What is the general formula for finding a line parallel to a given line through a specific point?

Tip: When finding parallel and perpendicular lines, remember that parallel lines share the same slope, while perpendicular lines have slopes that are negative reciprocals of each other.