The image contains a mathematical problem that asks for two tasks:

1. Write an equation for a line passing through the point (2, 6) that is parallel to the line $y = \frac{2}{7}x + 5$.
2. Write a second equation for a line passing through the given point (2, 6) that is perpendicular to the given line.

Let's solve these one by one:

1. Line parallel to $y = \frac{2}{7}x + 5$

• The slope of the given line is $m = \frac{2}{7}$.

• A parallel line will have the same slope. So, the slope of the new line is also $m = \frac{2}{7}$.

• We use the point-slope form of a line: $y - y_1 = m(x - x_1)$ where $(x_1, y_1) = (2, 6)$ is the point the line passes through and $m = \frac{2}{7}$.

  Plugging in the values: $y - 6 = \frac{2}{7}(x - 2)$

  Now, simplify this equation: $y - 6 = \frac{2}{7}x - \frac{4}{7}$ $y = \frac{2}{7}x - \frac{4}{7} + 6$ Convert 6 into a fraction: $y = \frac{2}{7}x - \frac{4}{7} + \frac{42}{7}$ $y = \frac{2}{7}x + \frac{38}{7}$

  So, the equation of the line parallel to $y = \frac{2}{7}x + 5$ passing through (2, 6) is: $y = \frac{2}{7}x + \frac{38}{7}$

2. Line perpendicular to $y = \frac{2}{7}x + 5$

• The slope of the given line is $m = \frac{2}{7}$.

• For a line to be perpendicular, its slope must be the negative reciprocal of the original slope. The negative reciprocal of $\frac{2}{7}$ is $-\frac{7}{2}$.

• Again, we use the point-slope form of a line with $m = -\frac{7}{2}$ and point $(2, 6)$.

  Using the point-slope form: $y - 6 = -\frac{7}{2}(x - 2)$

  Simplify this equation: $y - 6 = -\frac{7}{2}x + 7$ $y = -\frac{7}{2}x + 7 + 6$ $y = -\frac{7}{2}x + 13$

  So, the equation of the line perpendicular to $y = \frac{2}{7}x + 5$ passing through (2, 6) is: $y = -\frac{7}{2}x + 13$

Summary:

• The equation of the line parallel to $y = \frac{2}{7}x + 5$ passing through (2, 6) is: $y = \frac{2}{7}x + \frac{38}{7}$
• The equation of the line perpendicular to $y = \frac{2}{7}x + 5$ passing through (2, 6) is: $y = -\frac{7}{2}x + 13$

Would you like more details on any of these steps?

Here are some related questions for further practice:

1. What is the equation of a line parallel to $y = -3x + 7$ and passing through the point (1, -2)?
2. How do you find the equation of a line perpendicular to $y = 5x - 4$?
3. Can you determine the distance between two parallel lines with known equations?
4. How does the concept of negative reciprocal slopes apply to perpendicular lines in 3D geometry?
5. What happens to the slopes of lines that are neither parallel nor perpendicular?

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