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Let's solve the problem shown in the image step by step.

We are given the line:

\[
y = \frac{5}{8}x + 4
\]

### Part 1: Equation of the perpendicular line
To find the equation of the line that is **perpendicular** to this given line and passes through the point \((-5, -3)\), follow these steps:

1. **Find the slope of the perpendicular line**:  
   The slope of the given line is \(\frac{5}{8}\). For perpendicular lines, the slopes are negative reciprocals of each other. The negative reciprocal of \(\frac{5}{8}\) is \(-\frac{8}{5}\).
   
2. **Use the point-slope form** to write the equation of the perpendicular line:  
   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 the point the line passes through. Using \((-5, -3)\) and \(m = -\frac{8}{5}\):
   \[
   y - (-3) = -\frac{8}{5}(x - (-5))
   \]
   Simplify:
   \[
   y + 3 = -\frac{8}{5}(x + 5)
   \]
   Distribute the slope:
   \[
   y + 3 = -\frac{8}{5}x - 8
   \]
   Subtract 3 from both sides:
   \[
   y = -\frac{8}{5}x - 11
   \]
   So, the equation of the perpendicular line is:
   \[
   y = -\frac{8}{5}x - 11
   \]

### Part 2: Equation of the parallel line
For the line that is **parallel** to the given line and passes through the point \((-5, -3)\):

1. **Find the slope of the parallel line**:  
   Parallel lines have the same slope. So, the slope of the parallel line will also be \(\frac{5}{8}\).

2. **Use the point-slope form** to write the equation of the parallel line**:  
   Using the point \((-5, -3)\) and slope \(m = \frac{5}{8}\):
   \[
   y - (-3) = \frac{5}{8}(x - (-5))
   \]
   Simplify:
   \[
   y + 3 = \frac{5}{8}(x + 5)
   \]
   Distribute the slope:
   \[
   y + 3 = \frac{5}{8}x + \frac{25}{8}
   \]
   Subtract 3 (or \( \frac{24}{8} \)) from both sides:
   \[
   y = \frac{5}{8}x + \frac{25}{8} - \frac{24}{8}
   \]
   Simplify:
   \[
   y = \frac{5}{8}x + \frac{1}{8}
   \]
   So, the equation of the parallel line is:
   \[
   y = \frac{5}{8}x + \frac{1}{8}
   \]

### Final Results:
- Equation of the **perpendicular** line: \( y = -\frac{8}{5}x - 11 \)
- Equation of the **parallel** line: \( y = \frac{5}{8}x + \frac{1}{8} \)

Let me know if you'd like more details on any of these steps! Here are some questions you might consider:

1. What happens when two lines have the same slope?
2. How do you calculate the negative reciprocal of a fraction?
3. What is the point-slope form of a linear equation, and when is it useful?
4. How do parallel and perpendicular slopes relate to each other?
5. Can you explain why perpendicular lines always intersect at right angles?

**Tip**: The point-slope form is particularly useful when you know the slope of a line and a point on the line!

Find the equation of the line that is perpendicular and parallel to the line y = 5/8x + 4, and passes through the point (-5, -3).

Learn how to find the equation of a line parallel and perpendicular to y = 5/8x + 4, passing through (-5, -3). This problem covers key algebraic concepts including the point-slope form and relationships between line slopes.

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