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The image provides a set of exercises where you are asked to write the equation of a line passing through a given point \( P \), which is parallel to a given line. Let's break down each question:

### 13. \( P(0, -1), y = -2x + 3 \)
- The line is \( y = -2x + 3 \). The slope is \( m = -2 \).
- A parallel line must have the same slope, \( m = -2 \).
- Using the point-slope form: \( y - y_1 = m(x - x_1) \), where \( P(0, -1) \) gives \( x_1 = 0 \) and \( y_1 = -1 \):
  \[
  y - (-1) = -2(x - 0) \implies y + 1 = -2x \implies y = -2x - 1
  \]
- Final equation: \( y = -2x - 1 \).

### 14. \( P(3, 8), y = \frac{1}{5}(x + 4) \)
- The line is \( y = \frac{1}{5}(x + 4) \), so the slope is \( m = \frac{1}{5} \).
- A parallel line has the same slope, \( m = \frac{1}{5} \).
- Using point-slope form with \( P(3, 8) \):
  \[
  y - 8 = \frac{1}{5}(x - 3) \implies y = \frac{1}{5}(x - 3) + 8 = \frac{1}{5}x - \frac{3}{5} + 8
  \]
  Simplifying:
  \[
  y = \frac{1}{5}x + \frac{37}{5}
  \]
- Final equation: \( y = \frac{1}{5}x + \frac{37}{5} \).

### 15. \( P(-2, 6), x = -5 \)
- The line is \( x = -5 \), which is a vertical line.
- A line parallel to this one must also be vertical.
- Since the line passes through \( P(-2, 6) \), the equation is \( x = -2 \).

### 16. \( P(4, 0), -x + 2y = 12 \)
- First, rewrite the equation \( -x + 2y = 12 \) in slope-intercept form:
  \[
  2y = x + 12 \implies y = \frac{1}{2}x + 6
  \]
  So, the slope is \( m = \frac{1}{2} \).
- A parallel line has the same slope, \( m = \frac{1}{2} \).
- Using point-slope form with \( P(4, 0) \):
  \[
  y - 0 = \frac{1}{2}(x - 4) \implies y = \frac{1}{2}(x - 4) = \frac{1}{2}x - 2
  \]
- Final equation: \( y = \frac{1}{2}x - 2 \).

Would you like more details or assistance on graphing these lines to confirm their parallelism?

Here are some related questions to explore further:
1. How do you convert a line equation to slope-intercept form?
2. What is the geometric significance of parallel lines having the same slope?
3. How do you handle vertical and horizontal lines in this kind of exercise?
4. What steps are involved in applying the point-slope form?
5. How can you visually confirm that two lines are parallel by graphing?

**Tip**: When dealing with parallel lines, remember that their slopes are always equal, but their y-intercepts will differ unless they are the same line.

In Exercises 13-16, write an equation of the line passing through point P that is parallel to the given line.

Learn how to write the equation of a line that passes through a given point and is parallel to a given line. These exercises cover key algebra concepts, including the slope-intercept form, point-slope form, and parallel lines.

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