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From the image you uploaded, I can see the following questions related to lines and their slopes:

1. **Find the slopes of the lines parallel and perpendicular to each equation:**
   - \( a. \ x - y + 1 = 0 \)
   - \( b. \ 2x - 3y + 2 = 0 \)
   - \( c. \ x + 3 = 0 \)
   - \( d. \ -5x - 7y + 1 = 0 \)

2. **Determine the equation of the line in each of the following situations:**
   - It is parallel to the line \( x - y + 1 = 0 \) and passes through the origin.
   - It intersects the axes at points (-3, 0) and (0, 4).
   - It has a slope of 2 and passes through the point (1, -1).

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### Let's solve these step by step:

#### 1. **Slopes of the lines parallel and perpendicular to each equation:**

The general form of the equation of a line is \( Ax + By + C = 0 \), where the slope \( m = -\frac{A}{B} \).

- **(a) \( x - y + 1 = 0 \):**
  - Slope \( m = -\frac{1}{-1} = 1 \).
  - The slope of a parallel line is the same, so it is \( m = 1 \).
  - The slope of a perpendicular line is the negative reciprocal, so \( m = -1 \).

- **(b) \( 2x - 3y + 2 = 0 \):**
  - Slope \( m = -\frac{2}{-3} = \frac{2}{3} \).
  - The slope of a parallel line is \( m = \frac{2}{3} \).
  - The slope of a perpendicular line is \( m = -\frac{3}{2} \).

- **(c) \( x + 3 = 0 \) (a vertical line):**
  - Vertical lines have an undefined slope.
  - The slope of a line perpendicular to a vertical line is zero (horizontal line).

- **(d) \( -5x - 7y + 1 = 0 \):**
  - Slope \( m = -\frac{-5}{-7} = \frac{5}{7} \).
  - The slope of a parallel line is \( m = \frac{5}{7} \).
  - The slope of a perpendicular line is \( m = -\frac{7}{5} \).

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#### 2. **Equation of the line in each situation:**

- **Parallel to \( x - y + 1 = 0 \) and passes through the origin:**
  - The slope of the line \( x - y + 1 = 0 \) is 1 (as calculated earlier).
  - Using the point-slope form \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (0, 0) \):
    \[
    y - 0 = 1(x - 0) \implies y = x.
    \]
    The equation is \( y = x \).

- **Passes through (-3, 0) and (0, 4):**
  - The slope is \( m = \frac{4 - 0}{0 - (-3)} = \frac{4}{3} \).
  - Using the point-slope form with point \( (-3, 0) \):
    \[
    y - 0 = \frac{4}{3}(x + 3) \implies y = \frac{4}{3}x + 4.
    \]
    The equation is \( y = \frac{4}{3}x + 4 \).

- **Slope is 2 and passes through (1, -1):**
  - Using point-slope form with \( m = 2 \) and \( (x_1, y_1) = (1, -1) \):
    \[
    y - (-1) = 2(x - 1) \implies y + 1 = 2x - 2 \implies y = 2x - 3.
    \]
    The equation is \( y = 2x - 3 \).

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Would you like further details on any of these solutions?

### 5 Related Questions:
1. How do you find the equation of a line parallel to a given line?
2. How do you calculate the slope between two points?
3. What are the key properties of perpendicular lines?
4. How do you handle vertical and horizontal lines when determining slopes?
5. Can two lines with the same slope ever intersect?

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

Determine the slopes of the lines parallel and perpendicular to the given equations. Also, determine the equation of the line in different scenarios such as being parallel to a given line or passing through specific points.

This lesson involves determining the slopes of lines that are parallel and perpendicular to a given set of linear equations. It also covers how to find the equation of a line under various conditions, such as passing through specific points or being parallel to another line. Learn key formulas such as m = -A/B and point-slope form.

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