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The image you uploaded contains a few mathematical exercises focused on determining slopes of parallel and perpendicular lines, as well as finding equations of lines under certain conditions.

Here is a breakdown of the tasks and some explanations to help solve them:

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### 1. **Find the slopes of the lines parallel and perpendicular to each of the given equations:**

   a. \( x - y + 1 = 0 \)  
   b. \( 2x - 3y + 2 = 0 \)  
   c. \( x + 3 = 0 \)  
   d. \( -5x - 7y + 1 = 0 \)

**Steps to Solve:**

- Convert each equation into slope-intercept form \( y = mx + b \), where \( m \) is the slope.
- For parallel lines, the slope remains the same.
- For perpendicular lines, the slope \( m \) changes to \( -\frac{1}{m} \).

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### 2. **Determine the equation of the line in each of the following cases:**

   a. It is parallel to the line \( x - y + 1 = 0 \) and passes through the origin.

   b. It intersects the axes at points \( (-3, 0) \) and \( (0, 4) \).

   c. The slope is 2 and it passes through the point \( (1, -1) \).

**Steps to Solve:**

- For (a), since the line is parallel, it has the same slope. Use the point-slope form \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \( (x_1, y_1) \) is the given point (the origin in this case).
  
- For (b), you can use the intercept form of the line \( \frac{x}{a} + \frac{y}{b} = 1 \), where \( a \) and \( b \) are the intercepts on the x and y axes, respectively.
  
- For (c), apply the point-slope form \( y - y_1 = m(x - x_1) \) with the given point \( (1, -1) \) and the slope \( 2 \).

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Would you like step-by-step solutions for each of these problems? Also, here are 5 related questions to help further understanding:

1. How do you find the slope of a line from a general equation \( Ax + By + C = 0 \)?
2. What happens to the slope when two lines are perpendicular?
3. How can you find the equation of a line given two points on it?
4. How do you determine if two lines are parallel or perpendicular based on their equations?
5. How can intercept form help in determining the equation of a line?

**Tip**: Always remember that parallel lines have 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 each of the given equations, and find the equation of the line under specific conditions.

Learn how to find the slopes of parallel and perpendicular lines using the general equation Ax + By + C = 0 and determine the equation of a line given specific conditions. This problem covers linear equations, slope-intercept form, and point-slope form. Suitable for Grades 10-12.

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