https://r2.erweima.ai/mathbot/image/1726100549528rbfla58o.jpg

The image presents a set of mathematical problems about determining slopes and finding the equations of lines. The exercises seem to be based on the general form of the equation of a line: 

\[
Ax + By + C = 0
\]
with the slope \( m \) given by:

\[
m = -\frac{A}{B}
\]

Here is a translation and breakdown of the tasks from the image:

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### Problem 1: Finding Slopes of Parallel and Perpendicular Lines
You are asked to find the slopes of lines that are parallel and perpendicular to each equation provided.

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

For parallel lines, the slope remains the same as the original line, and for perpendicular lines, the slope is the negative reciprocal.

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### Problem 2: Writing the Equation of the Line in Specific Situations
You are asked to find the equation of a line for each of the following conditions:

1. A line that is **parallel** to \( x - y + 1 = 0 \) and passes through the origin.
2. A line that **intersects the axes** at the points (-3, 0) and (0, 4).
3. A line with **slope 2** that passes through the point (1, -1).

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Would you like a detailed step-by-step solution to any of these specific problems? 

Here are five related questions to consider:
1. What is the formula for the slope of a line in the general form \( Ax + By + C = 0 \)?
2. How can you determine the slope of a line perpendicular to a given line?
3. What is the relationship between the slopes of parallel lines?
4. How do you find the equation of a line given its slope and a point it passes through?
5. What is the process for finding the equation of a line that intersects the axes at known points?

**Tip:** To find the slope of a line from an equation in the form \( Ax + By + C = 0 \), use the formula \( m = -\frac{A}{B} \), where \( A \) and \( B \) are the coefficients of \( x \) and \( y \), respectively.

Determine the slopes of the lines parallel and perpendicular to each of the equations provided. Find the equation of a line in each of the given situations.

Learn how to determine the slopes of lines parallel and perpendicular to given equations. Solve problems on writing equations of lines based on specific conditions, such as passing through a point or being parallel to a given line. Suitable for students in Grades 9-12.

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