https://file.aiquickdraw.com/mathbot/file/17261910402478pocjx9d.jpg

The problem is asking you to identify relationships between three given lines. The equations of the lines are:

- (a) \( y = x - 2 \)
- (b) \( y = x + 3 \)
- (c) \( y = -x + 1 \)

### Step 1: Understanding the slopes of the lines
The general form of a line equation is \( y = mx + b \), where:
- \( m \) is the slope of the line.
- \( b \) is the y-intercept.

For each of the given lines:
- (a) \( y = x - 2 \): The slope is \( m = 1 \).
- (b) \( y = x + 3 \): The slope is \( m = 1 \).
- (c) \( y = -x + 1 \): The slope is \( m = -1 \).

### Step 2: Analyzing relationships
- **Parallel lines** have the same slope.
  - (a) and (b) both have a slope of \( 1 \), so they are parallel.
  - (a) and (c) have slopes of \( 1 \) and \( -1 \) respectively, so they are not parallel.
  - (b) and (c) also have different slopes, so they are not parallel.

- **Perpendicular lines** have slopes that are negative reciprocals of each other. If one line has a slope of \( m \), a line perpendicular to it will have a slope of \( -\frac{1}{m} \).
  - (a) has a slope of \( 1 \), and (c) has a slope of \( -1 \), which is the negative reciprocal of \( 1 \), so (a) and (c) are perpendicular.
  - (b) has a slope of \( 1 \), and (c) has a slope of \( -1 \), so (b) and (c) are also perpendicular.

### Step 3: Conclusion
Based on the analysis, the correct relationships are:
- (a) is parallel to (b)
- (a) is perpendicular to (c)
- (b) is perpendicular to (c)

Would you like further clarification on any of these concepts?

---

Here are 5 follow-up questions to deepen your understanding:
1. How can you determine if two lines are parallel based on their equations?
2. What is the geometric meaning of the y-intercept in a line equation?
3. Can two lines be neither parallel nor perpendicular? What is their relationship in such cases?
4. How would the equations change if the lines were shifted up or down on the graph?
5. What happens to the slope of a line if it becomes steeper or flatter?

**Tip:** The slope of a line determines its angle of inclination. A positive slope means the line rises from left to right, while a negative slope means it falls from left to right.

Identify any relationships that exist among the lines:
(a) y = x - 2
(b) y = x + 3
(c) y = -x + 1

Analyze the relationships between three linear equations: y = x - 2, y = x + 3, and y = -x + 1. Determine if the lines are parallel or perpendicular based on their slopes. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation