Determine whether the pair of lines is​ parallel, perpendicular, or neither.
y
equals
five sevenths x minus 2
​,
y
equals
negative five sevenths x plus 2
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Part 1
Choose the correct answer below.
perpendicular
neither parallel nor perpendicular
parallel

The equations of the two lines are:

1. \( y = \frac{5}{7}x - 2 \)
2. \( y = -\frac{5}{7}x + 2 \)

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

For the first equation, \( y = \frac{5}{7}x - 2 \), the slope \( m_1 = \frac{5}{7} \).
For the second equation, \( y = -\frac{5}{7}x + 2 \), the slope \( m_2 = -\frac{5}{7} \).

### Step 2: Analyze the slopes.
- **Parallel lines** have the same slope, \( m_1 = m_2 \).
- **Perpendicular lines** have slopes that are negative reciprocals of each other, meaning \( m_1 \times m_2 = -1 \).
- If neither of these conditions is met, the lines are **neither parallel nor perpendicular**.

### Step 3: Check for parallelism or perpendicularity.
- The slopes \( \frac{5}{7} \) and \( -\frac{5}{7} \) are not equal, so the lines are not parallel.
- The product of the slopes \( \frac{5}{7} \times -\frac{5}{7} = -\frac{25}{49} \), which is not equal to \(-1\), so the lines are not perpendicular.

### Conclusion:
The lines are **neither parallel nor perpendicular**.

Would you like more details or have any questions?

Here are some follow-up questions you might want to explore:
1. What are the conditions for lines to be parallel?
2. How do you find the slope of a line given an equation?
3. What is the significance of the y-intercept in a linear equation?
4. How do you determine if two lines are perpendicular from their equations?
5. How can you graphically verify if two lines are parallel or perpendicular?

**Tip:** Always check the slopes of two lines first when determining their relationship!

Determine whether the pair of lines is parallel, perpendicular, or neither. y = 5/7x - 2, y = -5/7x + 2.

Explore how to determine if the lines y = 5/7x - 2 and y = -5/7x + 2 are parallel, perpendicular, or neither. This problem covers key algebra concepts such as slopes and the conditions for line relationships. Suitable for Grades 8-10 students.

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